20_intractability - 4/14/08 Properties of Algorithms 7.8...

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4/14/08 1 Introduction to Computer Science Sedgewick and Wayne Copyright © 2007 http://www.cs.Princeton.EDU/IntroCS 7.8 Intractability 2 Q. Which algorithms are useful in practice? A working definition. [von Neumann 1953, Gödel 1956, Cobham 1964, Edmonds 1965, Rabin 1966] Model of computation = deterministic Turing machine. Measure running time as a function of input size n . Efficient = polynomial time for all inputs. Theory. Definition is broad and robust. Practice. Poly-time algorithms scale to huge problems. Properties of Algorithms Ex 1. Sorting n elements takes n 2 steps using insertion sort. Ex 2. Finding best TSP tour on n elements takes n ! steps using exhaustive search. constants a and b tend to be small a n b 3 Exponential Growth Exponential growth dwarfs technological change. Suppose you have a giant parallel computing device… With as many processors as electrons in the universe… And each processor has power of today's supercomputers… And each processor works for the life of the universe… Will not help solve 1,000 city TSP problem via brute force. quantity electrons in universe supercomputer instructions per second value 10 79 10 13 age of universe in seconds 10 17 estimated 1000! >> 10 1000 >> 10 79 × 10 13 × 10 17 4 TSP Competition 1. Michael Wu. 16705.073 2. Anthony Fazio and Mark Cerqueira. 16,952.972. Smallest insertion. 17265.628. Nearest insertion. 27868.710. Best so far. Jeff Bagdis, Spring '05. 16339.6373
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4/14/08 2 5 Q. Which problems can we solve in practice? A. Those with poly-time algorithms. Q. Which problems have poly-time algorithms? A. No easy answers. Focus of today's lecture. Properties of Problems 6 LSOLVE. Given a system of linear equations, find a solution. LP. Given a system of linear inequalities , find a solution. ILP. Given a system of linear inequalities, find a binary solution. 48 x 0 + 16 x 1 + 119 x 2 88 5 x 0 + 4 x 1 + 35 x 2 13 15 x 0 + 4 x 1 + 20 x 2 23 x 0 , x 1 , x 2 0 x 0 = 1 x 1 = 1 x 2 = 1 5 Three Fundamental Problems 0 x 0 + 1 x 1 + 1 x 2 = 4 2 x 0 + 4 x 1 2 x 2 = 2 0 x 0 + 3 x 1 + 15 x 2 = 36 x 0 = 1 x 1 = 2 x 2 = 2 x 1 + x 2 1 x 0 + x 2 1 x 0 + x 1 + x 2 2 x 0 = 0 x 1 = 1 x 2 = 1 each x i is either 0 or 1 7 LSOLVE. Given a system of linear equations, find a solution. LP. Given a system of linear inequalities, find a solution. ILP. Given a system of linear inequalities, find a binary solution. Q. Which of these problems have poly-time solutions? A. No easy answers. LSOLVE. Yes. Gaussian elimination solves n -by- n system in n 3 time. LP. Yes. Celebrated ellipsoid algorithm is poly-time. ILP. No poly-time algorithm known or believed to exist! Three Fundamental Problems ? 8 Search Problems Search problem. Given an instance I of a problem, find a solution S . Requirement. Must be able to efficiently check that S is a solution. poly-time in size of instance I or report none exists
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4/14/08 3 9 Search Problems Search problem. Given an instance I of a problem, find a solution S .
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This note was uploaded on 09/21/2009 for the course COMPUTER computer 1 taught by Professor Abedauthman during the Fall '08 term at Aarhus Universitet.

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20_intractability - 4/14/08 Properties of Algorithms 7.8...

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