6.006 Final Practice Problems with Solutions

Introduction to Algorithms: 6.006 Massachusetts Institute of Technology Professors Srini Devadas and Erik Demaine May 9, 2008 Handout 13 Final Practice Problems 1 Subset Sum You are given a sequence of n numbers (positive or negative): x 1 , x2 , . . . , x n Your job is to select a subset of these numbers of maximum total sum, subject to the constraint that you can’t select two elements that are adjacent (that is, if you pick xi then you cannot pick either xi−1 or xi+1 ). Explain how you can find, in time polynomial in n, the subset of maximum total sum. Solution: Let sumi be the maximum sum of the numbers x1 , x2 , . . . , xi given the adjacency constraint. sum0 = 0 sum1 = max(0, x1 ) sumi = max(sumi−2 + xi , sumi−1 ) This last step works because either we include xi , in which case we also want to include the best solution on up to i − 2, or we don’t include xi , in which case we can just use the best solution on i − 1. Our final answer is then just sumn . To calculate the set that gives the max sum, we could simply keep pointers back from i to either i − 1 or i − 2 depending on which one was bigger (or we could go back and check which was bigger). We follow those pointers, including appropriate numbers. Because there are n subproblems, and each subproblem takes O(1) time to solve, this runs in O(n) time. 2 2 Handout 13: Final Practice Problems Collecting Coins You are given an n-by-n grid, where each square (i, j) contains c(i, j) gold coins. Assume that c(i, j) ≥ 0 for all squares. You must start in the upper-left corner and end in the lower-right corner, and at each step you can only travel one square down or right. When you visit any square, including your starting or ending square, you may collect all of the coins on that square. Give an algorithm to find the maximum number of coins you can collect if you follow the optimal path. Solution: We will solve the following subproblems: let dp[i, j] be the maximum number of coins that it is possible to collect while ending at (i, j). We have the following recurrence: dp[i, j] = c(i, j) + max(dp[i − 1, j], dp[i, j − 1]) We also have the base case that when either i = 0 or j = 0, dp[i, j] = c(i, j). There are n2 subproblems, and each takes O(1) time to solve (because there are only two subprob­ lems to recurse on). Thus, the running time is O(n2 ). Handout 13: Final Practice Problems 3 3 True/False Decide whether these statements are True or False. You must briefly justify all your answers to receive full credit. 1. Any Dynamic Programming algorithm with n subproblems will run in O(n) time. True False Explain: Solution: False. The subproblems may take longer than constant time to compute, as was the case with with longest increasing subsequence. 2. Karatsuba’s method is based on the use of continued fractions. True False Explain: Solution: False. Karatsuba has nothing to do with continued fractions. 3. Newton’s Method for computing iteration. True False √ 2 essentially squares the number of correct digits at each Explain: √ Solution: False. Newton’s method for computing 2 has quadratic convergence, which means the number of correct digits doubles after each iteration. 4 Handout 13: Final Practice Problems 4 Numerics Suppose we are trying to compute √ 3 9 (the cube root of 9). Explain carefully how one iteration of Newton’s Method works for this problem, starting with an initial guess of x0 = 2. (Hint: the function to use is f (x) = x3 − 9. ) Be sure to derive carefully the value of x1 . Solution: f (x0 ) f � (x0 ) x3 − 9 = x0 − 0 2 3 ∗ x0 3 2 −9 =2− 3 ∗ 22 −1 =2− 12 25 = 12 x1 = x 0 − x1 x1 x1 x1