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<title data-rh="true">Top-down DP solution | mf</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://blog.mfocko.xyz/algorithms/recursion/pyramid-slide-down/top-down-dp/"><meta data-rh="true" property="og:locale" content="en"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-algorithms-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-algorithms-current"><meta data-rh="true" property="og:title" content="Top-down DP solution | mf"><meta data-rh="true" name="description" content="Top-down DP solution of the Pyramid Slide Down.
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<p><em>Top-down dynamic programming</em> is probably the most common approach, since (at
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least looks like) is the easiest to implement. The whole point is avoiding the
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unnecessary computations that we have already done.</p>
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<p>In our case, we can use our naïve solution and put a <em>cache</em> on top of it that
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will make sure, we don't do unnecessary calculations.</p>
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<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"><</span><span class="token generics class-name">Position</span><span class="token generics 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">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><
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<p>You have probably noticed that <code>record Position</code> have appeared. Since we are
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caching the already computed values, we need a “reasonable” key. In this case we
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share the cache only for one <em>run</em> (i.e. pyramid) of the <code>longestSlideDown</code>, so
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we can cache just with the indices within the pyramid, i.e. the <code>Position</code>.</p>
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<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
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immutable structure with implicitly defined <code>.equals()</code>, <code>.hashCode()</code>,
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<code>.toString()</code> and getters for the attributes.</p></div></div>
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<p>Because of the choice of <code>TreeMap</code>, we had to additionally define the ordering
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on it.</p>
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<p>In the <code>longestSlideDown</code> you can notice that the computation which used to be
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at the end of the naïve version above, is now wrapped in an <code>if</code> statement that
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checks for the presence of the position in the cache and computes the slide down
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just when it's needed.</p>
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<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>
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<p>If you think that evaluating time complexity for this approach is a bit more
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tricky, you are right. Keeping the cache in mind, it is not the easiest thing
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to do. However there are some observations that might help us figure this out:</p>
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<ol>
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<li>Slide down from each position is calculated only once.</li>
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<li>Once calculated, we use the result from the cache.</li>
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</ol>
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<p>Knowing this, we still cannot, at least easily, describe the time complexity of
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finding the best slide down from a specific position, <strong>but</strong> we can bound it
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from above for the <strong>whole</strong> run from the top. Now the question is how we can do
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that!</p>
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<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
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the pyramid:</p>
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<ol>
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<li>
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<p>We calculate and store it (using the partial results stored in cache). This
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is done only once.</p>
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<p>For each calculation we take 2 values from the cache and insert one value.
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Because we have chosen <code>TreeMap</code>, these 3 operations have logarithmic time
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complexity and therefore this step is equivalent to <span class="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>
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<p>However for the sake of simplicity, we are going to account only for the
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insertion, the reason is rather simple, if we include the 2 retrievals here,
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it will be interleaved with the next step, therefore it is easier to keep the
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retrievals in the following point.</p>
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<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's still not that easy, cause we're not having full
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cache right from the beginning, but the sum of those logarithms cannot be
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expressed in a nice way, so taking the upper bound, i.e. expecting the cache
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to be full at all times, is the best option for nice and readable complexity
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of the whole approach.</p></div></div>
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<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>
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</li>
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<li>
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<p>We retrieve it from the cache. Same as in first point, but only twice, so we
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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's done twice because of the <code>.containsKey()</code> in the <code>if</code> condition.</p></div></div>
|
|||
|
|
</li>
|
|||
|
|
</ol>
|
|||
|
|
<p>Okay, we have evaluated work done for each of the cells in the pyramid and now
|
|||
|
|
we need to put it together.</p>
|
|||
|
|
<p>Let's split the time complexity of our solution into two operands:</p>
|
|||
|
|
<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 &= 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 &= n \cdot 2 \cdot \left( 2 \cdot \log{n} \right) \\
|
|||
|
|
s &= 4 \cdot n \cdot \log{n}
|
|||
|
|
\end{align*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em"><span style="top:-3.91em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-2.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em"><span style="top:-3.91em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span><span style="top:-2.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em"><span></span></span></span></span></span></span></span></span></span></span></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'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'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 &= 3 \cdot n \cdot \log{n} \\
|
|||
|
|
s &= 2 \cdot n \cdot \log{n}
|
|||
|
|
\end{align*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em"><span style="top:-3.91em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span></span><span style="top:-2.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em"><span style="top:-3.91em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span><span style="top:-2.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mop">lo<span style="margin-right:0.01389em">g</span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em"><span></span></span></span></span></span></span></span></span></span></span></span><p>On the other hand we are summing both numbers together, therefore in the end it
|
|||
|
|
doesn't really matter.</p><p>(<em>Feel free to compare the sums of both “splits”.</em>)</p></div></div>
|
|||
|
|
<p>And so our final time complexity for the whole <em>top-down dynamic programming</em>
|
|||
|
|
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'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
|
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down. At the end we get the final result for <code>new Position(0, 0)</code>, so we need to
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compute everything below.</p>
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<p>That's how we obtain:</p>
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<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>
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<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>
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<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>
|
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<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're wondering whether it's correct because of the second <code>if</code> in our
|
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function, your guess is right. However we are expressing the complexity in the
|
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|
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Bachmann-Landau notation, so we care about the <strong>upper bound</strong>, not the exact
|
|||
|
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number.</p></div></div>
|
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<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
|
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|
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complexity of this approach. I'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>
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<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>
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<ol>
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<li id="user-content-fn-1">
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<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>
|
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</li>
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</ol>
|
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