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"use strict";(self.webpackChunkfi=self.webpackChunkfi||[]).push([[9414],{46617:(I,i,l)=>{l.r(i),l.d(i,{assets:()=>j,contentTitle:()=>g,default:()=>d,frontMatter:()=>a,metadata:()=>c,toc:()=>Z});var s=l(85893),m=l(11151);const a={id:"bfs-tree",title:"Distance boundaries from BFS tree on undirected graphs",description:"Short explanation of distance boundaries deduced from a BFS tree.\n",tags:["graphs","bfs"],last_update:{date:new Date("2022-04-30T00:00:00.000Z")}},g=void 0,c={id:"graphs/bfs-tree",title:"Distance boundaries from BFS tree on undirected graphs",description:"Short explanation of distance boundaries deduced from a BFS tree.\n",source:"@site/algorithms/10-graphs/2022-04-30-bfs-tree.md",sourceDirName:"10-graphs",slug:"/graphs/bfs-tree",permalink:"/algorithms/graphs/bfs-tree",draft:!1,unlisted:!1,editUrl:"https://github.com/mfocko/blog/tree/main/algorithms/10-graphs/2022-04-30-bfs-tree.md",tags:[{label:"graphs",permalink:"/algorithms/tags/graphs"},{label:"bfs",permalink:"/algorithms/tags/bfs"}],version:"current",lastUpdatedAt:1651276800,formattedLastUpdatedAt:"Apr 30, 2022",frontMatter:{id:"bfs-tree",title:"Distance boundaries from BFS tree on undirected graphs",description:"Short explanation of distance boundaries deduced from a BFS tree.\n",tags:["graphs","bfs"],last_update:{date:"2022-04-30T00:00:00.000Z"}},sidebar:"autogeneratedBar",previous:{title:"Iterative algorithms via iterators",permalink:"/algorithms/graphs/iterative-and-iterators"},next:{title:"Paths in Graphs",permalink:"/algorithms/category/paths-in-graphs"}},j={},Z=[{value:"Introduction",id:"introduction",level:2},{value:"Lower bound",id:"lower-bound",level:2},{value:"Proof by contradiction",id:"proof-by-contradiction",level:2}];function n(I){const i={admonition:"admonition",annotation:"annotation",em:"em",h2:"h2",img:"img",li:"li",math:"math",mi:"mi",mn:"mn",mo:"mo",mrow:"mrow",p:"p",semantics:"semantics",span:"span",strong:"strong",ul:"ul",...(0,m.a)(),...I.components};return(0,s.jsxs)(s.Fragment,{children:[(0,s.jsx)(i.h2,{id:"introduction",children:"Introduction"}),"\n",(0,s.jsxs)(i.p,{children:["As we have talked on the seminar, if we construct from some vertex ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mi,{children:"u"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"u"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.4306em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"u"})]})})]})," BFS tree on an undirected graph, we can obtain:"]}),"\n",(0,s.jsxs)(i.ul,{children:["\n",(0,s.jsxs)(i.li,{children:["lower bound of length of the shortest path between 2 vertices from the ",(0,s.jsx)(i.em,{children:"height difference"})]}),"\n",(0,s.jsxs)(i.li,{children:["upper bound of length of the shortest path between 2 vertices from the ",(0,s.jsx)(i.em,{children:"path through the root"})]}),"\n"]}),"\n",(0,s.jsx)(i.h2,{id:"lower-bound",children:"Lower bound"}),"\n",(0,s.jsx)(i.p,{children:"Consider the following graph:"}),"\n",(0,s.jsxs)(i.p,{children:[(0,s.jsx)(i.img,{src:l(79968).Z+"#gh-light-mode-only",width:"252",height:"539"}),"\n",(0,s.jsx)(i.img,{src:l(21949).Z+"#gh-dark-mode-only",width:"252",height:"539"})]}),"\n",(0,s.jsxs)(i.p,{children:["We run BFS from the vertex ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mi,{children:"a"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"a"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.4306em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"a"})]})})]})," and obtain the following BFS tree:"]}),"\n",(0,s.jsxs)(i.p,{children:[(0,s.jsx)(i.img,{src:l(62843).Z+"#gh-light-mode-only",width:"275",height:"347"}),"\n",(0,s.jsx)(i.img,{src:l(43770).Z+"#gh-dark-mode-only",width:"275",height:"347"})]}),"\n",(0,s.jsxs)(i.p,{children:["Let's consider pair of vertices ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mi,{children:"e"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"e"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.4306em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"e"})]})})]})," and ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mi,{children:"h"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"h"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6944em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"h"})]})})]}),". For them we can safely lay, from the BFS tree, following properties:"]}),"\n",(0,s.jsxs)(i.ul,{children:["\n",(0,s.jsxs)(i.li,{children:["lower bound: ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mn,{children:"2"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"2"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6444em"}}),(0,s.jsx)(i.span,{className:"mord",children:"2"})]})})]})]}),"\n",(0,s.jsxs)(i.li,{children:["upper bound: ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mn,{children:"4"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"4"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6444em"}}),(0,s.jsx)(i.span,{className:"mord",children:"4"})]})})]})]}),"\n"]}),"\n",(0,s.jsxs)(i.p,{children:["By having a look at the graph we started from, we can see that we have a path \u2039",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsxs)(i.mrow,{children:[(0,s.jsx)(i.mi,{children:"e"}),(0,s.jsx)(i.mo,{separator:"true",children:","}),(0,s.jsx)(i.mi,{children:"j"}),(0,s.jsx)(i.mo,{separator:"true",children:","}),(0,s.jsx)(i.mi,{children:"h"})]}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"e, j, h"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"e"}),(0,s.jsx)(i.span,{className:"mpunct",children:","}),(0,s.jsx)(i.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",style:{marginRight:"0.05724em"},children:"j"}),(0,s.jsx)(i.span,{className:"mpunct",children:","}),(0,s.jsx)(i.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"h"})]})})]}),"\u203a that has a length 2. Apart from that we can also notice there is another path from ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mi,{children:"e"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"e"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.4306em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"e"})]})})]})," to ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mi,{children:"h"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"h"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6944em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"h"})]})})]})," and that is \u2039",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsxs)(i.mrow,{children:[(0,s.jsx)(i.mi,{children:"e"}),(0,s.jsx)(i.mo,{separator:"true",children:","}),(0,s.jsx)(i.mi,{children:"a"}),(0,s.jsx)(i.mo,{separator:"true",children:","}),(0,s.jsx)(i.mi,{children:"c"}),(0,s.jsx)(i.mo,{separator:"true",children:","}),(0,s.jsx)(i.mi,{children:"i"}),(0,s.jsx)(i.mo,{separator:"true",children:","}),(0,s.jsx)(i.mi,{children:"d"}),(0,s.jsx)(i.mo,{separator:"true",children:","}),(0,s.jsx)(i.mi,{children:"h"})]}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"e, a, c, i, d, h"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"e"}),(0,s.jsx)(i.span,{className:"mpunct",children:","}),(0,s.jsx)(i.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"a"}),(0,s.jsx)(i.span,{className:"mpunct",children:","}),(0,s.jsx)(i.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"c"}),(0,s.jsx)(i.span,{className:"mpunct",children:","}),(0,s.jsx)(i.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"i"}),(0,s.jsx)(i.span,{className:"mpunct",children:","}),(0,s.jsx)(i.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"d"}),(0,s.jsx)(i.span,{className:"mpunct",children:","}),(0,s.jsx)(i.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"h"})]})})]}),"\u203a. And that path has a length of ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mn,{children:"5"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"5"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6444em"}}),(0,s.jsx)(i.span,{className:"mord",children:"5"})]})})]}),". Doesn't this break our statements at the beginning? (",(0,s.jsx)(i.em,{children:"I'm leaving that as an exercise ;)"}),")"]}),"\n",(0,s.jsx)(i.h2,{id:"proof-by-contradiction",children:"Proof by contradiction"}),"\n",(0,s.jsxs)(i.p,{children:["Let's keep the same graph, but break the lower bound, i.e. I have gotten a lower bound ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mn,{children:"2"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"2"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6444em"}}),(0,s.jsx)(i.span,{className:"mord",children:"2"})]})})]}),", but \u201cthere must be a shorter path\u201d! ;)"]}),"\n",(0,s.jsxs)(i.p,{children:["Now the more important question, is there a shorter path in that graph? The answer is no, there's no shorter path than the one with length ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mn,{children:"2"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"2"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6444em"}}),(0,s.jsx)(i.span,{className:"mord",children:"2"})]})})]}),". So what can we do about it? We'll add an edge to have a shorter path. Now we have gotten a lower bound of ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mn,{children:"2"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"2"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6444em"}}),(0,s.jsx)(i.span,{className:"mord",children:"2"})]})})]}),", which means the only shorter path we can construct has ",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsx)(i.mrow,{children:(0,s.jsx)(i.mn,{children:"1"})}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"1"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.6444em"}}),(0,s.jsx)(i.span,{className:"mord",children:"1"})]})})]})," edge and that is \u2039",(0,s.jsxs)(i.span,{className:"katex",children:[(0,s.jsx)(i.span,{className:"katex-mathml",children:(0,s.jsx)(i.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,s.jsxs)(i.semantics,{children:[(0,s.jsxs)(i.mrow,{children:[(0,s.jsx)(i.mi,{children:"e"}),(0,s.jsx)(i.mo,{separator:"true",children:","}),(0,s.jsx)(i.mi,{children:"h"})]}),(0,s.jsx)(i.annotation,{encoding:"application/x-tex",children:"e, h"})]})})}),(0,s.jsx)(i.span,{className:"katex-html","aria-hidden":"true",children:(0,s.jsxs)(i.span,{className:"base",children:[(0,s.jsx)(i.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"e"}),(0,s.jsx)(i.span,{className:"mpunct",children:","}),(0,s.jsx)(i.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.jsx)(i.span,{className:"mord mathnormal",children:"h"})]})})]}),"\u203a (no intermediary vertices). Let's do this!"]}),"\n",(0,s.jsxs)(i.p,{children:[(0,s.jsx)(i.img,{src:l(40050).Z+"#gh-light-mode-only",width:"252",height:"539"}),"\n",(0,s.jsx)(i.img,{src:l(7619).Z+"#gh-dark-mode-only",width:"252",height:"539"})]}),"\n",(0,s.jsx)(i.p,{children:"Okay, so we have a graph that breaks the rule we have laid. However, we need to run BFS to obtain the new BFS tree, since we have changed the graph."}),"\n",(0,s.jsxs)(i.admonition,{type:"tip",children:[(0,s.jsxs)(i.p,{children:["Do we need to run BFS after ",(0,s.jsx)(i.strong,{children:"every"})," change?"]}),(0,s.jsx)(i.p,{children:"\xadI am leaving that as an exercise ;)"})]}),"\n",(0,s.jsxs)(i.p,{children:[(0,s.jsx)(i.img,{src:l(9248).Z+"#gh-light-mode-only",width:"371",height:"347"}),"\n",(0,s.jsx)(i.img,{src:l(31350).Z+"#gh-dark-mode-only",width:"371",height:"347"})]}),"\n",(0,s.jsx)(i.p,{children:"Oops, we have gotten a new BFS tree, that has a height difference of 1."}),"\n",(0,s.jsx)(i.admonition,{type:"tip",children:(0,s.jsx)(i.p,{children:"Try to think about a way this can be generalized for shortening of minimal length 3 to minimal length 2 ;)"})})]})}function d(I={}){const{wrapper:i}={...(0,m.a)(),...I.components};return i?(0,s.jsx)(i,{...I,children:(0,s.jsx)(n,{...I})}):n(I)}},21949:(I,i,l)=>{l.d(i,{Z:()=>s});const 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