Question

In: Statistics and Probability

Let G = (V, E) be a directed graph, with source s ∈ V, sink t...

Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈ V, and nonnegative edge capacities {ce}. Give a polynomial-time algorithm to decide whether G has a unique minimum s-t cut (i.e., an s-t of capacity strictly less than that of all other s-t cuts).

Solutions

Expert Solution


Related Solutions

Given a graph G = (V,E), the source-sink pair (s,t) and capacity of edges {C_e ≥...
Given a graph G = (V,E), the source-sink pair (s,t) and capacity of edges {C_e ≥ 0 | e ∈ E}, design a polynomial-time algorithm to find a set of edges S, such that for every edge e ∈ S, increasing C_e will lead to an increase of max-flow value between s and t. Show the correctness of your algorithm.
You are given a directed graph G(V,E) with n vertices and m edges. Let S be...
You are given a directed graph G(V,E) with n vertices and m edges. Let S be the subset of vertices in G that are able to reach some cycle in G. Design an O(n + m) time algorithm to compute the set S. You can assume that G is given to you in the adjacency-list representation.
If G = (V, E) is a graph and x ∈ V , let G \...
If G = (V, E) is a graph and x ∈ V , let G \ x be the graph whose vertex set is V \ {x} and whose edges are those edges of G that don’t contain x. Show that every connected finite graph G = (V, E) with at least two vertices has at least two vertices x1, x2 ∈ V such that G \ xi is connected.
A DAG is a directed graph that contains no directed cycles. Define G = (V, E)...
A DAG is a directed graph that contains no directed cycles. Define G = (V, E) in which V is the set of all nodes as {v1, v2, ..., vi , ...vn} and E is the set of edges E = {ei,j = (vi , vj ) | vi , vj ∈ V} . A topological order of a directed graph G = (V, E) is an ordering of its nodes as {v1, v2, ..., vi , ...vn} so that...
Let G = (V, E) be a directed acyclic graph modeling a communication network. Each link...
Let G = (V, E) be a directed acyclic graph modeling a communication network. Each link e in E is associated with two parameters, w(e) and d(e), where w(e) is a non-negative number representing the cost of sending a unit-sized packet through e, and d(e) is an integer between 1 and D representing the time (or delay) needed for transmitting a packet through e. Design an algorithm to find a route for sending a packet between a given pair of...
. Provide a weighted directed graph G = (V, E, c) that includes three vertices a,...
. Provide a weighted directed graph G = (V, E, c) that includes three vertices a, b, and c, and for which the maximum-cost simple path P from a to b includes vertex c, but the subpath from a to c is not the maximum-cost path from a to c
# Problem Description Given a directed graph G = (V, E), find the number of connected...
# Problem Description Given a directed graph G = (V, E), find the number of connected components in G. # Input The graph has `n` vertices and `m` edges. There are m + 1 lines, the first line gives two numbers `n` and `m`, describing the number of vertices and edges. Each of the following lines contains two numbers `a` and `b` meaning there is an edge (a,b) belong to E. All the numbers in a line are separated by...
# Problem Description Given a directed graph G = (V,E) with edge length l(e) > 0...
# Problem Description Given a directed graph G = (V,E) with edge length l(e) > 0 for any e in E, and a source vertex s. Use Dijkstra’s algorithm to calculate distance(s,v) for all of the vertices v in V. (You can implement your own priority queue or use the build-in function for C++/Python) # Input The graph has `n` vertices and `m` edges. There are m + 1 lines, the first line gives three numbers `n`,`m` and `s`(1 <=...
Let G(V, E,w) be a weighted undirected graph, where V is the set of vertices, E...
Let G(V, E,w) be a weighted undirected graph, where V is the set of vertices, E is the set of edges, and w : E → R + is the weight of the edges (R + is the set of real positive numbers). Suppose T(G) is the set of all minimum spanning trees of G and is non-empty. If we know that the weight function w is a injection, i.e., no two edges in G have the same weight, then:...
Please use python: # Problem Description Given a directed graph G = (V, E), find the...
Please use python: # Problem Description Given a directed graph G = (V, E), find the number of connected components in G. # Input The graph has `n` vertices and `m` edges. There are m + 1 lines, the first line gives two numbers `n` and `m`, describing the number of vertices and edges. Each of the following lines contains two numbers `a` and `b` meaning there is an edge (a,b) belong to E. All the numbers in a line...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT