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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>.
<em>Slide down</em> is a maximum sum of consecutive numbers from the top to the bottom.</p><p>Let&#x27;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
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
will output <code>true</code>/<code>false</code> based on the expected output of our algorithm. Any
other differences will lie only in the solutions of the problem. You can see the
<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)">&quot;Test #1: &quot;</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
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)">&gt;=</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)">&lt;</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)">&gt;=</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&#x27;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
pyramid itself. Second one is the recursive “algorithm” that finds the slide
down.</p><p>It is a relatively simple solution… There&#x27;s nothing to do at the bottom of the
pyramid, so we just return the value in the <em>cell</em>. Otherwise we add it and try
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
are wondering about the time complexity of the proposed solution and, since it
really is a naïve solution, the time complexity is pretty bad. Let&#x27;s find the
worst case scenario.</p><p>Let&#x27;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&#x27;s not much to do here, so we can safely say that the time complexity of
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
doesn&#x27;t tell us anything, so let&#x27;s move on to the second overload where we are
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)">&gt;=</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)">&lt;</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)">&gt;=</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
and nothing else. Let&#x27;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&#x27;ll consider as
constant, but for the right part of the expression we take maximum from the left
and right paths. OK… So what happens? We evaluate the <code>longestSlideDown</code> while
choosing the under and right both. They are separate computations though, so we
are branching from each call of <code>longestSlideDown</code>, unless it&#x27;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 &amp; \text{, if } y = rows \\ 1 + 2 \cdot T(y + 1) &amp; \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">, 
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&#x27;s how we
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
subtrees.</li><li>Overall, if we were to represent the calls as a tree, we would get a full
binary tree of height <code>y</code>, in each node we do some work in constant time,
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&#x27;ve been more complicated to get an exact result. In the equation above
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&#x27;s start with a relatively simple <em>greedy</em>
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
optimal option at the moment.</p></div></div><p>We can try to adjust the naïve solution. The most problematic part are the
recursive calls. Let&#x27;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&#x27;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
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
we improved the time complexity tremendously. We just go down the pyramid all
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&#x27;s going on? Well, we have improved the time complexity, but greedy
algorithms are not the ideal solution to <strong>all</strong> problems. In this case there
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
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
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
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
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
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&#x27;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
least looks like) is the easiest to implement. The whole point is avoiding the
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
will make sure, we don&#x27;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)">&lt;</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)">&gt;</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=
caching the already computed values, we need a “reasonable” key. In this case we
share the cache only for one <em>run</em> (i.e. pyramid) of the <code>longestSlideDown</code>, so
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
immutable structure with implicitly defined <code>.equals()</code>, <code>.hashCode()</code>,
<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
on it.</p><p>In the <code>longestSlideDown</code> you can notice that the computation which used to be
at the end of the naïve version above, is now wrapped in an <code>if</code> statement that
checks for the presence of the position in the cache and computes the slide down
just when it&#x27;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
tricky, you are right. Keeping the cache in mind, it is not the easiest thing
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
finding the best slide down from a specific position, <strong>but</strong> we can bound it
from above for the <strong>whole</strong> run from the top. Now the question is how we can do
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
the pyramid:</p><ol><li><p>We calculate and store it (using the partial results stored in cache). This
is done only once.</p><p>For each calculation we take 2 values from the cache and insert one value.
Because we have chosen <code>TreeMap</code>, these 3 operations have logarithmic time
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&#x27;s still not that easy, cause we&#x27;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&#x27;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&#x27;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 &amp;= 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
lookups on the tree in each of them <strong>and</strong> we do it twice, cause each cell has
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 &amp;= n \cdot 2 \cdot \left( 2 \cdot \log{n} \right) \\ s &amp;= 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
of the results. This is not entirely true, since we have included the
<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&#x27;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 &amp;= 3 \cdot n \cdot \log{n} \\ s &amp;= 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
doesn&#x27;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>
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
it&#x27;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
have introduced cache. If you think that the memory complexity is linear to the
input, you are right. We start at the top and try to find each and every slide
down. At the end we get the final result for <code>new Position(0, 0)</code>, so we need to
compute everything below.</p><p>That&#x27;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
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
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
complexity of this approach. I&#x27;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
that the <em>core</em> of it stands on caching the calculations that have been already
done on the lower “levels” of the pyramid. Our bottom-up implementation will be
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
DP (unless the cached function has complicated parameters, in that case it might
get messy).</p><p>Bottom-up dynamic programming can be more effective, but may be more complicated
to implement right from the beginning.</p></div></div><p>Let&#x27;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&#x27;s nothing else to do… It&#x27;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
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
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
greedy approach is going from the top to the bottom without <strong>any</strong> knowledge of
what&#x27;s going on below. On the other hand, bottom-up DP is going from the bottom
(<em>DUH…</em>) and <strong>propagates</strong> the partial results to the top. The propagation is
what makes sure that at the top I don&#x27;t choose the best <strong>local</strong> choice, but
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
rows and then for each row, we copy it and adjust the partial results. Doing
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
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&#x27;re allocating an array for the pyramid <strong>again</strong> for our partial results, so
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&#x27;ve avoided that, but not
really.</p><p>C++ would allow us to <strong>copy</strong> the pyramid rightaway into the parameter, so we
would be able to directly change it. However it&#x27;s still a copy, even though we
don&#x27;t need to allocate anything ourselves. It&#x27;s just implicitly done for us.</p><p>Rust is more funny in this case. If the pyramids weren&#x27;t used after the call of
<code>longest_slide_down</code>, it would simply <strong>move</strong> them into the functions. If they
were used afterwards, the compiler would force you to either borrow it, or
<em>clone-and-move</em> for the function.</p><hr><p>Since we&#x27;re doing it in Java, we get a reference to the <em>original</em> array and we
can&#x27;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&#x27;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
the same problem using different approaches. Different approaches follow the
order in which you might come up with them, each approach influences its
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
<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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