advertisement

UML CS Analysis of Algorithms 91.503 (section 201) Spring, 2005 Homework #1 Assigned: Tuesday, 1/25 Due: Tuesday, 2/1 start of lecture This assignment covers textbook material in Chapter 15 (Dynamic Programming) plus review material from 91.404 (undergraduate algorithms). 1. (10 points) Function Order of Growth: Given the following 4 functions: 4 3 lg n n4 nlg 2 8 nlg lg 3 n n lg n2 List the 4 functions in order of nondecreasing asymptotic size. Justify your answer. 2. (10 points) Algorithm & Problem Bounds: Let P be a problem. Suppose that the worst-case asymptotic running time complexity of P is in ( n2 lg(n3lgn) ) and is also in O( n3 lgn ). That is, we know that there exists an algorithm solving P whose worst-case asymptotic running time complexity is in O( n3 lgn ) and we know that time proportional to n2 lg(n3lgn) is a lower bound on the worst-case asymptotic running time complexity of any algorithm solving P. Now, let A be an algorithm that solves P. For each of the following statements about A below [(1)-(4)] determine whether or not the statement is consistent with the information provided about the worst-case asymptotic time complexity of P. For example, for (1), determine whether or not it is possible for A to have worst-case time complexity in ( n2 lglgn) given that P’s worst-case time complexity is in O(n3 lgn) and is also in ( n2 lg(n3lgn)). Note that when working on (1) you can ignore statements [(2)-(4)]. 1) A has worst-case time complexity in (n2 lglgn). 2) A has worst-case time complexity in (n2 lg( n3/lgn )). 3) A has worst-case time complexity in( n3 lglgn ). 4) A has worst-case time complexity in O( n2 2lgn ). 3. (10 points) Recurrence: Find tight upper and lower asymptotic bounds on the solution to the following recurrence: T (n) 2T n 2 n You may assume that T(2) = 1 and n is even. page 1 of 2 Explain your answer. UML CS Analysis of Algorithms 91.503 (section 201) Spring, 2005 4. (10 points) Dynamic Programming: Determine an LCS of these two strings: <A, B, B, C, B, A, C, A> and <B, C, A, B, B, A, A, C>. Justify optimality of your answer. 5. (60 points) Dynamic Programming: Solve the following car painting problem efficiently using dynamic programming: given a sequence of n cars, each to be painted one of c different colors, and k adjacent positions each car can move, find a sequence with the minimum number of adjacent color changes. That is, each car in the final ordering can be at most k positions away from its original position. Example: Let n = 9, k = 2 and let 0, 1, 2 represent c = 3 colors. Let the following sequence represent the starting state: <0 1 1 2 0 0 1 2 1> The sequence < 1 1 0 0 0 2 2 1 1 > is an optimal solution. The number of adjacent color changes in this solution is 3: change from 1 to 0, change from 0 to 2 and change from 2 to 1. a. Characterize the structure of the optimal solution. b. Recursively define the value of an optimal solution. c. Compute the value of an optimal solution in a bottom-up fashion. d. Construct an optimal solution from the computed information. Include in your answer: - pseudocode written using the conventions on p. 19-20 of the textbook; - correctness justification; - upper bound on worst-case asymptotic running time. page 2 of 2