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 23.1, Problem 10E
Program Plan Intro
To show that the tree T is a minimum spanning tree for G with edge is subset of w’.
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Students have asked these similar questions
1. Prove the following known result on minimum spanning trees.
Theorem. Let G = (V, E) be a graph and let w : E → R+ be a weight function. Let A be a
subset of edges that is contained in some minimum spanning tree, and let (S, S) be a cut so
that (S, S)N A = 0. Prove that if e is a light edge in (S, S), then e is safe for A.
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2. Let S =
fi and a profit p; associated with it. Design a dynamic programing algorithm for computing
a set of compatible activities of maximum profit. You should start with defining some terms
and then write a recurrence relation for solving the problem. What is the time complexity of
your algorithm?
{1,2...n} be a set activities, where activity i E S has a start time si, a finish time
3.
4.
Given a directed acyclic graph G = (V, E) and two vertices s, te V, design an efficient
algorithm that computes the number of different directed paths from s to t.
Define the incidence matrix B of a directed graph with no self-loop to be an nxm matrix
with rows indexed by vertices, column indexed by edges such that
Bij
=
-1
1
0
if edge j leaves vertex i,
if edge jenters vertex i,
otherwise.
Let BT be the transpose of matrix B. Find out what the entries of the n x n matrix BBT stand for.
Given a graph that is a tree (connected and acyclic).
(1) Pick any vertex v.
(II) Compute the shortest path from v to every other vertex. Let w be the vertex with
the largest shortest path distance.
(III) Compute the shortest path from w to every other vertex. Let x be the vertex
with the largest shortest path distance.
Consider the path p from w to x. Which of the following are true
a. p is the longest path in the graph
b. p is the shortest path in the graph
c. p can be calculated in time linear in the number of edges/vertices
a,c
a,b
a,b,c
b.c
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- Given a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesarrow_forwardLet 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').arrow_forwardConsider a connected undirected graph G=(V,E) in which every edge e∈E has a distinct and nonnegative cost. Let T be an MST and P a shortest path from some vertex s to some other vertex t. Now suppose the cost of every edge e of G is increased by 1 and becomes ce+1. Call this new graph G′. Which of the following is true about G′ ? a) T must be an MST and P must be a shortest s - t path. b) T must be an MST but P may not be a shortest s - t path. c) T may not be an MST but P must be a shortest s - t path. d) T may not be an MST and P may not be a shortest s−t path. Pls use Kruskal's algorithm to reason about the MST.arrow_forward
- Please written by computer sourcearrow_forwardLet G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.arrow_forwardLet G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.arrow_forward
- You are given a graph G = (V, E) with positive edge weights, and a minimum spanning tree T = (V, E') with respect to these weights; you may assume G and T are given as adjacency lists. Now suppose the weight of a particular edge e in E is modified from w(e) to a new value w̃(e). You wish to quickly update the minimum spanning tree T to reflect this change, without recomputing the entire tree from scratch. There are four cases. In each case give a linear-time algorithm for updating the tree. Note, you are given the tree T and the edge e = (y, z) whose weight is changed; you are told its old weight w(e) and its new weight w~(e) (which you type in latex by widetilde{w}(e) surrounded by double dollar signs). In each case specify if the tree might change. And if it might change then give an algorithm to find the weight of the possibly new MST (just return the weight or the MST, whatever's easier). You can use the algorithms DFS, Explore, BFS, Dijkstra's, SCC, Topological Sort as…arrow_forward(V, E) be an undirected graph. Assume that the vertices 2. Let G represent documents. The edges are weighted so that w(i, j) is the dissimilarity between documents i and j. It is desired to partition the vertices into k > 3 disjoint clusters such that k ΣΣ υ(u, υ). i=1 (u,v)EE u,vEC; is minimized. The set C; is the set of documents in cluster i. Show that the e-approximate version of this problem is P-hard for all e, e > 0. Note that k is a fixed integer provided with each problem instance and may be different for different instances.arrow_forwardConsider a graph G, where each of the edges have different weights. Let T1 be the minimum-weight spanning tree produced by Kruskal's Algorithm, and let T2 be the minimum-weight spanning tree produced by Prim's Algorithm. I claim that T1 and T2 must be identical spanning trees - i.e., the exact same set of edges must appear in both trees. Determine whether this claim is TRUE or FALSE. If your answer is TRUE, see if you can figure out why the claim is true. If your answer is FALSE, see if you can come up with a counterexample.arrow_forward
- Please show steps clearlyarrow_forwardLet G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.arrow_forward1.6 If G is a simple graph of with number of vertices n > 0, show that deg(v) id(v) = od(v) = q. VEV VEVarrow_forward
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