Download CS231 Algorithms Problem Set 6 at Wellesley College and more Assignments Algorithms and Programming in PDF only on Docsity! 1 Wellesley College ◊ CS231 Algorithms ◊ November 1, 1996 Handout #24 PROBLEM SET 6 Due: Friday, November 8 Important: On Friday, November 8, you will receive a take-home exam. In order to give you maximal time to work on your exam, I require Problem Set 6to be turned in no later than 6pm on Friday, November 8. At that time, solutions to Problem Set 6 will be made available. Reading: CLR Chapter 16 (Dynamic Programming); CLR Chapter 17, Sections 17.1-- 17.3 (Greedy Algorithms) Suggested Problems: 16.1-1; 16.3-1; 16.3-6; 16-2; 16-3; 16-4; 17.2-2, 17.2-3, 17.2- 4, 17.2-5, 17.2-6; 17.3-1, 17.3-8; 17-1 Problem 1 [25] Consider the following recursive definition of the Fibonacci function: Fib(n) if n < 2 then return n else return Fib(n-1) + Fib(n-2) a [5] Draw a function call tree for Fib(6) and determine the number of times that Fib(2) is called. (In this case, a function call tree is a binary tree in which each node is labelled by Fib(i) and has left and right subtrees that are function call trees for Fib(i-1) and Fib(i- 2), respectively.) b [10] The above recursive definition of Fib has a running time that is exponential in n. Using memoization (with an auxiliary array), develop a O(n) recursive definition Fib- Memo of the Fibonacci function. Express your solution in pseudocode. c [10] Transform the top-down recursive strategy from part b into a bottom-up iterative strategy expressed in terms of a loop rather than a recursion. Express your solution in pseudocode. Problem 2 [10] CLR 16.3-1 (p. 319) Problem 3 [25] Solve CLR 16.3-5 (p. 319) using the two strategies described below. Assume that the input is stored in an array A[1..n]. For each strategy: • describe your algorithm (in English is fine) • briefly argue why is is correct • briefly argue why it takes O(n2 ) time. a[10] Develop a solution that uses the Θ(mn) LCS algorithm as a black box as part of the solution. Note: you can express a solution of this form in just a few lines! b[15] Develop a solution that does not use the LCS algorithm as a black box, but instead uses an auxiliary array M[1..n], where each M[i] stores the longest monotonically increasing sequence in A[i..n] that begins with A[i]. 2 Problem 4[25] CLR 16-2 (p. 325) Problem 5[15] CLR 17.2-4 (p. 337) Extra Credit Problem [20] CLR 16.3-6 (p. 319)
