Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 24.5, Problem 2E
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To explanation the BFS moreover proof of every shortest path and shortest path tress in Graph is unique.
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Let G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-set
4. Let G (V, E) be a directed graph. Suppose we have performed a DFS traversal of
G, and for each vertex v, we know its pre and post numbers. Show the following:
(a) If for a pair of vertices u, v € V, pre(u) < pre(v) < post(v) < post(u), then there
is a directed path from u to v in G.
(b) If for a pair of vertices u, v € V, pre(u) < post(u) < pre(v) < post(v), then there
is no directed path from u to v in G.
Let G = (V, E) be an undirected graph with vertices V and edges E. Let w(e) denote the weight of e E E. Let T C E be a
spanning tree of G.
Select all of the following that imply that T is not a minimum spanning tree (MST) for G. Incorrect choices will be penalized.
There exists e'
(u, v) g T, u, v E V such that w(e') w(e').
O There exists e' g T such that w(e') w(e) for all e E E.
O There exists e'
(u, v) É T, u, v E V such that w(e') < w(e) for all e on the shortest path from u to v in T.
O There exists e E T, e' ¢ T with w(e) < w(e').
Chapter 24 Solutions
Introduction to Algorithms
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