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For any weighted graph G = ( V , E , w ) and integer k , define Gk to be the graph that results

For any weighted graph G =(V, E,w) and integer k, define Gk to be the graph that results fromremoving every edge in G having weight k or larger.Given a connected undirected weighted graph G =(V, E,w), where every edge has a uniqueinteger weight, describe an O(|E| log |E|)-time algorithm to determine the largest value of k suchthat Gk is not connected.Solution: Construct an array A containing the |E| distinct edge weights in G, and sort it inO(|E| log |E|) time, e.g., using merge sort. We will binary search to find k. Specifically, consideran edge weight k0 in A (initially the median edge weight), and run a reachability algorithm (e.g.,Full-BFS or Full-DFS) to compute the reachability of an arbitrary vertex x V in O(|E|) time.If exactly |V | vertices are reachable from x, then Gk0 is connected and k > k0; recurse on strictlylarger values for k0. Otherwise, Gk0 is not connected, so k k0; recurse on non-strictly smallervalues for k0. By dividing the search range by a constant fraction at each step (i.e., by alwayschoosing the median index weight of the unsearched space), binary search will terminate afterO(log |E|) steps, identifying the largest value of k such that Gk is not connected. This algorithmtakes O(|E| log |E|) time to sort, and computes reachability of a vertex in O(|E|) time, O(log |E|)times, so this algorithm runs in O(|E| log |E|) time in total.

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