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<a href="https://www.codewars.com/kata/551f23362ff852e2ab000037" target="_blank" rel="noopener noreferrer">Pyramid Slide Down</a>.</p><p>We are given a 2D array of integers and we are to find the <em>slide down</em>.
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<em>Slide down</em> is a maximum sum of consecutive numbers from the top to the bottom.</p><p>Let's have a look at few examples. Consider the following pyramid:</p><div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token plain"> 3</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 7 4</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 2 4 6</span><br></span><span class="token-line" style="color:#000000"><span class="token plain">8 5 9 3</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div><p>This pyramid has following slide down:</p><div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token plain"> *3</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> *7 4</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 2 *4 6</span><br></span><span class="token-line" style="color:#000000"><span class="token plain">8 5 *9 3</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div><p>And its value is <code>23</code>.</p><p>We can also have a look at a <em>bigger</em> example:</p><div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token plain"> 75</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 95 64</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 17 47 82</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 18 35 87 10</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 20 4 82 47 65</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 19 1 23 3 34</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 88 2 77 73 7 63 67</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 99 65 4 28 6 16 70 92</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 41 4
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them in <em>Java</em><sup id="fnref-1"><a href="#fn-1" class="footnote-ref">1</a></sup>.</p></div></div><p>For all of the following solutions I will be using basic <code>main</code> function that
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will output <code>true</code>/<code>false</code> based on the expected output of our algorithm. Any
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other differences will lie only in the solutions of the problem. You can see the
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<code>main</code> here:</p><div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token keyword" style="color:rgb(0, 0, 255)">public</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">static</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">void</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">main</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token class-name" style="color:rgb(38, 127, 153)">String</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token plain"> args</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token class-name" style="color:rgb(38, 127, 153)">System</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">out</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token function" style="color:rgb(0, 0, 255)">print</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token string" style="color:rgb(163, 21, 21)">"Test #1: "</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token punctuation" style="color:rgb(4, 81, 165)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token class-name" style="color:rgb(38, 127, 153)">System</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">out</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token function" style="color:rgb(0, 0, 255)">println</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token function" style="color:rgb(0, 0, 255)">longestSlideDown</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token keyword" style="color:rgb(0, 0, 255)">new</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"> </span><span class="token number" style="color:rgb(9, 134, 88)">3</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">}</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"> </span><span class="token number" style="color:rgb(9, 134, 88)">7</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><spa
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the one with maximum sum.</p><div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token keyword" style="color:rgb(0, 0, 255)">public</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">static</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">longestSlideDown</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> row</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> col</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token plain">row </span><span class="token operator" style="color:rgb(0, 0, 0)">>=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">length </span><span class="token operator" style="color:rgb(0, 0, 0)">||</span><span class="token plain"> col </span><span class="token operator" style="color:rgb(0, 0, 0)"><</span><span class="token plain"> </span><span class="token number" style="color:rgb(9, 134, 88)">0</span><span class="token plain"> </span><span class="token operator" style="color:rgb(0, 0, 0)">||</span><span class="token plain"> col </span><span class="token operator" style="color:rgb(0, 0, 0)">>=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token comment" style="color:rgb(0, 128, 0)">// BASE: We have gotten out of bounds, there's no reasonable value to</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token comment" style="color:rgb(0, 128, 0)">// return, so we just return the ‹MIN_VALUE› to ensure that it cannot</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token comment" style="color:rgb(0, 128, 0)">// be maximum.</span><span class="token plai
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pyramid itself. Second one is the recursive “algorithm” that finds the slide
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down.</p><p>It is a relatively simple solution… There's nothing to do at the bottom of the
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pyramid, so we just return the value in the <em>cell</em>. Otherwise we add it and try
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to slide down the available cells below the current row.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="time-complexity">Time complexity<a href="#time-complexity" class="hash-link" aria-label="Direct link to Time complexity" title="Direct link to Time complexity"></a></h3><p>If you get the source code and run it yourself, it runs rather fine… I hope you
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are wondering about the time complexity of the proposed solution and, since it
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really is a naïve solution, the time complexity is pretty bad. Let's find the
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worst case scenario.</p><p>Let's start with the first overload:</p><div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token keyword" style="color:rgb(0, 0, 255)">public</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">static</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">longestSlideDown</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">return</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">longestSlideDown</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token number" style="color:rgb(9, 134, 88)">0</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token number" style="color:rgb(9, 134, 88)">0</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token punctuation" style="color:rgb(4, 81, 165)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"></span><span class="token punctuation" style="color:rgb(4, 81, 165)">}</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div><p>There's not much to do here, so we can safely say that the time complexity of
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this function is bounded by <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span>, where <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span></span></span></span></span> is our second overload. This
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doesn't tell us anything, so let's move on to the second overload where we are
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going to define the <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span> function.</p><div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token keyword" style="color:rgb(0, 0, 255)">public</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">static</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">longestSlideDown</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> row</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> col</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token plain">row </span><span class="token operator" style="color:rgb(0, 0, 0)">>=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">length </span><span class="token operator" style="color:rgb(0, 0, 0)">||</span><span class="token plain"> col </span><span class="token operator" style="color:rgb(0, 0, 0)"><</span><span class="token plain"> </span><span class="token number" style="color:rgb(9, 134, 88)">0</span><span class="token plain"> </span><span class="token operator" style="color:rgb(0, 0, 0)">||</span><span class="token plain"> col </span><span class="token operator" style="color:rgb(0, 0, 0)">>=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><s
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and nothing else. Let's dissect them!</p><p>First <code>return</code> statement is the base case, so it has a constant time complexity.</p><p>Second one a bit tricky. We add two numbers together, which we'll consider as
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constant, but for the right part of the expression we take maximum from the left
|
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and right paths. OK… So what happens? We evaluate the <code>longestSlideDown</code> while
|
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choosing the under and right both. They are separate computations though, so we
|
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are branching from each call of <code>longestSlideDown</code>, unless it's a base case.</p><p>What does that mean for us then? We basically get</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>, if </mtext><mi>y</mi><mo>=</mo><mi>r</mi><mi>o</mi><mi>w</mi><mi>s</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>⋅</mo><mi>T</mi><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>, otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">T(y) = \begin{cases} 1 & \text{, if } y = rows \\ 1 + 2 \cdot T(y + 1) & \text{, otherwise} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">, if </span></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">ro</span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mord mathnormal">s</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">,
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regardless of being in the base case. Those are the <code>1</code>s in both cases.</li><li>If we are not in the base case, we move one row down <strong>twice</strong>. That's how we
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obtained <code>2 *</code> and <code>y + 1</code> in the <em>otherwise</em> case.</li><li>We move row-by-row, so we move down <code>y</code>-times and each call splits to two
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subtrees.</li><li>Overall, if we were to represent the calls as a tree, we would get a full
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binary tree of height <code>y</code>, in each node we do some work in constant time,
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therefore we can just sum the ones.</li></ol><div class="theme-admonition theme-admonition-warning alert alert--danger admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>danger</div><div class="admonitionContent_S0QG"><p>It would've been more complicated to get an exact result. In the equation above
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we are assuming that the width of the pyramid is bound by the height.</p></div></div><p>Hopefully we can agree that this is not the best we can do. 😉</p><h2 class="anchor anchorWithStickyNavbar_LWe7" id="greedy-solution">Greedy solution<a href="#greedy-solution" class="hash-link" aria-label="Direct link to Greedy solution" title="Direct link to Greedy solution"></a></h2><p>We will try to optimize it a bit. Let's start with a relatively simple <em>greedy</em>
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approach.</p><div class="theme-admonition theme-admonition-info alert alert--info admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Greedy algorithms</div><div class="admonitionContent_S0QG"><p><em>Greedy algorithms</em> can be described as algorithms that decide the action on the
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optimal option at the moment.</p></div></div><p>We can try to adjust the naïve solution. The most problematic part are the
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recursive calls. Let's apply the greedy approach there:</p><div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token keyword" style="color:rgb(0, 0, 255)">public</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">static</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">longestSlideDown</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> row</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> col</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token plain">row </span><span class="token operator" style="color:rgb(0, 0, 0)">==</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">length </span><span class="token operator" style="color:rgb(0, 0, 0)">-</span><span class="token plain"> </span><span class="token number" style="color:rgb(9, 134, 88)">1</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token comment" style="color:rgb(0, 128, 0)">// BASE: We're at the bottom</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">return</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token plain">col</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="co
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we simply go left.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="time-complexity-1">Time complexity<a href="#time-complexity-1" class="hash-link" aria-label="Direct link to Time complexity" title="Direct link to Time complexity"></a></h3><p>We have switched from <em>adding the maximum</em> to <em>following the “bigger” path</em>, so
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we improved the time complexity tremendously. We just go down the pyramid all
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the way to the bottom. Therefore we are getting:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>r</mi><mi>o</mi><mi>w</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(rows)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">ro</span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mord mathnormal">s</span><span class="mclose">)</span></span></span></span></span></div><p>We have managed to convert our exponential solution into a linear one.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="running-the-tests">Running the tests<a href="#running-the-tests" class="hash-link" aria-label="Direct link to Running the tests" title="Direct link to Running the tests"></a></h3><p>However, if we run the tests, we notice that the second test failed:</p><div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token plain">Test #1: passed</span><br></span><span class="token-line" style="color:#000000"><span class="token plain">Test #2: failed</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div><p>What's going on? Well, we have improved the time complexity, but greedy
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algorithms are not the ideal solution to <strong>all</strong> problems. In this case there
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may be a solution that is bigger than the one found using the greedy algorithm.</p><p>Imagine the following pyramid:</p><div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token plain"> 1</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 2 3</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 5 6 7</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> 8 9 10 11</span><br></span><span class="token-line" style="color:#000000"><span class="token plain">99 13 14 15 16</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div><p>We start at the top:</p><ol><li>Current cell: <code>1</code>, we can choose from <code>2</code> and <code>3</code>, <code>3</code> looks better, so we
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choose it.</li><li>Current cell: <code>3</code>, we can choose from <code>6</code> and <code>7</code>, <code>7</code> looks better, so we
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choose it.</li><li>Current cell: <code>7</code>, we can choose from <code>10</code> and <code>11</code>, <code>11</code> looks better, so we
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choose it.</li><li>Current cell: <code>11</code>, we can choose from <code>15</code> and <code>16</code>, <code>16</code> looks better, so
|
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we choose it.</li></ol><p>Our final sum is: <code>1 + 3 + 7 + 11 + 16 = 38</code>, but in the bottom left cell we
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have a <code>99</code> that is bigger than our whole sum.</p><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_S0QG"><p>Dijkstra's algorithm is a greedy algorithm too, try to think why it is correct.</p></div></div><h2 class="anchor anchorWithStickyNavbar_LWe7" id="top-down-dp">Top-down DP<a href="#top-down-dp" class="hash-link" aria-label="Direct link to Top-down DP" title="Direct link to Top-down DP"></a></h2><p><em>Top-down dynamic programming</em> is probably the most common approach, since (at
|
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least looks like) is the easiest to implement. The whole point is avoiding the
|
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unnecessary computations that we have already done.</p><p>In our case, we can use our naïve solution and put a <em>cache</em> on top of it that
|
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will make sure, we don't do unnecessary calculations.</p><div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token comment" style="color:rgb(0, 128, 0)">// This “structure” is required, since I have decided to use ‹TreeMap› which</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"></span><span class="token comment" style="color:rgb(0, 128, 0)">// requires the ordering on the keys. It represents one position in the pyramid.</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"></span><span class="token keyword" style="color:rgb(0, 0, 255)">record</span><span class="token plain"> </span><span class="token class-name" style="color:rgb(38, 127, 153)">Position</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> row</span><span class="token punctuation" style="color:rgb(4, 81, 165)">,</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> col</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">implements</span><span class="token plain"> </span><span class="token class-name" style="color:rgb(38, 127, 153)">Comparable</span><span class="token generics punctuation" style="color:rgb(4, 81, 165)"><</span><span class="token generics class-name" style="color:rgb(38, 127, 153)">Position</span><span class="token generics punctuation" style="color:rgb(4, 81, 165)">></span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">public</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">compareTo</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token class-name" style="color:rgb(38, 127, 153)">Position</span><span class="token plain"> r</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token plain">row </span><span class="token operator" style="color:rgb(0, 0, 0)">!=</span><span class="token plain"> r</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">row</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">return</span><span class="token plain"> </span><span class="token class-name" style="color:rgb(38, 127, 153)">Integer</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token function" style="color:rgb(0, 0, 255)">valueOf</span><span class=
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caching the already computed values, we need a “reasonable” key. In this case we
|
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share the cache only for one <em>run</em> (i.e. pyramid) of the <code>longestSlideDown</code>, so
|
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we can cache just with the indices within the pyramid, i.e. the <code>Position</code>.</p><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Record</div><div class="admonitionContent_S0QG"><p><em>Record</em> is relatively new addition to the Java language. It is basically an
|
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immutable structure with implicitly defined <code>.equals()</code>, <code>.hashCode()</code>,
|
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|
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<code>.toString()</code> and getters for the attributes.</p></div></div><p>Because of the choice of <code>TreeMap</code>, we had to additionally define the ordering
|
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on it.</p><p>In the <code>longestSlideDown</code> you can notice that the computation which used to be
|
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|
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at the end of the naïve version above, is now wrapped in an <code>if</code> statement that
|
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|
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checks for the presence of the position in the cache and computes the slide down
|
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just when it's needed.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="time-complexity-2">Time complexity<a href="#time-complexity-2" class="hash-link" aria-label="Direct link to Time complexity" title="Direct link to Time complexity"></a></h3><p>If you think that evaluating time complexity for this approach is a bit more
|
|||
|
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tricky, you are right. Keeping the cache in mind, it is not the easiest thing
|
|||
|
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to do. However there are some observations that might help us figure this out:</p><ol><li>Slide down from each position is calculated only once.</li><li>Once calculated, we use the result from the cache.</li></ol><p>Knowing this, we still cannot, at least easily, describe the time complexity of
|
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|
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finding the best slide down from a specific position, <strong>but</strong> we can bound it
|
|||
|
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from above for the <strong>whole</strong> run from the top. Now the question is how we can do
|
|||
|
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that!</p><p>Overall we are doing the same things for almost<sup id="fnref-2"><a href="#fn-2" class="footnote-ref">2</a></sup> all of the positions within
|
|||
|
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the pyramid:</p><ol><li><p>We calculate and store it (using the partial results stored in cache). This
|
|||
|
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is done only once.</p><p>For each calculation we take 2 values from the cache and insert one value.
|
|||
|
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Because we have chosen <code>TreeMap</code>, these 3 operations have logarithmic time
|
|||
|
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complexity and therefore this step is equivalent to <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mo>⋅</mo><msub><mrow><mi>log</mi><mo></mo></mrow><mn>2</mn></msub><mi>n</mi></mrow><annotation encoding="application/x-tex">3 \cdot \log_2{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em"><span style="top:-2.4559em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span></span></span>.</p><p>However for the sake of simplicity, we are going to account only for the
|
|||
|
|
insertion, the reason is rather simple, if we include the 2 retrievals here,
|
|||
|
|
it will be interleaved with the next step, therefore it is easier to keep the
|
|||
|
|
retrievals in the following point.</p><div class="theme-admonition theme-admonition-caution alert alert--warning admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>caution</div><div class="admonitionContent_S0QG"><p>You might have noticed it's still not that easy, cause we're not having full
|
|||
|
|
cache right from the beginning, but the sum of those logarithms cannot be
|
|||
|
|
expressed in a nice way, so taking the upper bound, i.e. expecting the cache
|
|||
|
|
to be full at all times, is the best option for nice and readable complexity
|
|||
|
|
of the whole approach.</p></div></div><p>Our final upper bound of this work is therefore <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo></mo></mrow><mn>2</mn></msub><mi>n</mi></mrow><annotation encoding="application/x-tex">\log_2{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em"><span style="top:-2.4559em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span></span></span>.</p></li><li><p>We retrieve it from the cache. Same as in first point, but only twice, so we
|
|||
|
|
get <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>⋅</mo><msub><mrow><mi>log</mi><mo></mo></mrow><mn>2</mn></msub><mi>n</mi></mrow><annotation encoding="application/x-tex">2 \cdot \log_2{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em"><span style="top:-2.4559em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span></span></span>. </p><div class="theme-admonition theme-admonition-caution alert alert--warning admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>caution</div><div class="admonitionContent_S0QG"><p>It's done twice because of the <code>.containsKey()</code> in the <code>if</code> condition.</p></div></div></li></ol><p>Okay, we have evaluated work done for each of the cells in the pyramid and now
|
|||
|
|
we need to put it together.</p><p>Let's split the time complexity of our solution into two operands:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(r + s)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">s</span><span class="mclose">)</span></span></span></span></span></div><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span></span></span></span> will represent the <em>actual</em> calculation of the cells and <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">s</span></span></span></span></span> will represent
|
|||
|
|
the additional retrievals on top of the calculation.</p><p>We calculate the values only <strong>once</strong>, therefore we can safely agree on:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>r</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*} r &= n \cdot \log{n} \\ \end{align*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5em;vertical-align:-0.5em"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1em"><span style="top:-3.16em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1em"><span style="top:-3.16em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>What about the <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">s</span></span></span></span></span> though? Key observation here is the fact that we have 2
|
|||
|
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lookups on the tree in each of them <strong>and</strong> we do it twice, cause each cell has
|
|||
|
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at most 2 parents:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>s</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>n</mi><mo>⋅</mo><mn>2</mn><mo>⋅</mo><mrow><mo fence="true">(</mo><mn>2</mn><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>s</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>4</mn><mo>⋅</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*} s &= n \cdot 2 \cdot \left( 2 \cdot \log{n} \right) \\ s &= 4 \cdot n \cdot \log{n} \end{align*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em"><span style="top:-3.91em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-2.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em"><span style="top:-3.91em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span><span style="top:-2.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em"><span></span></span></span></span></span></span></span></span></span></span></s
|
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|
of the results. This is not entirely true, since we have included the
|
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<code>.containsKey()</code> and <code>.get()</code> from the <code>return</code> statement in the second part.</p><p>If we were to represent this more precisely, we could've gone with:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>r</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>3</mn><mo>⋅</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>s</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mo>⋅</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*} r &= 3 \cdot n \cdot \log{n} \\ s &= 2 \cdot n \cdot \log{n} \end{align*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em"><span style="top:-3.91em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span></span><span style="top:-2.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em"><span style="top:-3.91em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span><span style="top:-2.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>On the other hand we are summing both numbers together, therefore in the end it
|
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doesn't really matter.</p><p>(<em>Feel free to compare the sums of both “splits”.</em>)</p></div></div><p>And so our final time complexity for the whole <em>top-down dynamic programming</em>
|
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|
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approach is:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mi>s</mi><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi><mo>+</mo><mn>4</mn><mo>⋅</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>5</mn><mo>⋅</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(r + s) \\ \mathcal{O}(n \cdot \log{n} + 4 \cdot n \cdot \log{n}) \\ \mathcal{O}(5 \cdot n \cdot \log{n}) \\ \mathcal{O}(n \cdot \log{n})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">s</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.4445em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.4445em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" sty
|
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it's better than the <em>naïve</em> one.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="memory-complexity">Memory complexity<a href="#memory-complexity" class="hash-link" aria-label="Direct link to Memory complexity" title="Direct link to Memory complexity"></a></h3><p>With this approach we need to talk about the memory complexity too, because we
|
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have introduced cache. If you think that the memory complexity is linear to the
|
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input, you are right. We start at the top and try to find each and every slide
|
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down. At the end we get the final result for <code>new Position(0, 0)</code>, so we need to
|
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compute everything below.</p><p>That's how we obtain:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span></div><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span></span> represents the total amount of cells in the pyramid, i.e.</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow><mrow><mrow><mi mathvariant="monospace">p</mi><mi mathvariant="monospace">y</mi><mi mathvariant="monospace">r</mi><mi mathvariant="monospace">a</mi><mi mathvariant="monospace">m</mi><mi mathvariant="monospace">i</mi><mi mathvariant="monospace">d</mi><mi mathvariant="monospace">.</mi><mi mathvariant="monospace">l</mi><mi mathvariant="monospace">e</mi><mi mathvariant="monospace">n</mi><mi mathvariant="monospace">g</mi><mi mathvariant="monospace">t</mi><mi mathvariant="monospace">h</mi></mrow><mo>−</mo><mn>1</mn></mrow></munderover><mrow><mi mathvariant="monospace">p</mi><mi mathvariant="monospace">y</mi><mi mathvariant="monospace">r</mi><mi mathvariant="monospace">a</mi><mi mathvariant="monospace">m</mi><mi mathvariant="monospace">i</mi><mi mathvariant="monospace">d</mi></mrow><mrow><mo fence="true">[</mo><mi>y</mi><mo fence="true">]</mo></mrow><mrow><mi mathvariant="monospace">.</mi><mi mathvariant="monospace">l</mi><mi mathvariant="monospace">e</mi><mi mathvariant="monospace">n</mi><mi mathvariant="monospace">g</mi><mi mathvariant="monospace">t</mi><mi mathvariant="monospace">h</mi></mrow></mrow><annotation encoding="application/x-tex">\sum_{y=0}^{\mathtt{pyramid.length} - 1} \mathtt{pyramid}\left[y\right]\mathtt{.length}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2709em;vertical-align:-1.4032em"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8677em"><span style="top:-1.8829em;margin-left:0em"><span class="pstrut" style="height:3.05em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">y</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em"><span class="pstrut" style="height:3.05em"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3666em;margin-left:0em"><span class="pstrut" style="height:3.05em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathtt mtight">pyramid.length</span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4032em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span
|
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function, your guess is right. However we are expressing the complexity in the
|
|||
|
|
Bachmann-Landau notation, so we care about the <strong>upper bound</strong>, not the exact
|
|||
|
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number.</p></div></div><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Can this be optimized?</div><div class="admonitionContent_S0QG"><p>Yes, it can! Try to think about a way, how can you minimize the memory
|
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complexity of this approach. I'll give you a hint:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>r</mi><mi>o</mi><mi>w</mi><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(rows)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">ro</span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mord mathnormal">s</span><span class="mclose">)</span></span></span></span></span></div></div></div><h2 class="anchor anchorWithStickyNavbar_LWe7" id="bottom-up-dp">Bottom-up DP<a href="#bottom-up-dp" class="hash-link" aria-label="Direct link to Bottom-up DP" title="Direct link to Bottom-up DP"></a></h2><p>If you try to think in depth about the top-down DP solution, you might notice
|
|||
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that the <em>core</em> of it stands on caching the calculations that have been already
|
|||
|
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done on the lower “levels” of the pyramid. Our bottom-up implementation will be
|
|||
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using this fact!</p><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_S0QG"><p>As I have said in the <em>top-down DP</em> section, it is the easiest way to implement
|
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DP (unless the cached function has complicated parameters, in that case it might
|
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get messy).</p><p>Bottom-up dynamic programming can be more effective, but may be more complicated
|
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to implement right from the beginning.</p></div></div><p>Let's see how we can implement it:</p><div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token keyword" style="color:rgb(0, 0, 255)">public</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">static</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token plain"> </span><span class="token function" style="color:rgb(0, 0, 255)">longestSlideDown</span><span class="token punctuation" style="color:rgb(4, 81, 165)">(</span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(4, 81, 165)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token comment" style="color:rgb(0, 128, 0)">// In the beginning we declare new array. At this point it is easier to just</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token comment" style="color:rgb(0, 128, 0)">// work with the one dimension, i.e. just allocating the space for the rows.</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token plain"> slideDowns </span><span class="token operator" style="color:rgb(0, 0, 0)">=</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">new</span><span class="token plain"> </span><span class="token keyword" style="color:rgb(0, 0, 255)">int</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:rgb(4, 81, 165)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token punctuation" style="color:rgb(4, 81, 165)">]</span><span class="token punctuation" style="color:rgb(4, 81, 165)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token comment" style="color:rgb(0, 128, 0)">// Bottom row gets just copied, there's nothing else to do… It's the base</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> </span><span class="token comment" style="color:rgb(0, 128, 0)">// case.</span><span class="token plain"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> slideDowns</span><span class="token punctuation" style="color:rgb(4, 81, 165)">[</span><span class="token
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might be more beneficial to see right next to the “offending” lines.</p><p>As you can see, in this approach we go from the other side<sup id="fnref-3"><a href="#fn-3" class="footnote-ref">3</a></sup>, the bottom of
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the pyramid and propagate the partial results up.</p><div class="theme-admonition theme-admonition-info alert alert--info admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span><mdxadmonitiontitle>How is this different from the <em>greedy</em> solution???</mdxadmonitiontitle></div><div class="admonitionContent_S0QG"><p>If you try to compare them, you might find a very noticable difference. The
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greedy approach is going from the top to the bottom without <strong>any</strong> knowledge of
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what's going on below. On the other hand, bottom-up DP is going from the bottom
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(<em>DUH…</em>) and <strong>propagates</strong> the partial results to the top. The propagation is
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what makes sure that at the top I don't choose the best <strong>local</strong> choice, but
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the best <strong>overall</strong> result I can achieve.</p></div></div><h3 class="anchor anchorWithStickyNavbar_LWe7" id="time-complexity-3">Time complexity<a href="#time-complexity-3" class="hash-link" aria-label="Direct link to Time complexity" title="Direct link to Time complexity"></a></h3><p>Time complexity of this solution is rather simple. We allocate an array for the
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rows and then for each row, we copy it and adjust the partial results. Doing
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this we get:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>r</mi><mi>o</mi><mi>w</mi><mi>s</mi><mo>+</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(rows + 2n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">ro</span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span></div><p>Of course, this is an upper bound, since we iterate through the bottom row only
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once.</p><h3 class="anchor anchorWithStickyNavbar_LWe7" id="memory-complexity-1">Memory complexity<a href="#memory-complexity-1" class="hash-link" aria-label="Direct link to Memory complexity" title="Direct link to Memory complexity"></a></h3><p>We're allocating an array for the pyramid <strong>again</strong> for our partial results, so
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we get:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span></div><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_S0QG"><p>If we were writing this in C++ or Rust, we could've avoided that, but not
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really.</p><p>C++ would allow us to <strong>copy</strong> the pyramid rightaway into the parameter, so we
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would be able to directly change it. However it's still a copy, even though we
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don't need to allocate anything ourselves. It's just implicitly done for us.</p><p>Rust is more funny in this case. If the pyramids weren't used after the call of
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<code>longest_slide_down</code>, it would simply <strong>move</strong> them into the functions. If they
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were used afterwards, the compiler would force you to either borrow it, or
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<em>clone-and-move</em> for the function.</p><hr><p>Since we're doing it in Java, we get a reference to the <em>original</em> array and we
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can't do whatever we want with it.</p></div></div><h2 class="anchor anchorWithStickyNavbar_LWe7" id="summary">Summary<a href="#summary" class="hash-link" aria-label="Direct link to Summary" title="Direct link to Summary"></a></h2><p>And we've finally reached the end. We have seen 4 different “solutions”<sup id="fnref-4"><a href="#fn-4" class="footnote-ref">4</a></sup> of
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the same problem using different approaches. Different approaches follow the
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order in which you might come up with them, each approach influences its
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successor and represents the way we can enhance the existing implementation.</p><hr><div class="theme-admonition theme-admonition-info alert alert--info admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>source</div><div class="admonitionContent_S0QG"><p>You can find source code referenced in the text
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<a href="/files/ib002/recursion/pyramid-slide-down.tar.gz" target="_blank" rel="noopener noreferrer">here</a>.</p></div></div><div class="footnotes"><hr><ol><li id="fn-1">cause why not, right!?<a href="#fnref-1" class="footnote-backref">↩</a></li><li id="fn-2">except the bottom row<a href="#fnref-2" class="footnote-backref">↩</a></li><li id="fn-3">definitely not an RHCP reference 😉<a href="#fnref-3" class="footnote-backref">↩</a></li><li id="fn-4">one was not correct, thus the quotes<a href="#fnref-4" class="footnote-backref">↩</a></li></ol></div></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-tags-row row margin-bottom--sm"><div class="col"><b>Tags:</b><ul class="tags_jXut padding--none margin-left--sm"><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/java">java</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/recursion">recursion</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/exponential">exponential</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/greedy">greedy</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/dynamic-programming">dynamic-programming</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/top-down-dp">top-down-dp</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/bottom-up-dp">bottom-up-dp</a></li></ul></div></div><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://gitlab.com/mfocko/blog/tree/main/ib002/04-recursion/2023-08-17-pyramid-slide-down.md" target="_blank" rel="noreferrer noopener" class="theme-edit-this-page"><svg fill="currentColor" height="20" width="20" viewBox="0 0 40 40" class="iconEdit_Z9Sw" aria-hidden="true"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="col lastUpdated_vwxv"><span class="theme-last-updated">Last updated<!-- --> on <b><time datetime="2023-09-07T17:49:31.000Z">Sep 7, 2023</time></b></span></div></div></footer></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/ib002/recursion/karel-1"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Recursion and backtracking with Robot Karel</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/ib002/category/red-black-trees"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Red-Black Trees</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#problem" class="table-of-contents__link toc-highlight">Problem</a></li><li><a href="#solving-the-problem" class="table-of-contents__link toc-highlight">Solving the problem</a></li><li><a href="#naïve-solution" class="table-of-contents__link toc-highlight">Naïve solution</a><ul><li><a href="#time-complexity" class="table-of-contents__link toc-highlight">Time complexity</a></li></ul></li><li><a href="#greedy-solution" class="table-of-contents__link toc-highlight">Greedy solution</a><ul><li><a href="#time-complexity-1" class="table-of-contents__link toc-highlight">Time complexity</a></li><li><a href="#running-the-tests" class="table-of-contents__link toc-highlight">Running the tests</a></li></ul></li><li><a href="#top-down-dp" class="table-of-contents__link toc-highlight">Top-down DP</a><ul><li><a href="#time-complexity-2" class="table-of-contents__link toc-highlight">Time complexity</a></li><li><a href="#memory-complexity" class="table-of-contents__link toc-highlight">Memory complexity</a></li></ul></li><li><a href="#bottom-up-dp" class="table-of-contents__link toc-highlight">Bottom-up DP</a><ul><li><a href="#time-compl
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