widest path problem widest path problem problem finding path designated weighted weight minimum weight edge path widest path problem known shortest path problem maximum capacity path problem possible adapt shortest path widest paths modifying use distance instead path length cases faster possible instance represents connections internet weight edge represents connection widest path problem problem finding end end path internet maximum possible smallest edge weight path known capacity path applications network routing widest path problem important component method deciding winner election applied digital analysis maximum flows closely related problem path problem asks path maximum weight edges applications include transportation planning widest path problem transformed path problem vice versa reversing sense weight comparisons performed replacing edge weight widest path found path maximum spanning tree path found path minimum spanning tree directed straightforward finding widest path weight minimum weight edge known simply smaller edges search path remaining edges breadth search depth search based test exists linear time finding widest path use maximum spanning tree main idea apply linear time path finding median edge weight smaller edges contract larger edges according path exist resulting smaller use shortest paths order form composite aerial photographs combine multiple images overlapping areas widest path problem applies images transformed common coordinate system remaining task select curve passes region overlap divides images copied images copied image unlike methods average images produces valid photographic image region photographed weight edges grid estimate visually apparent edge find shortest path weights path conventional shortest path causes system find difficult points allowing trade greater visibility image lesser visibility solution path problem opposite corners grid find weak distance chains grid represents pair line segments chain weight edge represents distance needed pass pair segments edge weights positive distances pairs points maximum edge weights paths form conversely finite space comes distances way data structure constructed minimum spanning tree allows distance pair constant time lowest common ancestor tree root tree represents heaviest minimum spanning tree edge children root trees constructed minimum spanning tree formed removing heaviest edge leaves tree represent input distance equals weight tree lowest common ancestor minimum spanning tree edges tree constructed linear time directed directed maximum spanning tree solution instead different known choice use depends start destination path fixed paths start destination found simultaneously pairs pairs widest path problem applications method choosing winner elections voters rank candidates preference order method complete directed represent candidates connected edge edge directed winner loser contest candidates connects labeled margin victory contest method widest paths pairs winner candidate wider paths opponent vice versa results election method consistent method candidate wins contests automatically wins election generally allows winner selected situations method fails method organizations including widest path pairs dense directed ones arise voting application fastest known approach takes time fast matrix best known matrix time bound instead reference implementation method uses modified version simpler takes time sparse efficient repeatedly apply single source widest path single source edges weights modified version designated start linear time key idea conventional version sequence distances order considered sequence edge weights priority queue implemented bucket queue array numbers number edges array cell contains distance weight edge position order method allows widest path problem solved quickly instance edge weights represented time bounds list apply problem single source single destination suggest service vehicles emergency vehicles use paths returning service base application time return important response time service occurs vehicle process returning path weight edge maximum travel time point edge possible service plan route maximum possible delay service arrival responding vehicle lee use paths model dominant reaction chains networks model weight edge free energy reaction represented edge application widest paths arises ford maximum flow problem repeatedly flow maximum capacity path network flow leads small bound log number needed find maximum flow edge capacities assumed analysis depend finding path exact maximum capacity path capacity constant factor maximum combining idea shortest path method leads maximum flow running time log possible find maximum capacity paths paths single source single destination efficiently models allow comparisons input edge weights maintains set edges known contain edge optimal path initially set edges iteration splits ordered sequence approximately equal size number partition chosen way split points found repeated median finding time edge index containing edge uses modified based results determine linear time contains edge weight replaces determined contain weight starts iteration new set number split increases step number proportional function log total time log model edge weight machine use repeated replaced list splitting technique allowing split smaller sets single step leading linear overall time bound point sets variant path problem considered sets points plane problem path problem solved efficiently finding minimum spanning tree path tree path problem complicated path desired hop length paths hop length approximately total length path solution geometric number theory moat problem asks paths prime numbers bounded length exist constant pair points infinite point set defined path edge length
Widest path problem