CMPSC 465, Spring 2009, Homework 8
Due: Friday,April 10, 7 pm. to Dr.Berman).
Page 292, exercise 7.
Iviewthe matrix as distances between nodes 0, 1, 2, 3, 4. Amatrix
Rkallows intermediate nodes 0, ... , k−1.
original
0
6
•
•
3
2
0
•
•
•
•
3
0
2
•
1
2
4
0
•
8
•
•
3
0
→
allow 0
0
6
•
•
3
2
0
•
•
5
•
3
0
2
•
1
2
4
0
4
8
14
•
3
0
→
allow 1
0
6
•
•
3
2
0
•
•
5
5
3
0
2
8
1
2
4
0
4
8
14
•
3
0
→
allow 2, 3
0
6
•
•
3
2
0
•
•
5
3
3
0
2
6
1
2
4
0
4
4
5
7
3
0
→
allow 4
0
6
10
6
3
2
0
12
8
5
3
3
0
2
6
1
2
4
0
4
4
5
7
3
0
Page 292, exercise 9.
Floyd’salgorithm givesincorrect results if there exists a negative cost
cycle. In that case, the minimum cost path does not exist, but Floyd
returns an answer.E.g. we may have two nodes, d(0, 1) =1, d(1, 0) =−2.
Page 298, exercise 5.
False. For example, when the nodes have very similar search probability
then the optimum shape is not changed. We may have nodes 0, 1, 2 with
probabilities 0. 34, 0.33, 0.33. With 0 at the root the average search
cost is 1.99 and with 1 at the root the average is 1.67.
Page 299, exercise 10.
a. Let A1be 2000×1matrix, A2be 1×2000 matrix, and A3be 2000×1matrix.
Evaluating (A1×A2)×A3costs 2000×1×2000+2000×2000×1=8,000,000.
Evaluating A1×(A2×A3) costs 1×2000×1+2000×1×1=4,000.
b. A‘‘way’’tocompute the product nmatrices is a binary tree with nleaves. As dis-
cussed on page 294, this is the n-th Catalan number,
c(n)=
2n
n
1
n+1
with order of growth Θ(4nn−1.5).
c. Desig a dynamic programming solution.
The input will be the array of matrix dimentions, Aiis a D[i−1]×D[i]matrix.
We define a recursive subproblem: to find the minimum cost C[i,j]ofmultiplying
matrices Ai×...×Aj,where i≤j.
The basic case is C[i,i]=0, no cost to multiply when there is only one matrix.
The recursive relationship holds when i<j:
