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fill="currentColor"></path></svg></a></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="/algorithms/category/recursion/"><span itemprop="name">Recursion</span></a><meta itemprop="position" content="1"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="/algorithms/recursion/pyramid-slide-down/"><span itemprop="name">Introduction to dynamic programming</span></a><meta itemprop="position" content="2"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link" itemprop="name">Naïve solution</span><meta itemprop="position" content="3"></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><header><h1>Naïve solution</h1></header><p>Our naïve solution consists of trying out all the possible slides and finding
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the one with maximum sum.</p>
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<div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token keyword" style="color:hsl(301, 63%, 40%)">public</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">static</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">>=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"> col </span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"> col </span><span class="token operator" style="color:hsl(221, 87%, 60%)">>=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// BASE: We have gotten out of bounds, there's no reasonable value to</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// 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:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// be maximum.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">Integer</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token constant" style="color:hsl(35, 99%, 36%)">MIN_VALUE</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">==</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// BASE: Bottom of the pyramid, we just return the value, there's</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// nowhere to slide anymore.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// Otherwise we account for the current position and return maximum of the</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// available “slides”.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">Math</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token function" style="color:hsl(221, 87%, 60%)">max</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> col </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">public</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">static</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// We start the slide in the top cell of the pyramid.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</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>As you can see, we have 2 overloads:</p>
|
||
<div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</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>First one is used as a <em>public interface</em> to the solution, you just pass in the
|
||
pyramid itself. Second one is the recursive “algorithm” that finds the slide
|
||
down.</p>
|
||
<p>It is a relatively simple solution… There'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>
|
||
<h2 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></h2>
|
||
<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's find the
|
||
worst case scenario.</p>
|
||
<p>Let's start with the first overload:</p>
|
||
<div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token keyword" style="color:hsl(301, 63%, 40%)">public</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">static</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p>There's not much to do here, so we can safely say that the time complexity of
|
||
this function is bounded by <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>, where <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> is our second overload. This
|
||
doesn't tell us anything, so let's move on to the second overload where we are
|
||
going to define the <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> function.</p>
|
||
<div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token keyword" style="color:hsl(301, 63%, 40%)">public</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">static</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">int</span><span class="token plain"> col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">>=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"> col </span><span class="token operator" style="color:hsl(221, 87%, 60%)"><</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">||</span><span class="token plain"> col </span><span class="token operator" style="color:hsl(221, 87%, 60%)">>=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// BASE: We have gotten out of bounds, there's no reasonable value to</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// 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:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// be maximum.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">Integer</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token constant" style="color:hsl(35, 99%, 36%)">MIN_VALUE</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">if</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">==</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token plain">length </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// BASE: Bottom of the pyramid, we just return the value, there's</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// nowhere to slide anymore.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// Otherwise we account for the current position and return maximum of the</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)">// available “slides”.</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">return</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">row</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token class-name" style="color:hsl(35, 99%, 36%)">Math</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">.</span><span class="token function" style="color:hsl(221, 87%, 60%)">max</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> col</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token function" style="color:hsl(221, 87%, 60%)">longestSlideDown</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> row </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> col </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">}</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>Fun fact is that the whole “algorithm” consists of just 2 <code>return</code> statements
|
||
and nothing else. Let's dissect them!</p>
|
||
<p>First <code>return</code> statement is the base case, so it has a constant time complexity.</p>
|
||
<p>Second one a bit tricky. We add two numbers together, which we'll consider as
|
||
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's a base case.</p>
|
||
<p>What does that mean for us then? We basically get</p>
|
||
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>, if </mtext><mi>y</mi><mo>=</mo><mi>r</mi><mi>o</mi><mi>w</mi><mi>s</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>⋅</mo><mi>T</mi><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>, otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">T(y) =
|
||
\begin{cases}
|
||
1 & \text{, if } y = rows \\
|
||
1 + 2 \cdot T(y + 1) & \text{, otherwise}
|
||
\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">, if </span></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">ro</span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mord mathnormal">s</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">, otherwise</span></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></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>
|
||
<p>That looks rather easy to compute, isn't it? If you sum it up, you'll get:</p>
|
||
<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>r</mi><mi>o</mi><mi>w</mi><mi>s</mi><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mrow><mi>r</mi><mi>o</mi><mi>w</mi><mi>s</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(rows) \in \mathcal{O}(2^{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 mathnormal" style="margin-right:0.13889em">T</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 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:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ro</span><span class="mord mathnormal mtight" style="margin-right:0.02691em">w</span><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>
|
||
<p>If you wonder why, I'll try to describe it intuitively:</p>
|
||
<ol>
|
||
<li>In each call to <code>longestSlideDown</code> we do some work in constant time,
|
||
regardless of being in the base case. Those are the <code>1</code>s in both cases.</li>
|
||
<li>If we are not in the base case, we move one row down <strong>twice</strong>. That's how we
|
||
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-danger admonition_xJq3 alert alert--danger"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><p>It would'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></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="/algorithms/tags/java/">java</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/recursion/">recursion</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/exponential/">exponential</a></li></ul></div></div><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://github.com/mfocko/blog/tree/main/algorithms/04-recursion/2023-08-17-pyramid-slide-down/01-naive.md" target="_blank" rel="noopener noreferrer" 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 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