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52+Slides--More+on+Complexity

# 52+Slides--More+on+Complexity - CS103 HO#52 Slides-More on...

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CS103 HO#52 Slides--More on Complexity 5/23/11 1 The Language Class P P is closed under complement. PROOF: If L P, there is a deterministic Turing machine M that decides L in polynomial time. Construct a machine M' that is M with the accepting and rejecting states swapped. M' will halt in exactly the same number of steps that M would take and it would accept L. Is NOTCONNECTED the same as CONNECTED? CONNECTED = NOTCONNECTED {strings not of the form G } = { w | w is not an undirected graph or w = G for a graph G that is not connected } The question of whether a string is an encoded graph is decidable in polynomial time, so malformed cases don't affect the polynomial time decidability of CONNECTED. Thus it is reasonable to use NOTCONNECTED as a shorthand for CONNECTED. CONNECTED = { G | G is an undirected graph and G is connected } NOTCONNECTED = { G | G is an undirected graph and G is not connected } The Language Class P C A B D The Königsberg Bridges An Eulerian path through a graph traverses each edge exactly once. An Eulerian circuit through a graph is an Eulerian path that starts and ends at the same node. (Note: a Hamiltonian circuit visits each vertex exactly once.) What is the complexity of deciding whether a graph has an Eulerian circuit? The Language Class P EC = { G | G is an undirected graph and G contains an Eulerian circuit. } Euler showed that • a connected graph has an Eulerian path that is not a circuit if and only if it contains exactly 2 vertices of odd degree • a connected graph has an Eulerian circuit if and only if all vertices have even degree. M = "On input G, a graph with vertices V and edges E: 1. If CONNECTED (G) rejects, reject . 2. For each vertex v in V: 3. Count the number of edges that have v as one endpoint but not both. 4. If the count is odd, exit the loop and reject . 5. If all counts are even, accept ." The Language Class P EC = { G | G is an undirected graph and G contains an Eulerian circuit. } M = "On input G, a graph with vertices V and edges E: 1. If CONNECTED (G) rejects, reject .

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52+Slides--More+on+Complexity - CS103 HO#52 Slides-More on...

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