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the one with maximum sum.</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 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 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 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">&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: We have gotten out of bounds, there&#x27;s no reasonable value to</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// 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:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// be maximum.</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<
<p>As you can see, we have 2 overloads:</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 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 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"> </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 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 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>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&#x27;s nothing to do at the bottom of the
pyramid, so we just return the value in the <em>cell</em>. Otherwise we add it and try
to slide down the available cells below the current row.</p>
<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&#x27;s find the
worst case scenario.</p>
<p>Let&#x27;s start with the first overload:</p>
<div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#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 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 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"> </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 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>There&#x27;s not much to do here, so we can safely say that the time complexity of
this function is bounded by <span class="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&#x27;t tell us anything, so let&#x27;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-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 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 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 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">&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: We have gotten out of bounds, there&#x27;s no reasonable value to</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// 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:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// be maximum.</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<
<p>Fun fact is that the whole “algorithm” consists of just 2 <code>return</code> statements
and nothing else. Let&#x27;s dissect them!</p>
<p>First <code>return</code> statement is the base case, so it has a constant time complexity.</p>
<p>Second one a bit tricky. We add two numbers together, which we&#x27;ll consider as
constant, but for the right part of the expression we take maximum from the left
and right paths. OK… So what happens? We evaluate the <code>longestSlideDown</code> while
choosing the under and right both. They are separate computations though, so we
are branching from each call of <code>longestSlideDown</code>, unless it&#x27;s a base case.</p>
<p>What does that mean for us then? We basically get</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>, if </mtext><mi>y</mi><mo>=</mo><mi>r</mi><mi>o</mi><mi>w</mi><mi>s</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>⋅</mo><mi>T</mi><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>, otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">T(y) =
\begin{cases}
1 &amp; \text{, if } y = rows \\
1 + 2 \cdot T(y + 1) &amp; \text{, otherwise}
\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal" style="margin-right:0.13889em">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">, if </span></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">ro</span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mord mathnormal">s</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">, 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&#x27;t it? If you sum it up, you&#x27;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&#x27;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&#x27;s how we
obtained <code>2 *</code> and <code>y + 1</code> in the <em>otherwise</em> case.</li>
<li>We move row-by-row, so we move down <code>y</code>-times and each call splits to two
subtrees.</li>
<li>Overall, if we were to represent the calls as a tree, we would get a full
binary tree of height <code>y</code>, in each node we do some work in constant time,
therefore we can just sum the ones.</li>
</ol>
<div class="theme-admonition theme-admonition-warning 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>warning</div><div class="admonitionContent_BuS1"><p>It would&#x27;ve been more complicated to get an exact result. In the equation above
we are assuming that the width of the pyramid is bound by the height.</p></div></div>
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