blog/assets/js/70a4540f.312c2807.js

"use strict";(self.webpackChunkfi=self.webpackChunkfi||[]).push([[9249],{44493:(e,s,n)=>{n.r(s),n.d(s,{assets:()=>m,contentTitle:()=>l,default:()=>h,frontMatter:()=>i,metadata:()=>r,toc:()=>c});var a=n(85893),t=n(11151);const i={id:"naive",slug:"/recursion/pyramid-slide-down/naive",title:"Na\xefve solution",description:"Na\xefve solution of the Pyramid Slide Down.\n",tags:["java","recursion","exponential"],last_update:{date:new Date("2023-08-17T00:00:00.000Z")}},l=void 0,r={id:"recursion/2023-08-17-pyramid-slide-down/naive",title:"Na\xefve solution",description:"Na\xefve solution of the Pyramid Slide Down.\n",source:"@site/algorithms/04-recursion/2023-08-17-pyramid-slide-down/01-naive.md",sourceDirName:"04-recursion/2023-08-17-pyramid-slide-down",slug:"/recursion/pyramid-slide-down/naive",permalink:"/algorithms/recursion/pyramid-slide-down/naive",draft:!1,unlisted:!1,editUrl:"https://github.com/mfocko/blog/tree/main/algorithms/04-recursion/2023-08-17-pyramid-slide-down/01-naive.md",tags:[{label:"java",permalink:"/algorithms/tags/java"},{label:"recursion",permalink:"/algorithms/tags/recursion"},{label:"exponential",permalink:"/algorithms/tags/exponential"}],version:"current",lastUpdatedAt:1692230400,formattedLastUpdatedAt:"Aug 17, 2023",sidebarPosition:1,frontMatter:{id:"naive",slug:"/recursion/pyramid-slide-down/naive",title:"Na\xefve solution",description:"Na\xefve solution of the Pyramid Slide Down.\n",tags:["java","recursion","exponential"],last_update:{date:"2023-08-17T00:00:00.000Z"}},sidebar:"autogeneratedBar",previous:{title:"Introduction to dynamic programming",permalink:"/algorithms/recursion/pyramid-slide-down"},next:{title:"Greedy solution",permalink:"/algorithms/recursion/pyramid-slide-down/greedy"}},m={},c=[{value:"Time complexity",id:"time-complexity",level:2}];function o(e){const s={admonition:"admonition",annotation:"annotation",code:"code",em:"em",h2:"h2",li:"li",math:"math",mi:"mi",mn:"mn",mo:"mo",mrow:"mrow",mstyle:"mstyle",msup:"msup",mtable:"mtable",mtd:"mtd",mtext:"mtext",mtr:"mtr",ol:"ol",p:"p",pre:"pre",semantics:"semantics",span:"span",strong:"strong",...(0,t.a)(),...e.components};return(0,a.jsxs)(a.Fragment,{children:[(0,a.jsx)(s.p,{children:"Our na\xefve solution consists of trying out all the possible slides and finding\nthe one with maximum sum."}),"\n",(0,a.jsx)(s.pre,{children:(0,a.jsx)(s.code,{className:"language-java",children:"public static int longestSlideDown(int[][] pyramid, int row, int col) {\n    if (row >= pyramid.length || col < 0 || col >= pyramid[row].length) {\n        // BASE: We have gotten out of bounds, there's no reasonable value to\n        // return, so we just return the \u2039MIN_VALUE\u203a to ensure that it cannot\n        // be maximum.\n        return Integer.MIN_VALUE;\n    }\n\n    if (row == pyramid.length - 1) {\n        // BASE: Bottom of the pyramid, we just return the value, there's\n        // nowhere to slide anymore.\n        return pyramid[row][col];\n    }\n\n    // Otherwise we account for the current position and return maximum of the\n    // available \u201cslides\u201d.\n    return pyramid[row][col] + Math.max(\n            longestSlideDown(pyramid, row + 1, col),\n            longestSlideDown(pyramid, row + 1, col + 1));\n}\n\npublic static int longestSlideDown(int[][] pyramid) {\n    // We start the slide in the top cell of the pyramid.\n    return longestSlideDown(pyramid, 0, 0);\n}\n"})}),"\n",(0,a.jsx)(s.p,{children:"As you can see, we have 2 overloads:"}),"\n",(0,a.jsx)(s.pre,{children:(0,a.jsx)(s.code,{className:"language-java",children:"int longestSlideDown(int[][] pyramid);\nint longestSlideDown(int[][] pyramid, int row, int col);\n"})}),"\n",(0,a.jsxs)(s.p,{children:["First one is used as a ",(0,a.jsx)(s.em,{children:"public interface"})," to the solution, you just pass in the\npyramid itself. Second one is the recursive \u201calgorithm\u201d that finds the slide\ndown."]}),"\n",(0,a.jsxs)(s.p,{children:["It is a relatively simple solution\u2026 There's nothing to do at the bottom of the\npyramid, so we just return the value in the ",(0,a.jsx)(s.em,{children:"cell"}),". Otherwise we add it and try\nto slide down the available cells below the current row."]}),"\n",(0,a.jsx)(s.h2,{id:"time-complexity",children:"Time complexity"}),"\n",(0,a.jsx)(s.p,{children:"If you get the source code and run it yourself, it runs rather fine\u2026 I hope you\nare wondering about the time complexity of the proposed solution and, since it\nreally is a na\xefve solution, the time complexity is pretty bad. Let's find the\nworst case scenario."}),"\n",(0,a.jsx)(s.p,{children:"Let's start with the first overload:"}),"\n",(0,a.jsx)(s.pre,{children:(0,a.jsx)(s.code,{className:"language-java",children:"public static int longestSlideDown(int[][] pyramid) {\n    return longestSlideDown(pyramid, 0, 0);\n}\n"})}),"\n",(0,a.jsxs)(s.p,{children:["There's not much to do here, so we can safely say that the time complexity of\nthis function is bounded by ",(0,a.jsxs)(s.span,{className:"katex",children:[(0,a.jsx)(s.span,{className:"katex-mathml",children:(0,a.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,a.jsxs)(s.semantics,{children:[(0,a.jsxs)(s.mrow,{children:[(0,a.jsx)(s.mi,{children:"T"}),(0,a.jsx)(s.mo,{stretchy:"false",children:"("}),(0,a.jsx)(s.mi,{children:"n"}),(0,a.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,a.jsx)(s.annotation,{encoding:"application/x-tex",children:"T(n)"})]})})}),(0,a.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,a.jsxs)(s.span,{className:"base",children:[(0,a.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,a.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"T"}),(0,a.jsx)(s.span,{className:"mopen",children:"("}),(0,a.jsx)(s.span,{className:"mord mathnormal",children:"n"}),(0,a.jsx)(s.span,{className:"mclose",children:")"})]})})]}),", where ",(0,a.jsxs)(s.span,{className:"katex",children:[(0,a.jsx)(s.span,{className:"katex-mathml",children:(0,a.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,a.jsxs)(s.semantics,{children:[(0,a.jsx)(s.mrow,{children:(0,a.jsx)(s.mi,{children:"T"})}),(0,a.jsx)(s.annotation,{encoding:"application/x-tex",children:"T"})]})})}),(0,a.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,a.jsxs)(s.span,{className:"base",children:[(0,a.jsx)(s.span,{className:"strut",style:{height:"0.6833em"}}),(0,a.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"T"})]})})]})," is our second overload. This\ndoesn't tell us anything, so let's move on to the second overload where we are\ngoing to define the ",(0,a.jsxs)(s.span,{className:"katex",children:[(0,a.jsx)(s.span,{className:"katex-mathml",children:(0,a.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,a.jsxs)(s.semantics,{children:[(0,a.jsxs)(s.mrow,{children:[(0,a.jsx)(s.mi,{children:"T"}),(0,a.jsx)(s.mo,{stretchy:"false",children:"("}),(0,a.jsx)(s.mi,{children:"n"}),(0,a.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,a.jsx)(s.annotation,{encoding:"application/x-tex",children:"T(n)"})]})})}),(0,a.jsx)(s.span,{className:"katex-html","aria-hidden":"true",children:(0,a.jsxs)(s.span,{className:"base",children:[(0,a.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,a.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"T"}),(0,a.jsx)(s.span,{className:"mopen",children:"("}),(0,a.jsx)(s.span,{className:"mord mathnormal",children:"n"}),(0,a.jsx)(s.span,{className:"mclose",children:")"})]})})]})," function."]}),"\n",(0,a.jsx)(s.pre,{children:(0,a.jsx)(s.code,{className:"language-java",children:"public static int longestSlideDown(int[][] pyramid, int row, int col) {\n    if (row >= pyramid.length || col < 0 || col >= pyramid[row].length) {\n        // BASE: We have gotten out of bounds, there's no reasonable value to\n        // return, so we just return the \u2039MIN_VALUE\u203a to ensure that it cannot\n        // be maximum.\n        return Integer.MIN_VALUE;\n    }\n\n    if (row == pyramid.length - 1) {\n        // BASE: Bottom of the pyramid, we just return the value, there's\n        // nowhere to slide anymore.\n        return pyramid[row][col];\n    }\n\n    // Otherwise we account for the current position and return maximum of the\n    // available \u201cslides\u201d.\n    return pyramid[row][col] + Math.max(\n            longestSlideDown(pyramid, row + 1, col),\n            longestSlideDown(pyramid, row + 1, col + 1));\n}\n"})}),"\n",(0,a.jsxs)(s.p,{children:["Fun fact is that the whole \u201calgorithm\u201d consists of just 2 ",(0,a.jsx)(s.code,{children:"return"})," statements\nand nothing else. Let's dissect them!"]}),"\n",(0,a.jsxs)(s.p,{children:["First ",(0,a.jsx)(s.code,{children:"return"})," statement is the base case, so it has a constant time complexity."]}),"\n",(0,a.jsxs)(s.p,{children:["Second one a bit tricky. We add two numbers together, which we'll consider as\nconstant, but for the right part of the expression we take maximum from the left\nand right paths. OK\u2026 So what happens? We evaluate the ",(0,a.jsx)(s.code,{children:"longestSlideDown"})," while\nchoosing the under and right both. They are separate computations though, so we\nare branching from each call of ",(0,a.jsx)(s.code,{children:"longestSlideDown"}),", unless it's a base case."]}),"\n",(0,a.jsx)(s.p,{children:"What does that mean for us then? We basically get"}),"\n",(0,a.jsx)(s.span,{className:"katex-display",children:(0,a.jsxs)(s.span,{className:"katex",children:[(0,a.jsx)(s.span,{className:"katex-mathml",children:(0,a.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,a.jsxs)(s.semantics,{children:[(0,a.jsxs)(s.mrow,{children:[(0,a.jsx)(s.mi,{children:"T"}),(0,a.jsx)(s.mo,{stretchy:"false",children:"("}),(0,a.jsx)(s.mi,{children:"y"}),(0,a.jsx)(s.mo,{stretchy:"false",children:")"}),(0,a.jsx)(s.mo,{children:"="}),(0,a.jsxs)(s.mrow,{children:[(0,a.jsx)(s.mo,{fence:"true",children:"{"}),(0,a.jsxs)(s.mtable,{rowspacing:"0.36em",columnalign:"left 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If you sum it up, you'll get:"}),"\n",(0,a.jsx)(s.span,{className:"katex-display",children:(0,a.jsxs)(s.span,{className:"katex",children:[(0,a.jsx)(s.span,{className:"katex-mathml",children:(0,a.jsx)(s.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,a.jsxs)(s.semantics,{children:[(0,a.jsxs)(s.mrow,{children:[(0,a.jsx)(s.mi,{children:"T"}),(0,a.jsx)(s.mo,{stretchy:"false",children:"("}),(0,a.jsx)(s.mi,{children:"r"}),(0,a.jsx)(s.mi,{children:"o"}),(0,a.jsx)(s.mi,{children:"w"}),(0,a.jsx)(s.mi,{children:"s"}),(0,a.jsx)(s.mo,{stretchy:"false",children:")"}),(0,a.jsx)(s.mo,{children:"\u2208"}),(0,a.jsx)(s.mi,{mathvariant:"script",children:"O"}),(0,a.jsx)(s.mo,{stretchy:"false",children:"("}),(0,a.jsxs)(s.msup,{children:[(0,a.jsx)(s.mn,{children:"2"}),(0,a.jsxs)(s.mrow,{children:[(0,a.jsx)(s.mi,{children:"r"}),(0,a.jsx)(s.mi,{children:"o"}),(0,a.jsx)(s.mi,{children:"w"}),(0,a.jsx)(s.mi,{children:"s"})]})]}),(0,a.jsx)(s.mo,{stretchy:"false",children:")"})]}),(0,a.jsx)(s.annotation,{encoding:"application/x-tex",children:"T(rows) \\in \\mathcal{O}(2^{rows})"})]})})}),(0,a.jsxs)(s.span,{className:"katex-html","aria-hidden":"true",children:[(0,a.jsxs)(s.span,{className:"base",children:[(0,a.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,a.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"T"}),(0,a.jsx)(s.span,{className:"mopen",children:"("}),(0,a.jsx)(s.span,{className:"mord mathnormal",children:"ro"}),(0,a.jsx)(s.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"}),(0,a.jsx)(s.span,{className:"mord mathnormal",children:"s"}),(0,a.jsx)(s.span,{className:"mclose",children:")"}),(0,a.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,a.jsx)(s.span,{className:"mrel",children:"\u2208"}),(0,a.jsx)(s.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,a.jsxs)(s.span,{className:"base",children:[(0,a.jsx)(s.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,a.jsx)(s.span,{className:"mord mathcal",style:{marginRight:"0.02778em"},children:"O"}),(0,a.jsx)(s.span,{className:"mopen",children:"("}),(0,a.jsxs)(s.span,{className:"mord",children:[(0,a.jsx)(s.span,{className:"mord",children:"2"}),(0,a.jsx)(s.span,{className:"msupsub",children:(0,a.jsx)(s.span,{className:"vlist-t",children:(0,a.jsx)(s.span,{className:"vlist-r",children:(0,a.jsx)(s.span,{className:"vlist",style:{height:"0.7144em"},children:(0,a.jsxs)(s.span,{style:{top:"-3.113em",marginRight:"0.05em"},children:[(0,a.jsx)(s.span,{className:"pstrut",style:{height:"2.7em"}}),(0,a.jsx)(s.span,{className:"sizing reset-size6 size3 mtight",children:(0,a.jsxs)(s.span,{className:"mord mtight",children:[(0,a.jsx)(s.span,{className:"mord mathnormal mtight",children:"ro"}),(0,a.jsx)(s.span,{className:"mord mathnormal mtight",style:{marginRight:"0.02691em"},children:"w"}),(0,a.jsx)(s.span,{className:"mord mathnormal mtight",children:"s"})]})})]})})})})})]}),(0,a.jsx)(s.span,{className:"mclose",children:")"})]})]})]})}),"\n",(0,a.jsx)(s.p,{children:"If you wonder why, I'll try to describe it intuitively:"}),"\n",(0,a.jsxs)(s.ol,{children:["\n",(0,a.jsxs)(s.li,{children:["In each call to ",(0,a.jsx)(s.code,{children:"longestSlideDown"})," we do some work in constant time,\nregardless of being in the base case. Those are the ",(0,a.jsx)(s.code,{children:"1"}),"s in both cases."]}),"\n",(0,a.jsxs)(s.li,{children:["If we are not in the base case, we move one row down ",(0,a.jsx)(s.strong,{children:"twice"}),". That's how we\nobtained ",(0,a.jsx)(s.code,{children:"2 *"})," and ",(0,a.jsx)(s.code,{children:"y + 1"})," in the ",(0,a.jsx)(s.em,{children:"otherwise"})," case."]}),"\n",(0,a.jsxs)(s.li,{children:["We move row-by-row, so we move down ",(0,a.jsx)(s.code,{children:"y"}),"-times and each call splits to two\nsubtrees."]}),"\n",(0,a.jsxs)(s.li,{children:["Overall, if we were to represent the calls as a tree, we would get a full\nbinary tree of height ",(0,a.jsx)(s.code,{children:"y"}),", in each node we do some work in constant time,\ntherefore we can just sum the ones."]}),"\n"]}),"\n",(0,a.jsx)(s.admonition,{type:"warning",children:(0,a.jsx)(s.p,{children:"It would've been more complicated to get an exact result. In the equation above\nwe are assuming that the width of the pyramid is bound by the height."})}),"\n",(0,a.jsxs)(s.p,{children:["Hopefully we can agree that this is not the best we can do. ","\ud83d\ude09"]})]})}function h(e={}){const{wrapper:s}={...(0,t.a)(),...e.components};return s?(0,a.jsx)(s,{...e,children:(0,a.jsx)(o,{...e})}):o(e)}},11151:(e,s,n)=>{n.d(s,{Z:()=>r,a:()=>l});var a=n(67294);const t={},i=a.createContext(t);function l(e){const s=a.useContext(i);return a.useMemo((function(){return"function"==typeof e?e(s):{...s,...e}}),[s,e])}function r(e){let s;return s=e.disableParentContext?"function"==typeof e.components?e.components(t):e.components||t:l(e.components),a.createElement(i.Provider,{value:s},e.children)}}}]);