Homework 4 Solution

# Thus line 3 is executed n times since graph is

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Unformatted text preview: ns in line 3 are executed once. Thus line 3 is executed n times (since graph is connected, and there are n vertices), and hence at most C ′ n opertations, for some constant C ′ , are incurred by line 3. The for loop in line 4 is executed deg(x) = 2m x ∈V times, and each time that the for loop is iterated, a constant number of operations are performed. Thus, the number of steps incurred by the for loop in line 4 is at most C ′′ m for some constant C ′′ . This shows that the total number of operations required by BFS is at most C (n + m) for some constant C . In time-complexity, we use O(m + n), denoting that the time-complexity of BFS is 1 linear in m and n (as opposed to square, cubic, exp, or log). (For example, a linear-time algorithm is obviously better than an exponential-time algorithm, and the diﬀerence is huge when the problem size is big.) Question 2. (2 points) (a) Exercise 2.3.3. Use Kruskal’s algorithm to ﬁnd a minimum weight spanning tree for a graph G with 5 vertices whose edge weights are given by the matrix below: 0 3 5 11 9 3 0 3 9 8 5 3 0 ∞ 10 11 9 ∞ 0 7 9 8 10 7 0 Note: For i = j , the ij -th entry represents the weight of the edge joining vertices vi and vj . (Since loops are irrelevant in spanning trees, we use 0 for the diagonal entries.) (b) Exercise 2.3.5. (altered) The following matrix gives the edge weights a ﬁve-vertex digraph: 0 10 20 ∞ 17 7 0 5 22 33 14 13 0 15 27 30 ∞ 17 0 10 ∞ 15 12 8 0 For i = j , the ij -th entry represents the weight of the edge (vi , vj ). Use Dijkstra’s algorithm to ﬁnd a shortest paths spanning tree, showing shortest paths from v1 to all other vertices. Find also d(v1 , vi ), for i = 1, 2, . . . , 5. Solution. Assume the vertices are labeled as a, b, c, d, e. The minimum weight spanning tree I found is given below. a 11 00 11 00 11 00 b 3,1 00 11 11 00 11 00 8,5 d 1 0 1 0 1 0 7,4 5,3 3,2 11 00 11 00 11 00 1 0 1 0 1 0 c e The dotted edge(s) are those examined but not added to the tree....
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## This note was uploaded on 01/13/2014 for the course MAD 5305 taught by Professor Suen during the Spring '12 term at University of South Florida - Tampa.

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