lec1 - CS151 Complexity Theory Lecture 1 March 29, 2011...

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CS151 Complexity Theory Lecture 1 March 29, 2011
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March 29, 2011 2 Complexity Theory Classify problems according to the computational resources required running time storage space parallelism randomness rounds of interaction, communication, others… Attempt to answer: what is computationally feasible with limited resources ?
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March 29, 2011 3 Complexity Theory Contrast with decidability: What is computable? answer: some things are not We care about resources! leads to many more subtle questions fundamental open problems
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March 29, 2011 4 The central questions Is finding a solution as easy as recognizing one? P = NP? Is every efficient sequential algorithm parallelizable ? P = NC? Can every efficient algorithm be converted into one that uses a tiny amount of memory ? P = L? Are there small Boolean circuits for all problems that require exponential running time? EXP P/poly? Can every efficient randomized algorithm be converted into a deterministic algorithm one? P = BPP?
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March 29, 2011 5 Central Questions We think we know the answers to all of these questions … but no one has been able to prove that even a small part of this “world-view” is correct. If we’re wrong on any one of these then computer science will change dramatically
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March 29, 2011 6 Introduction You already know about two complexity classes P = the set of problems decidable in polynomial time NP = the set of problems with witnesses that can be checked in polynomial time … and notion of NP-completeness Useful tool Deep mathematical problem : P = NP? Course should be both useful and mathematically interesting
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March 29, 2011 7 A question Given: polynomial f(x 1 , x 2 , …, x n ) as arithmetic formula (fan-out 1): Question : is f identically zero? - * x 1 x 2 * + - x 3 x n * multiplication (fan-in 2) addition (fan-in 2) negation (fan-in 1)
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March 29, 2011 8 A question Given : multivariate polynomial f(x 1 , x 2 , …, x n ) as an arithmetic formula. Question : is f identically zero? Challenge: devise a deterministic poly- time algorithm for this problem.
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March 29, 2011 9 A randomized algorithm Given : multivariate degree r poly. f(x 1 , x 2 , …, x d ) note: r = deg(f) · size of formula Algorithm : pick small number of random points if f is zero on all of these points, answer “yes” otherwise answer “no” (low-degree non-zero polynomial evaluates to zero on only a small fraction of its domain) No efficient deterministic algorithm known
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March 29, 2011 10 Derandomization Here is a deterministic algorithm that works under the assumption that there exist hard problems, say SAT. solve SAT on all instances of length log n encode using error-correcting code (variant of a Reed-Muller code) 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1
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March 29, 2011 11 Derandomization run randomized alg. using these strings in place of random evaluation points if f is zero on all of these points, answer “yes”
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lec1 - CS151 Complexity Theory Lecture 1 March 29, 2011...

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