# lec1 - Complexity Theory Classify problems according to the...

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1 CS151 Complexity Theory Lecture 1 March 29, 2011 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 ? 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 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? 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 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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2 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) 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. 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 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
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lec1 - Complexity Theory Classify problems according to the...

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