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7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 1-1z" 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">Top-down DP 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"><h1>Top-down dynamic programming</h1>
<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:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic">// 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:#393A34"><span class="token plain"></span><span class="token comment" style="color:#999988;font-style:italic">// 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:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">record</span><span class="token plain"> </span><span class="token class-name">Position</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> row</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> col</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">implements</span><span class="token plain"> </span><span class="token class-name">Comparable</span><span class="token generics punctuation" style="color:#393A34">&lt;</span><span class="token generics class-name">Position</span><span class="token generics punctuation" style="color:#393A34">&gt;</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">public</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">compareTo</span><span class="token punctuation" style="color:#393A34">(</span><span class="token class-name">Position</span><span class="token plain"> r</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">if</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">row </span><span class="token operator" style="color:#393A34">!=</span><span class="token plain"> r</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">row</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> </span><span class="token class-name">Integer</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">valueOf</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">row</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">compareTo</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">r</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">row</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">if</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">col </span><span class="token operator" style="color:#393A34">!=</span><span class="token plain"> r</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">col</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> </span><span class="token class-name">Integer</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">valueOf</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">col</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">compareTo</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">r</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">col</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">public</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">static</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">longestSlideDown</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token punctuation" style="color:#393A34">[</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token class-name">TreeMap</span><span class="token generics punctuation" style="color:#393A34">&lt;</span><span class="token generics class-name">Position</span><span class="token generics punctuation" style="color:#393A34">,</span><span class="token generics"> </span><span class="token generics class-name">Integer</span><span class="token generics punctuation" style="color:#393A34">&gt;</span><span class="token plain"> cache</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token class-name">Position</span><span class="token plain"> position</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> row </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> position</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">row</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> col </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> position</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">col</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">if</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">row </span><span class="token operator" style="color:#393A34">&gt;=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">length </span><span class="token operator" style="color:#393A34">||</span><span class="token plain"> col </span><span class="token operator" style="color:#393A34">&lt;</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">||</span><span class="token plain"> col </span><span class="token operator" style="color:#393A34">&gt;=</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">row</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">length</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// BASE: out of bounds</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> </span><span class="token class-name">Integer</span><span class="token punctuation" style="color:#393A34">.</span><span class="token constant" style="color:#36acaa">MIN_VALUE</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">if</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">row </span><span class="token operator" style="color:#393A34">==</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">length </span><span class="token operator" style="color:#393A34">-</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// BASE: bottom of the pyramid</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">position</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">row</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">position</span><span class="token punctuation" style="color:#393A34">.</span><span class="token plain">col</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">if</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token operator" style="color:#393A34">!</span><span class="token plain">cache</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">containsKey</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">position</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// We haven&#x27;t computed the position yet, so we run the same “formula” as</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// in the naïve version »and« we put calculated slide into the cache.</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// Next time we want the slide down from given position, it will be just</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// retrieved from the cache.</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> slideDown </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token class-name">Math</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">max</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token function" style="color:#d73a49">longestSlideDown</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> cache</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">new</span><span class="token plain"> </span><span class="token class-name">Position</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">row </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> col</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token function" style="color:#d73a49">longestSlideDown</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> cache</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">new</span><span class="token plain"> </span><span class="token class-name">Position</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">row </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> col </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> cache</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">put</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">position</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">row</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">col</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> slideDown</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> cache</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">get</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">position</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">public</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">static</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">longestSlideDown</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">int</span><span class="token punctuation" style="color:#393A34">[</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> pyramid</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// At the beginning we need to create a cache and share it across the calls.</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token class-name">TreeMap</span><span class="token generics punctuation" style="color:#393A34">&lt;</span><span class="token generics class-name">Position</span><span class="token generics punctuation" style="color:#393A34">,</span><span class="token generics"> </span><span class="token generics class-name">Integer</span><span class="token generics punctuation" style="color:#393A34">&gt;</span><span class="token plain"> cache </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">new</span><span class="token plain"> </span><span class="token class-name">TreeMap</span><span class="token generics punctuation" style="color:#393A34">&lt;</span><span class="token generics punctuation" style="color:#393A34">&gt;</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">longestSlideDown</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">pyramid</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> cache</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">new</span><span class="token plain"> </span><span class="token class-name">Position</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</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>You have probably noticed that <code>record Position</code> have appeared. Since we are
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 admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><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>
<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 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><a href="#user-content-fn-1" id="user-content-fnref-1" data-footnote-ref="true" aria-describedby="footnote-label">1</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="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>.</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 admonition_xJq3 alert alert--warning"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><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="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>.</p>
</li>
<li>
<p>We retrieve it from the cache. Same as in first point, but only twice, so we
get <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>.</p>
<div class="theme-admonition theme-admonition-caution admonition_xJq3 alert alert--warning"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><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>
<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>
<p><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> will represent the <em>actual</em> calculation of the cells and <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> 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>
<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>
<p>What about the <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> 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>
<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></span>
<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><p>You might&#x27;ve noticed that lookups actually take more time than the construction
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><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><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>
<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" 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 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></span></span>
<p>As you can see, this is worse than our <em>greedy</em> solution that was incorrect, but
it&#x27;s better than the <em>naïve</em> one.</p>
<h2 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></h2>
<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>
<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>
<p><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> represents the total amount of cells in the pyramid, i.e.</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><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 class="mord"><span class="mord mathtt">pyramid</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathtt">.length</span></span></span></span></span></span>
<div class="theme-admonition theme-admonition-caution admonition_xJq3 alert alert--warning"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><p>If you&#x27;re wondering whether it&#x27;s correct because of the second <code>if</code> in our
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 admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><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><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>
<section data-footnotes="true" class="footnotes"><h2 class="anchor anchorWithStickyNavbar_LWe7 sr-only" id="footnote-label">Footnotes<a href="#footnote-label" class="hash-link" aria-label="Direct link to Footnotes" title="Direct link to Footnotes">​</a></h2>
<ol>
<li id="user-content-fn-1">
<p>except the bottom row <a href="#user-content-fnref-1" data-footnote-backref="" aria-label="Back to reference 1" class="data-footnote-backref">↩</a></p>
</li>
</ol>
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