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approach.< / p >
< 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 > Greedy algorithms< / div > < div class = "admonitionContent_BuS1" > < p > < em > Greedy algorithms< / em > can be described as algorithms that decide the action on the
optimal option at the moment.< / p > < / div > < / div >
< p > We can try to adjust the naïve solution. The most problematic part are the
recursive calls. Let' s apply the greedy approach there:< / p >
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< div class = "language-java codeBlockContainer_Ckt0 theme-code-block" style = "--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)" > < div class = "codeBlockContent_biex" > < pre tabindex = "0" class = "prism-code language-java codeBlock_bY9V thin-scrollbar" style = "background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)" > < code class = "codeBlockLines_e6Vv" > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token keyword" style = "color:hsl(301, 63%, 40%)" > public< / span > < span class = "token plain" > < / span > < span class = "token keyword" style = "color:hsl(301, 63%, 40%)" > static< / span > < span class = "token plain" > < / span > < span class = "token keyword" style = "color:hsl(301, 63%, 40%)" > int< / span > < span class = "token plain" > < / span > < span class = "token function" style = "color:hsl(221, 87%, 60%)" > longestSlideDown< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > (< / span > < span class = "token keyword" style = "color:hsl(301, 63%, 40%)" > int< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > [< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > ]< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > [< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > ]< / span > < span class = "token plain" > pyramid< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > ,< / span > < span class = "token plain" > < / span > < span class = "token keyword" style = "color:hsl(301, 63%, 40%)" > int< / span > < span class = "token plain" > row< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > ,< / span > < span class = "token plain" > < / span > < span class = "token keyword" style = "color:hsl(301, 63%, 40%)" > int< / span > < span class = "token plain" > col< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > )< / span > < span class = "token plain" > < / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > {< / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token keyword" style = "color:hsl(301, 63%, 40%)" > if< / span > < span class = "token plain" > < / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > (< / span > < span class = "token plain" > row < / span > < span class = "token operator" style = "color:hsl(221, 87%, 60%)" > ==< / span > < span class = "token plain" > pyramid< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > .< / span > < span class = "token plain" > length < / span > < span class = "token operator" style = "color:hsl(221, 87%, 60%)" > -< / span > < span class = "token plain" > < / span > < span class = "token number" style = "color:hsl(35, 99%, 36%)" > 1< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > )< / span > < span class = "token plain" > < / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > {< / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token comment" style = "color:hsl(230, 4%, 64%)" > // BASE: We' re at the bottom< / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token keyword" style = "color:hsl(301, 63%, 40%)" > return< / span > < span class = "token plain" > pyramid< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > [< / span > < span class = "token plain" > row< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > ]< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > [< / span > < span class = "token plain" > col< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > ]< / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > ;< / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token punctuation" style = "color:hsl(119, 34%, 47%)" > }< / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > <
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< p > OK, if we cannot go right < strong > or< / strong > the right path adds smaller value to the sum,
we simply go left.< / 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 > We have switched from < em > adding the maximum< / em > to < em > following the “bigger” path< / em > , so
we improved the time complexity tremendously. We just go down the pyramid all
the way to the bottom. Therefore we are getting:< / 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 >
< p > We have managed to convert our exponential solution into a linear one.< / p >
< h2 class = "anchor anchorWithStickyNavbar_LWe7" id = "running-the-tests" > Running the tests< a href = "#running-the-tests" class = "hash-link" aria-label = "Direct link to Running the tests" title = "Direct link to Running the tests" > < / a > < / h2 >
< p > However, if we run the tests, we notice that the second test failed:< / p >
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< div class = "codeBlockContainer_Ckt0 theme-code-block" style = "--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)" > < div class = "codeBlockContent_biex" > < pre tabindex = "0" class = "prism-code language-text codeBlock_bY9V thin-scrollbar" style = "background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)" > < code class = "codeBlockLines_e6Vv" > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > Test #1: passed< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > Test #2: failed< / 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 >
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< p > What' s going on? Well, we have improved the time complexity, but greedy
algorithms are not the ideal solution to < strong > all< / strong > problems. In this case there
may be a solution that is bigger than the one found using the greedy algorithm.< / p >
< p > Imagine the following pyramid:< / p >
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< div class = "codeBlockContainer_Ckt0 theme-code-block" style = "--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)" > < div class = "codeBlockContent_biex" > < pre tabindex = "0" class = "prism-code language-text codeBlock_bY9V thin-scrollbar" style = "background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)" > < code class = "codeBlockLines_e6Vv" > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > 1< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > 2 3< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > 5 6 7< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > 8 9 10 11< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > 99 13 14 15 16< / 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 >
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< p > We start at the top:< / p >
< ol >
< li > Current cell: < code > 1< / code > , we can choose from < code > 2< / code > and < code > 3< / code > , < code > 3< / code > looks better, so we
choose it.< / li >
< li > Current cell: < code > 3< / code > , we can choose from < code > 6< / code > and < code > 7< / code > , < code > 7< / code > looks better, so we
choose it.< / li >
< li > Current cell: < code > 7< / code > , we can choose from < code > 10< / code > and < code > 11< / code > , < code > 11< / code > looks better, so we
choose it.< / li >
< li > Current cell: < code > 11< / code > , we can choose from < code > 15< / code > and < code > 16< / code > , < code > 16< / code > looks better, so
we choose it.< / li >
< / ol >
< p > Our final sum is: < code > 1 + 3 + 7 + 11 + 16 = 38< / code > , but in the bottom left cell we
have a < code > 99< / code > that is bigger than our whole sum.< / 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 > Dijkstra' s algorithm is a greedy algorithm too, try to think why it is correct.< / p > < / div > < / div > < / div > < footer class = "theme-doc-footer docusaurus-mt-lg" > < div class = "theme-doc-footer-tags-row row margin-bottom--sm" > < div class = "col" > < b > Tags:< / b > < ul class = "tags_jXut padding--none margin-left--sm" > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/java/" > java< / a > < / li > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/greedy/" > greedy< / a > < / li > < / ul > < / div > < / div > < div class = "theme-doc-footer-edit-meta-row row" > < div class = "col" > < a href = "https://github.com/mfocko/blog/tree/main/algorithms/04-recursion/2023-08-17-pyramid-slide-down/02-greedy.md" target = "_blank" rel = "noopener noreferrer" class = "theme-edit-this-page" > < svg fill = "currentColor" height = "20" width = "20" viewBox = "0 0 40 40" class = "iconEdit_Z9Sw" aria-hidden = "true" > < g > < path d = "m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z" > < / path > < / g > < / svg > Edit this page< / a > < / div > < div class = "col lastUpdated_vwxv" > < span class = "theme-last-updated" > Last updated<!-- --> on < b > < time datetime = "2023-08-17T00:00:00.000Z" > Aug 17, 2023< / time > < / b > < / span > < / div > < / div > < / footer > < / article > < nav class = "pagination-nav docusaurus-mt-lg" aria-label = "Docs pages" > < a class = "pagination-nav__link pagination-nav__link--prev" href = "/algorithms/recursion/pyramid-slide-down/naive/" > < div class = "pagination-nav__sublabel" > Previous< / div > < div class = "pagination-nav__label" > Naïve solution< / div > < / a > < a class = "pagination-nav__link pagination-nav__link--next" href = "/algorithms/recursion/pyramid-slide-down/top-down-dp/" > < div class = "pagination-nav__sublabel" > Next< / div > < div class = "pagination-nav__label" > Top-down DP solution< / div > < / a > < / nav > < / div > < / div > < div class = "col col--3" > < div class = "tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop" > < ul class = "table-of-contents table-of-contents__left-border" > < li > < a href = "#time-complexity" class = "table-of-contents__link toc-highlight" > Time complexity< / a > < / li > < li > < a href = "#running-the-tests" class = "table-of-contents__link toc-highlight" > Running the tests< / a > < / li > < / ul > < / div > < / div > < / div > < / div > < / main > < / div > < / div > < / div > < footer class = "footer footer--dark" > < div class = "container container-fluid" > < div class = "row footer__links" > < div class = "col footer__col" > < div class = "footer__title" > Git< / div > < ul class = "footer__items clean-list" > < li class = "footer__item" > < a href = "https://github.com/mfocko" target = "_blank" rel = "noopener noreferrer" class = "footer__link-item" > GitHub< svg width = "13.5" height = "13.5" aria-hidden = "true" viewBox = "0 0 24 24" class = "iconExternalLink_nPIU" > < path fill = "currentColor" d = "M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z" > < / path > < / svg > < / a > < / li > < li class = "footer__item" > < a href = "https://gitlab.com/mfocko" target = "_blank" rel = "noopener noreferrer" class = "footer__link-item" > GitLab< svg width = "13.5" height = "13.5" aria-hidden = "true" viewBox = "0 0 24 24" class = "iconExternalLink_nPIU" > < path fill = "currentColor" d = "M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z" > < / path > < / svg > < / a > < / li > < li class = "footer__item" > < a href = "https://git.mfocko.xyz/mfocko" target = "_blank" rel = "noopener noreferrer" class = "footer_
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