CSE 101 Final Exam. Spring 2017

Transcription

1 CSE 101 Final Exam Spring 2017 Instructions: Do not open until the exam starts. The exam will run for 180 minutes. The problems are roughly sorted in increasing order of difficulty. Answer all questions completely. You are free to make use of any result in the textbook or proved in class. You may use up to 12 1-sided pages of notes, and may not use the textbook nor any electronic aids. Write your solutions in the space provided, the pages at the end of this handout, or on the scratch paper provided (be sure to label it with your name). If you have solutions written anywhere other than the provided space be sure to indicate where they are to be found. For several problems on this exam an algorithm is asked for with a specific runtime. Providing a working algorithm with worse runtime will usually receive some partial credit. Some questions will also ask you for justifications of your stated runtime or of the algorithm correctness. You will not be required to provide these things unless asked explicitly, so please be sure to read the problem statements carefully. Please sit in the seat specified on your exam booklet. Name: ID Number: Seat: Problem Total Score 1

2 Question 1 (Strongly Connected Components, 15 points). Compute the strongly connected components of the graph below: 2

3 Question 2 (Knapsack, 20 points). Compute the collection of distinct items from the list below with total weight at most 10, and total value as large as possible subject to this constraint. Item Weight Value A 1 1 B 2 5 C 3 6 D 4 7 E 5 11 F 6 14 Note: although simply listing the correct items will get full credit, an incorrect solution is unlikely to get partial credit unless you show your work. 3

4 Question 3 (Subsequence Decision, 15 points). Given an algorithm that given two sequences a 1,..., a m and b 1,..., b n determines whether or not the a s are a subsequence of the b s. In particular, determine whether or not there is x 1 < x 2 <... < x m so that b xi = a i for all i. For full credit, your algorithm should run in time O(n) or better. 4

5 Question 4 (Multiple Order Statistics, 15 points). In the multiple order statistics problem, you are given a list L of length n and given an integer m dividing n and asked to find the (n/m) th, (2n/m) th,..., ((m 1)n/m) th and n th smallest elements of L. For example if n = 6 and m = 2, you should return both the 3 rd smallest element of L and the largest element of L. For full credit, your algorithm should run in time O(n log(m)) or better. 5

6 Question 5 (Road Trip, 15 points). Karkat is planning a road trip. He has a map of the country given by a weighted, undirected graph G = (V, E) where the (positive) weights are the lengths of the roads. He is trying to get from vertex s to vertex t. However, his car has a limited gas tank and can only travel at most m miles on a single tank. However, there are k > 0 vertices of G that are labelled as gas stations, where he can refill his tank. In other words, Karkat needs to find a path from s to t that has length at most m between gas stations. Give an algorithm that given G, m and the set of gas stations, finds the shortest such path. For full credit, your algorithm should run in time O(k( V + E ) log( V )) or better. 6

7 Question 6 (Exact Path Length, 20 points). Given a DAG G with edge weights (given as potentially large integers) and two specified vertices s and t, the Exact Path Length problem asks whether or not there exists a path from s to t in G with path length exactly equal to some given integer N. Prove that the Exact Path Length problem is NP-Complete. 7

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