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8 4 5 18 6 8 (a) Graph G 12 3 8 (E) 6 (b) Tree Ti 12 F E 5 18 6 (c) Tree T₂ B

8 4 5 18 6 8 (a) Graph G 12 3 8 (E) 6 (b) Tree Ti 12 F E 5 18 6 (c) Tree T₂ B

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 74EQ
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= Question 1 (SPTree Verification You are given a directed weighted graph G (V, E, W), a vertex s E V and a tree T such that each vertex vi E V maps to a node in T. The root of T is s. For each node n in T we define the predecessor n. as its parent in the tree. By definition, s. =Nil for the root node s. For any other node n₁.π = n2, there must be a direct edge (n2, n₁) EE. You need to develop an algorithm that tests whether T is a shortest-path tree from the single-source s to all destinations in G. In particular, the algorithm should return true if T is a shortest path tree and false, otherwise. Notice that your algorithm should be more efficient than the Bellman-Ford algorithm which runs in O(VE). A E 5 18 6 8 (a) Graph G 12 F 3 B A 8 E D 6 (b) Tree T₁ 12 F B A 8 E 5 18 6 (c) Tree T₂ F B An example is given in the figure above. Assume that for the graph G we would like to test two trees rooted at A. The first tree, Ti is a shortest path tree. Your algorithm should return true when running on G, T₁. On the other hand, the tree T2 is not a shortest path tree and your algorithm should return false.
Transcribed Image Text:= Question 1 (SPTree Verification You are given a directed weighted graph G (V, E, W), a vertex s E V and a tree T such that each vertex vi E V maps to a node in T. The root of T is s. For each node n in T we define the predecessor n. as its parent in the tree. By definition, s. =Nil for the root node s. For any other node n₁.π = n2, there must be a direct edge (n2, n₁) EE. You need to develop an algorithm that tests whether T is a shortest-path tree from the single-source s to all destinations in G. In particular, the algorithm should return true if T is a shortest path tree and false, otherwise. Notice that your algorithm should be more efficient than the Bellman-Ford algorithm which runs in O(VE). A E 5 18 6 8 (a) Graph G 12 F 3 B A 8 E D 6 (b) Tree T₁ 12 F B A 8 E 5 18 6 (c) Tree T₂ F B An example is given in the figure above. Assume that for the graph G we would like to test two trees rooted at A. The first tree, Ti is a shortest path tree. Your algorithm should return true when running on G, T₁. On the other hand, the tree T2 is not a shortest path tree and your algorithm should return false.
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