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"use strict";(self.webpackChunkfi=self.webpackChunkfi||[]).push([[1246],{3905:(a,e,t)=>{t.d(e,{Zo:()=>l,kt:()=>k});var n=t(7294);function s(a,e,t){return e in a?Object.defineProperty(a,e,{value:t,enumerable:!0,configurable:!0,writable:!0}):a[e]=t,a}function m(a,e){var t=Object.keys(a);if(Object.getOwnPropertySymbols){var n=Object.getOwnPropertySymbols(a);e&&(n=n.filter((function(e){return Object.getOwnPropertyDescriptor(a,e).enumerable}))),t.push.apply(t,n)}return t}function r(a){for(var e=1;e<arguments.length;e++){var t=null!=arguments[e]?arguments[e]:{};e%2?m(Object(t),!0).forEach((function(e){s(a,e,t[e])})):Object.getOwnPropertyDescriptors?Object.defineProperties(a,Object.getOwnPropertyDescriptors(t)):m(Object(t)).forEach((function(e){Object.defineProperty(a,e,Object.getOwnPropertyDescriptor(t,e))}))}return a}function p(a,e){if(null==a)return{};var t,n,s=function(a,e){if(null==a)return{};var t,n,s={},m=Object.keys(a);for(n=0;n<m.length;n++)t=m[n],e.indexOf(t)>=0||(s[t]=a[t]);return 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contradiction",id:"proof-by-contradiction",level:2}],l={toc:o},h="wrapper";function c(a){let{components:e,...m}=a;return(0,s.kt)(h,(0,n.Z)({},l,m,{components:e,mdxType:"MDXLayout"}),(0,s.kt)("h2",{id:"introduction"},"Introduction"),(0,s.kt)("p",null,"As we have talked on the seminar, if we construct from some vertex ",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"u")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"u")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"u")))))," BFS tree on an undirected graph, we can obtain:"),(0,s.kt)("ul",null,(0,s.kt)("li",{parentName:"ul"},"lower bound of length of the shortest path between 2 vertices from the ",(0,s.kt)("em",{parentName:"li"},"height difference")),(0,s.kt)("li",{parentName:"ul"},"upper bound of length of the shortest path between 2 vertices from the ",(0,s.kt)("em",{parentName:"li"},"path through the root"))),(0,s.kt)("h2",{id:"lower-bound"},"Lower bound"),(0,s.kt)("p",null,"Consider the following graph:"),(0,s.kt)("p",null,(0,s.kt)("img",{src:t(491).Z+"#gh-light-mode-only",width:"252",height:"539"}),"\n",(0,s.kt)("img",{src:t(9257).Z+"#gh-dark-mode-only",width:"252",height:"539"})),(0,s.kt)("p",null,"We run BFS from the vertex ",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"a")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"a")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"a")))))," and obtain the following BFS tree:"),(0,s.kt)("p",null,(0,s.kt)("img",{src:t(1281).Z+"#gh-light-mode-only",width:"275",height:"347"}),"\n",(0,s.kt)("img",{src:t(820).Z+"#gh-dark-mode-only",width:"275",height:"347"})),(0,s.kt)("p",null,"Let's consider pair of vertices ",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"e")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"e")))))," and ",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"h")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"h")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"))))),". For them we can safely lay, from the BFS tree, following properties:"),(0,s.kt)("ul",null,(0,s.kt)("li",{parentName:"ul"},"lower bound: ",(0,s.kt)("span",{parentName:"li",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mn",{parentName:"mrow"},"2")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,s.kt)("span",{parentName:"span",className:"mord"},"2")))))),(0,s.kt)("li",{parentName:"ul"},"upper bound: ",(0,s.kt)("span",{parentName:"li",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mn",{parentName:"mrow"},"4")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"4")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,s.kt)("span",{parentName:"span",className:"mord"},"4"))))))),(0,s.kt)("p",null,"By having a look at the graph we started from, we can see that we have a path \u2039",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"e"),(0,s.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,s.kt)("mi",{parentName:"mrow"},"j"),(0,s.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,s.kt)("mi",{parentName:"mrow"},"h")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e, j, h")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,s.kt)("span",{parentName:"span",className:"mpunct"},","),(0,s.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05724em"}},"j"),(0,s.kt)("span",{parentName:"span",className:"mpunct"},","),(0,s.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.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,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"e")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"e")))))," to ",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"h")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"h")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"h")))))," and that is \u2039",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"e"),(0,s.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,s.kt)("mi",{parentName:"mrow"},"a"),(0,s.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,s.kt)("mi",{parentName:"mrow"},"c"),(0,s.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,s.kt)("mi",{parentName:"mrow"},"i"),(0,s.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,s.kt)("mi",{parentName:"mrow"},"d"),(0,s.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,s.kt)("mi",{parentName:"mrow"},"h")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e, a, c, i, d, h")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,s.kt)("span",{parentName:"span",className:"mpunct"},","),(0,s.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"a"),(0,s.kt)("span",{parentName:"span",className:"mpunct"},","),(0,s.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"c"),(0,s.kt)("span",{parentName:"span",className:"mpunct"},","),(0,s.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"i"),(0,s.kt)("span",{parentName:"span",className:"mpunct"},","),(0,s.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"d"),(0,s.kt)("span",{parentName:"span",className:"mpunct"},","),(0,s.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"))))),"\u203a. And that path has a length of ",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mn",{parentName:"mrow"},"5")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"5")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,s.kt)("span",{parentName:"span",className:"mord"},"5"))))),". Doesn't this break our statements at the beginning? (",(0,s.kt)("em",{parentName:"p"},"I'm leaving that as an exercise ;)"),")"),(0,s.kt)("h2",{id:"proof-by-contradiction"},"Proof by contradiction"),(0,s.kt)("p",null,"Let's keep the same graph, but break the lower bound, i.e. I have gotten a lower bound ",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mn",{parentName:"mrow"},"2")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,s.kt)("span",{parentName:"span",className:"mord"},"2"))))),", but \u201cthere must be a shorter path\u201d! ;)"),(0,s.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,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mn",{parentName:"mrow"},"2")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,s.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,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mn",{parentName:"mrow"},"2")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"2")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,s.kt)("span",{parentName:"span",className:"mord"},"2"))))),", which means the only shorter path we can construct has ",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mn",{parentName:"mrow"},"1")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"1")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,s.kt)("span",{parentName:"span",className:"mord"},"1")))))," edge and that is \u2039",(0,s.kt)("span",{parentName:"p",className:"math math-inline"},(0,s.kt)("span",{parentName:"span",className:"katex"},(0,s.kt)("span",{parentName:"span",className:"katex-mathml"},(0,s.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,s.kt)("semantics",{parentName:"math"},(0,s.kt)("mrow",{parentName:"semantics"},(0,s.kt)("mi",{parentName:"mrow"},"e"),(0,s.kt)("mo",{parentName:"mrow",separator:"true"},","),(0,s.kt)("mi",{parentName:"mrow"},"h")),(0,s.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"e, h")))),(0,s.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,s.kt)("span",{parentName:"span",className:"base"},(0,s.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"e"),(0,s.kt)("span",{parentName:"span",className:"mpunct"},","),(0,s.kt)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.1667em"}}),(0,s.kt)("span",{parentName:"span",className:"mord mathnormal"},"h"))))),"\u203a (no intermediary vertices). Let's do this!"),(0,s.kt)("p",null,(0,s.kt)("img",{src:t(340).Z+"#gh-light-mode-only",width:"252",height:"539"}),"\n",(0,s.kt)("img",{src:t(4110).Z+"#gh-dark-mode-only",width:"252",height:"539"})),(0,s.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,s.kt)("admonition",{type:"tip"},(0,s.kt)("p",{parentName:"admonition"},"Do we need to run BFS after ",(0,s.kt)("strong",{parentName:"p"},"every")," change?"),(0,s.kt)("p",{parentName:"admonition"},"\xadI am leaving that as an exercise ;)")),(0,s.kt)("p",null,(0,s.kt)("img",{src:t(6256).Z+"#gh-light-mode-only",width:"371",height:"347"}),"\n",(0,s.kt)("img",{src:t(8680).Z+"#gh-dark-mode-only",width:"371",height:"347"})),(0,s.kt)("p",null,"Oops, we have gotten a new BFS tree, that has a height difference of 1."),(0,s.kt)("admonition",{type:"tip"},(0,s.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},9257:(a,e,t)=>{t.d(e,{Z:()=>n});const n=t.p+"assets/images/bfs_graph_dark-f7a3a78eaf9de049469b4c64e0712867.png"},491:(a,e,t)=>{t.d(e,{Z:()=>n});const n=t.p+"assets/images/bfs_graph_light-6e21a942bccd92bcce6840da7c3cb056.png"},4110:(a,e,t)=>{t.d(e,{Z:()=>n});const n=t.p+"assets/images/bfs_graph_with_additional_edge_dark-c2da6c052b067785e877b4654a13f328.png"},340:(a,e,t)=>{t.d(e,{Z:()=>n});const n=t.p+"assets/images/bfs_graph_with_additional_edge_light-799673ba333298d16327abe67c90507e.png"},820:(a,e,t)=>{t.d(e,{Z:()=>n});const n=t.p+"assets/images/bfs_tree_dark-34f32262c6e4ffc14983b3ebf9a2f5a9.png"},1281:(a,e,t)=>{t.d(e,{Z:()=>n});const n=t.p+"assets/images/bfs_tree_light-61d6723c3c587d565b6280b8b8eca211.png"},8680:(a,e,t)=>{t.d(e,{Z:()=>n});const n=t.p+"assets/images/bfs_tree_with_additional_edge_dark-127aa5b00988d4569669a92f5d841dbf.png"},6256:(a,e,t)=>{t.d(e,{Z:()=>n});const n=t.p+"assets/images/bfs_tree_with_additional_edge_light-93df97e1f9f1883467248532809374d0.png"}}]);
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