blog/assets/js/2a09abcd.ae41ee68.js
github-actions[bot] 70bdf5ed11 deploy: d91860e0f7
2023-09-07 18:30:53 +00:00

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"use strict";(self.webpackChunkfi=self.webpackChunkfi||[]).push([[1246],{3905:(I,a,i)=>{i.d(a,{Zo:()=>g,kt:()=>b});var t=i(7294);function m(I,a,i){return a in I?Object.defineProperty(I,a,{value:i,enumerable:!0,configurable:!0,writable:!0}):I[a]=i,I}function n(I,a){var i=Object.keys(I);if(Object.getOwnPropertySymbols){var t=Object.getOwnPropertySymbols(I);a&&(t=t.filter((function(a){return Object.getOwnPropertyDescriptor(I,a).enumerable}))),i.push.apply(i,t)}return i}function e(I){for(var a=1;a<arguments.length;a++){var i=null!=arguments[a]?arguments[a]:{};a%2?n(Object(i),!0).forEach((function(a){m(I,a,i[a])})):Object.getOwnPropertyDescriptors?Object.defineProperties(I,Object.getOwnPropertyDescriptors(i)):n(Object(i)).forEach((function(a){Object.defineProperty(I,a,Object.getOwnPropertyDescriptor(i,a))}))}return I}function l(I,a){if(null==I)return{};var i,t,m=function(I,a){if(null==I)return{};var i,t,m={},n=Object.keys(I);for(t=0;t<n.length;t++)i=n[t],a.indexOf(i)>=0||(m[i]=I[i]);return 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contradiction",id:"proof-by-contradiction",level:2}],g={toc:s},Z="wrapper";function c(I){let{components:a,...n}=I;return(0,m.kt)(Z,(0,t.Z)({},g,n,{components:a,mdxType:"MDXLayout"}),(0,m.kt)("h2",{id:"introduction"},"Introduction"),(0,m.kt)("p",null,"As we have talked on the seminar, if we construct from some vertex ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"u")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"u")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"u")))))," BFS tree on an undirected graph, we can obtain:"),(0,m.kt)("ul",null,(0,m.kt)("li",{parentName:"ul"},"lower bound of length of the shortest path between 2 vertices from the ",(0,m.kt)("em",{parentName:"li"},"height difference")),(0,m.kt)("li",{parentName:"ul"},"upper bound of length of the shortest path between 2 vertices from the ",(0,m.kt)("em",{parentName:"li"},"path through the root"))),(0,m.kt)("h2",{id:"lower-bound"},"Lower bound"),(0,m.kt)("p",null,"Consider the following graph:"),(0,m.kt)("p",null,(0,m.kt)("img",{src:i(6547).Z+"#gh-light-mode-only",width:"252",height:"539"}),"\n",(0,m.kt)("img",{src:i(1338).Z+"#gh-dark-mode-only",width:"252",height:"539"})),(0,m.kt)("p",null,"We run BFS from the vertex ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"a")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"a")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"a")))))," and obtain the following BFS tree:"),(0,m.kt)("p",null,(0,m.kt)("img",{src:i(6069).Z+"#gh-light-mode-only",width:"275",height:"347"}),"\n",(0,m.kt)("img",{src:i(6325).Z+"#gh-dark-mode-only",width:"275",height:"347"})),(0,m.kt)("p",null,"Let's consider pair of vertices ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"e")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"e")))))," and ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"h")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"h")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"))))),". For them we can safely lay, from the BFS tree, following properties:"),(0,m.kt)("ul",null,(0,m.kt)("li",{parentName:"ul"},"lower bound: ",(0,m.kt)("span",{parentName:"li",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mn",{parentName:"mrow"},"2")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"2")))))),(0,m.kt)("li",{parentName:"ul"},"upper bound: ",(0,m.kt)("span",{parentName:"li",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mn",{parentName:"mrow"},"4")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"4")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"4"))))))),(0,m.kt)("p",null,"By having a look at the graph we started from, we can see that we have a path \u2039",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"e"),(0,m.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,m.kt)("mi",{parentName:"mrow"},"j"),(0,m.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,m.kt)("mi",{parentName:"mrow"},"h")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e, j, h")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,m.kt)("span",{parentName:"span",className:"mpunct"},","),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05724em"}},"j"),(0,m.kt)("span",{parentName:"span",className:"mpunct"},","),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"))))),"\u203a that has a length 2. Apart from that we can also notice there is another path from ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"e")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"e")))))," to ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"h")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"h")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"h")))))," and that is \u2039",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"e"),(0,m.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,m.kt)("mi",{parentName:"mrow"},"a"),(0,m.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,m.kt)("mi",{parentName:"mrow"},"c"),(0,m.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,m.kt)("mi",{parentName:"mrow"},"i"),(0,m.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,m.kt)("mi",{parentName:"mrow"},"d"),(0,m.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,m.kt)("mi",{parentName:"mrow"},"h")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e, a, c, i, d, h")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,m.kt)("span",{parentName:"span",className:"mpunct"},","),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"a"),(0,m.kt)("span",{parentName:"span",className:"mpunct"},","),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"c"),(0,m.kt)("span",{parentName:"span",className:"mpunct"},","),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"i"),(0,m.kt)("span",{parentName:"span",className:"mpunct"},","),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"d"),(0,m.kt)("span",{parentName:"span",className:"mpunct"},","),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"))))),"\u203a. And that path has a length of ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mn",{parentName:"mrow"},"5")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"5")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"5"))))),". Doesn't this break our statements at the beginning? (",(0,m.kt)("em",{parentName:"p"},"I'm leaving that as an exercise ;)"),")"),(0,m.kt)("h2",{id:"proof-by-contradiction"},"Proof by contradiction"),(0,m.kt)("p",null,"Let's keep the same graph, but break the lower bound, i.e. I have gotten a lower bound ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mn",{parentName:"mrow"},"2")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"2"))))),", but \u201cthere must be a shorter path\u201d! ;)"),(0,m.kt)("p",null,"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,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mn",{parentName:"mrow"},"2")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"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,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mn",{parentName:"mrow"},"2")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"2"))))),", which means the only shorter path we can construct has ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mn",{parentName:"mrow"},"1")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"1")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"1")))))," edge and that is \u2039",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"e"),(0,m.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,m.kt)("mi",{parentName:"mrow"},"h")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e, h")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,m.kt)("span",{parentName:"span",className:"mpunct"},","),(0,m.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"))))),"\u203a (no intermediary vertices). Let's do this!"),(0,m.kt)("p",null,(0,m.kt)("img",{src:i(9907).Z+"#gh-light-mode-only",width:"252",height:"539"}),"\n",(0,m.kt)("img",{src:i(2830).Z+"#gh-dark-mode-only",width:"252",height:"539"})),(0,m.kt)("p",null,"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."),(0,m.kt)("admonition",{type:"tip"},(0,m.kt)("p",{parentName:"admonition"},"Do we need to run BFS after ",(0,m.kt)("strong",{parentName:"p"},"every")," change?"),(0,m.kt)("p",{parentName:"admonition"},"\xadI am leaving that as an exercise ;)")),(0,m.kt)("p",null,(0,m.kt)("img",{src:i(4570).Z+"#gh-light-mode-only",width:"371",height:"347"}),"\n",(0,m.kt)("img",{src:i(1722).Z+"#gh-dark-mode-only",width:"371",height:"347"})),(0,m.kt)("p",null,"Oops, we have gotten a new BFS tree, that has a height difference of 1."),(0,m.kt)("admonition",{type:"tip"},(0,m.kt)("p",{parentName:"admonition"},"Try to think about a way this can be generalized for shortening of minimal length 3 to minimal length 2 ;)")))}c.isMDXComponent=!0},1338:(I,a,i)=>{i.d(a,{Z:()=>t});const 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