CS6212 Youssef

June 6, 2011

Homework 3

Due Date: June 20, 2011

Problem 1: (20 points)

Let a₁ < a₂ < a₃ < a₄ < a₅ < a₆ be 6 numbers, and let

	p1 = 1/20	p2 = 3/20	p3 = 1/20	p4 = 4/20	p5 = 1/20	p6=3/20
q0 = 0	q1 = 2/20	q2 = 1/20	q3 = 1/20	q4 = 1/20	q5 = 1/20	q6 = 1/20

where p_i is the probability of accessing a_i for i=1,2,3,4,5,6, q₀ is the probability of searching for an item < a₁, q₆ is the probability of searching for an item > a₆, and q_i the probability of searching for an item between a_i and a_i+1 for i = 1, 2, 3, 4, 5. You are to construct an optimal binary search tree.

a)     Write down a table for the w_ij's, and a table for the C_ij's and their corresponding r_ij's as defined in the algorithm for optimal binary search trees.

b)     Contruct the optimal binary search tree.


Problem 2: (20 points)

Let A[1..n] be an array of positive numbers, and let k be an integer, k=1,2,...,n. A linear k-partition of A is any sequence of subarrays of the form A[1..n₁], A[n₁+1..n₂], ... , A[n_k-1+1..n], for some n₁, n₂, ... , n_k-1 where 1<= n₁ < n₂ < n₃ < ... < n_k-1 < n. The weight of a subarray is the products of the numbers in that subarray, and the weight of a linear k-partition is defined to be the sum of the weights of the k subarrays of the partition.

Give a dynamic programming algorithm to compute the maximum-weight k-partition of A. (Hint: Take W_i^r as the weight of the maximum-weight r-partition of the array A[1..i], and develop a recurrence relation for W_i^r.) Analyze the time complexity of your algorithm.


Problem 3: (20 points)

You are a consultant for the president of a corporation, which is planning to form a high-caliber advisory committee made up of a cross section of the company. The company has a hierarchical structure; that is, the supervisor relation forms a tree rooted at the president, where each node represents the immediate supervisor of its children nodes. The personnel office has ranked each employee x with a competence rating C(x), which is a real number. In order to form a small but most competent advisory committee, the following conditions must be met:

• No two members have the same rank (i.e., not in the same level of the tree).
• Every member must have his/her boss (i.e., parent node) included in the committee. (This rule does not apply to the president because he/she does not have a boss.)
• The sum of the competence rankings of the committee members is maximum.

a)     Develop a dynamic programming algorithm to make up the committee that maximizes the sum of the competence ratings of its members. (Note that there is no restriction on the number of members on the committee.) Analyze the time complexity of your algorithm.

b) Develop a greedy algorithm for the same problem, and show by a counter example that the greedy solution is not necessarily optimal.


Problem 4: (20 points)

Give a single-source shortest path algorithm for weighted graphs where all the edges have the same weight, such that your algorithm uses a graph traversal technique and is faster than the greedy based single-source shortest path algorithm. Analyze the time complexity of your algorithm.


Problem 5: (20 points)

Let G = (V,E) be the following undirected graph: V = {1,2,3,4,5,6,7,8,9,10,11,12,13}, and E = {(1,2) , (1,3) , (2,4) , (3,4) , (4,5) , (4,7) , (5,6) , (6,7) , (8,6), (10,6) , (8,9) , (10,9) , (7 , 11) , (7,12), (11,13), (12,13)}.

a)     Do a depth-first search on G from node 1, showing the depth-first search tree and the backward edges. Compute the DFN and L of every node.

b)     Identify the articulation points of G.

c)     Do a breadth-first search on G starting from node 1, and show the tree edges and the crossed edges.