lecture17ALGsummary - Algorithm Theory Summary 9.7 of the...

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  1 Algorithm Theory Summary 9.7 of the Weiss text, plus all other  course work
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  2 Problems, algorithms, complexity The algorithm as a solution to a problem Some algorithms are more efficient than  others – complexity theory. Some problems are harder than others. Some problems have NO efficient  algorithms to solve them. Some problems have no solution at all!
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  3 For future work CSE 460 Computability and formal  Languages CSE 830 Algorithms  CSE 835 Graph Algorithms
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  4 Problems and algorithms The  complexity of a PROBLEM  is the  complexity of the best algorithm for solving  that problem. Finding the max of an unordered array is  θ (n) Sorting an unordered array using comparisons is  θ (N log N) “best” can change in terms of average, worst, or  best case when performance depends on the input  [or its order]
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  5 Polynomial Time: class P An algorithm runs in Polynomial time if: T(N)=O(N k ) for some constant k.   An algorithm is considered “efficient” in  these slides if it runs in polynomial time. Polynomial time is better than  exponential runtime, after all.
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  6 Some problems are  undecidable/impossible. There is no algorithm  to decide whether  or not an input WFF of the predicate  logic is a tautology or not. There can be no compiler that can  decide whether or not an arbitrary C++  program has an infinite loop. (See 9.7.1  Weiss.)
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  7 Fermat’s last theorem is just  one such WFF Consider only the domain of integers (Vx)(Vy)(Vz)(Vn) [ (n>2)   (x^n + y^n !=  z^n) ] i.e. there are no “Pythagorean triples”  for n>2   (3^2 + 4^2 = 5^2 for n=2) Fermat claimed to have proved it in the 17 th  century. It was posed by  Diophantus in the 3 rd  century. Andrew Wiles proved it in 1995 after  working on it for 7 years almost full time. (He was interested in the  problem since age 10.) There are several important points in this story.
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  8 Some problems are very hard
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This note was uploaded on 07/25/2008 for the course CSE 331 taught by Professor M.mccullen during the Spring '08 term at Michigan State University.

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lecture17ALGsummary - Algorithm Theory Summary 9.7 of the...

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