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<title data-rh="true">Bottom-up 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/bottom-up-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="Bottom-up DP solution | mf"><meta data-rh="true" name="description" content="Bottom-up DP solution of the Pyramid Slide Down.
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<p>If you try to think in depth about the top-down DP solution, you might notice
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that the <em>core</em> of it stands on caching the calculations that have been already
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done on the lower “levels” of the pyramid. Our bottom-up implementation will be
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using this fact!</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>tip</div><div class="admonitionContent_BuS1"><p>As I have said in the <em>top-down DP</em> section, it is the easiest way to implement
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DP (unless the cached function has complicated parameters, in that case it might
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get messy).</p><p>Bottom-up dynamic programming can be more effective, but may be more complicated
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to implement right from the beginning.</p></div></div>
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<p>Let's see how we can implement it:</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 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">// In the beginning we declare new array. At this point it is easier to just</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// work with the one dimension, i.e. just allocating the space for the rows.</span><span class="token plain"></span><br></span><span class="token-line" style="color:#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"> slideDowns </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 keyword" style="color:#00009f">int</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">length</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" style="display:inline-block"></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">// Bottom row gets just copied, there's nothing else to do… It's the base</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">// case.</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> slideDowns</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">length </span><span class="token operator" style="color:#393A34">-</span><span class="token plain"> </spa
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<p>I've tried to explain the code as much as possible within the comments, since it
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might be more beneficial to see right next to the “offending” lines.</p>
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<p>As you can see, in this approach we go from the other side<sup><a href="#user-content-fn-1" id="user-content-fnref-1" data-footnote-ref="true" aria-describedby="footnote-label">1</a></sup>, the bottom of
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the pyramid and propagate the partial results up.</p>
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<div class="theme-admonition theme-admonition-info admonition_xJq3 alert alert--info"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>How is this different from the <em>greedy</em> solution???</div><div class="admonitionContent_BuS1"><p>If you try to compare them, you might find a very noticable difference. The
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greedy approach is going from the top to the bottom without <strong>any</strong> knowledge of
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what's going on below. On the other hand, bottom-up DP is going from the bottom
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(<em>DUH…</em>) and <strong>propagates</strong> the partial results to the top. The propagation is
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what makes sure that at the top I don't choose the best <strong>local</strong> choice, but
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the best <strong>overall</strong> result I can achieve.</p></div></div>
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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>Time complexity of this solution is rather simple. We allocate an array for the
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rows and then for each row, we copy it and adjust the partial results. Doing
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this we get:</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>r</mi><mi>o</mi><mi>w</mi><mi>s</mi><mo>+</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(rows + 2n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">ro</span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span>
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<p>Of course, this is an upper bound, since we iterate through the bottom row only
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once.</p>
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<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>
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<p>We're allocating an array for the pyramid <strong>again</strong> for our partial results, so
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we get:</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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<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>If we were writing this in C++ or Rust, we could've avoided that, but not
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really.</p><p>C++ would allow us to <strong>copy</strong> the pyramid rightaway into the parameter, so we
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would be able to directly change it. However it's still a copy, even though we
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don't need to allocate anything ourselves. It's just implicitly done for us.</p><p>Rust is more funny in this case. If the pyramids weren't used after the call of
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<code>longest_slide_down</code>, it would simply <strong>move</strong> them into the functions. If they
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were used afterwards, the compiler would force you to either borrow it, or
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<em>clone-and-move</em> for the function.</p><hr><p>Since we're doing it in Java, we get a reference to the <em>original</em> array and we
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can't do whatever we want with it.</p></div></div>
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<h1>Summary</h1>
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<p>And we've finally reached the end. We have seen 4 different “solutions”<sup><a href="#user-content-fn-2" id="user-content-fnref-2" data-footnote-ref="true" aria-describedby="footnote-label">2</a></sup> of
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the same problem using different approaches. Different approaches follow the
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order in which you might come up with them, each approach influences its
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successor and represents the way we can enhance the existing implementation.</p>
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<hr>
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<div class="theme-admonition theme-admonition-info admonition_xJq3 alert alert--info"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>source</div><div class="admonitionContent_BuS1"><p>You can find source code referenced in the text
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<a href="/files/algorithms/recursion/pyramid-slide-down.tar.gz" target="_blank" rel="noopener noreferrer">here</a>.</p></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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<p>definitely not an RHCP reference <!-- -->😉 <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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<p>one was not correct, thus the quotes <a href="#user-content-fnref-2" data-footnote-backref="" aria-label="Back to reference 2" class="data-footnote-backref">↩</a></p>
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