lecture2 - FIRST COMPLEXITY CLASS CSci 5403 Definition:...

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1 COMPLEXITY THEORY CSci 5403 LECTURE II: TIME COMPLEXITY AND THE CENTRAL QUESTIONS P = DTIME(n c ) c N FIRST COMPLEXITY CLASS P is the class of problems that can be solved in polynomial time: they are efficiently decidable Definition: DTIME(t(n)) = { L | L is a language decided by a O(t(n)) time Turing Machine } WHY P? Many t(n) time variants of TMs can be simulated In time O(t(n) c ) by a single-tape TM, including: TMs with double-unbounded tape: O(t(n)) Multi-tape TMs: O(t 2 (n)) RAM Machines: O(t 6 (n)) Polynomials are the simplest class of functions closed under composition, multiplication, etc… They “grow slowly” compared to exponentials EXAMPLE: STCONN A directed graph with n nodes… is a set of vertices V = {1,2,…,n} and a set of edges E µ V £ V. 1 2 3 4 can be encoded as an adjacency matrix has a path from s to t if there are nodes i 1 , …, i k such that { (s,i 1 ), (i 1 , i 2 ),…, (i k , t) } µ E. STCONN = { ʪ G,s,t ʫ | G has a path from s to t } A ij = 1, if (i,j) 2 E 0, otherwise
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2 STABLE MARRIAGES A High School has N boys and M N girls. Each has a ranked list of dates for the 1951 Senior Prom. Albert Bob Charlie Alice Betty Carol B,C,A A,C,B A,B,C B,A,C C,A,B C,B,A An unstable couple prefer each other to their current dates. STABLE = { B,G | There is a pairing with no unstable couple} STABLE MARRIAGES in P Albert Bob Charlie Alice Betty Carol B,C,A A,C,B A,B,C B,A,C C,A,B C,B,A 1. Each boy asks his “first choice” to the prom. 2. Each girl “accepts” her best offer – for now. 3. Repeat until every boy has a date: a. Each boy with no date asks the next girl on his list. b. Each girl “accepts” her new best offer – for now.
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lecture2 - FIRST COMPLEXITY CLASS CSci 5403 Definition:...

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