Data Structures and Algorithms: Lower Bounds and Problem Complexity - Prof. C. Huang, Study notes of Computer Science

Download Data Structures and Algorithms: Lower Bounds and Problem Complexity - Prof. C. Huang and more Study notes Computer Science in PDF only on Docsity! 1 CSCE 350: Data Structures and Algorithms Chin-Tser Huang huangct@cse.sc.edu University of South Carolina 04/20/2009 2 Announcement Reading assignment: Chapter 11.3 Homework 7 is returned and discussed today Highest 4 Average 3.64 Homework 8 is assigned on April 17 and is due on Wednesday, April 22 in class Have your answers neatly typed Explain your answer clearly and adequately by showing the steps Study guide for final exam has been distributed in last Friday’s class 04/20/2009 3 Methods of Obtaining Lower Bound Trivial lower bounds Information-theoretic arguments Problem reduction 04/20/2009 4 Lower Bounds by Problem Reduction Idea: If problem P is at least as hard as problem Q, then a lower bound for Q is also a lower bound for P. Hence, find problem Q with a known lower bound that can be reduced to problem P in question. 04/20/2009 5 Lower Bounds by Problem Reduction Example: Euclidean MST problem Given a set of n points in the plane, construct a tree with minimum total length that connects the given points. (considered as problem P) To get a lower bound for this problem, reduce the element uniqueness problem to it. (considered as problem Q) If an algorithm faster than n log n exists for Euclidean MST, then one exists for element uniqueness also. Aha! A contradiction! Therefore, any algorithm for Euclidean MST must take Ω(n log n) time. 04/20/2009 6 Classifying Problem Complexity As we discussed, problems that can be solved in polynomial time are usually called tractable, otherwise are called intractable, now the question to ask is Is there a polynomial-time algorithm that solves the problem? Possible answers: yes no because it can be proved that all algorithms take exponential time because it can be proved that no algorithm exists at all to solve this problem don’t know don’t know, but if such algorithm were to be found, then it would provide a means of solving many other problems in polynomial time