1999 S

1. (10%) The linear time string matching algorithm proposed by Knuth, Morris, and Pratt finds the occurrences of a pattern P[1..m] in a text T[1..n] by utilizing the prefix function π os P.
 * (a) (5%) Write the definition of π.
 * (b) (5%) Let m=10 and P=ababcababa. Give the values of π[i]. for i=1, 2, ..., 10.

2. (10%) What is the largest k such that if you can multiply 3 by 3 matrices using k multiplications (not assuming commutativity of multiplication), then you can multiply n by n matrices in time o($$n^{log_2 7}$$)? What would the running time of this algorithm be? You should justify your answers.

3. (15%) Use Ford-Fulkerson's method for the maximum flow problem to find a maximum matching for the following bipartite graph. While applying Ford-Fulkerson's method, you should describe the steps in detail instead of calling it as a procedure. What is the time complexity of your algorith?

4. (10%) Let G=(V,E) be a weighted directed graph. Let V={1, 2, ..., n} and $$s_{i,j}$$ be the non-negative distance between vertex i and vertex. Let $$x_{a,b}^{(q)}$$, 1≤a,b≤n and q≤n, be the distance of a shortest path from vertex a to vertex b with all intermediate vertices in the set $$M_q$$={n-q+1, n-q+2, ..., n}. (In case q≤0, $$M_q$$ is an empty set.) Give a recurrence for $$x_{a,b}^{(q)}$$.

5. (10%) Let A[1..n] be an array storing all the existing telephone numbers in Taipei area. Give an O(n) time algorithm to sort the elements of A.

6. (10%) Consider the recurrence: f(n)=f(g(n))+c, where c is a constant. You have seen two algorithms whose costs fit this recurrence but which have two growth rates. In linear search g(n)=n-1 and in binary search g(n)=n/2. The respective growth rates of f are O(n) and O(lgn). If we want to design a search algorithm that has O(lglglgn) time, what g should we look for?

7. (10%)
 * (a) (5%) Describe an approximation algorithm with ratio bound 2 for the vertex cover problem.
 * (b) (5%) Prove that your algorithm has a ratio bound 2.

8. (15%) We all know the SATISFIABILITY problem is NP-compete. The k-SATISFIABILITY problem is similar to the SATISFIABILITY problem with more restriction: Every clause must contain exactly k literals. Prove that 3-SATISFIABILITY problem is NP-complete.

9. (10%) Huffman gave a greedy algorithm that constructs an optimal prefix code called Huffman code. Prove the correctness of the greedy algorithm.