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Published byHector Watkins Modified over 8 years ago
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1 CPSC 320: Intermediate Algorithm Design and Analysis July 30, 2014
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2 Course Outline Introduction and basic concepts Asymptotic notation Greedy algorithms Graph theory Amortized analysis Recursion Divide-and-conquer algorithms Randomized algorithms Dynamic programming algorithms NP-completeness
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3 Schedule Monday: BC Day, no classes Wednesday: Quiz 5 (NP and NP completeness) Review: Amortized Analysis Friday: Survey (bonus marks) Review: probably Graph Theory and Dynamic Programming
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4 NP Completeness
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5 Recap We are working on problems, not algorithms Our focus now is on decision problems (yes/no), not optimization problems We distinguish “finding” a solution and “checking” a solution Classes: P: decision problems that are solvable in polynomial time NP: decision problems for which a given certificate can be checked in polynomial time NP hard: at least as “hard” as any problem in NP NP complete (NPC): both NP and NP hard
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6 NP complete NPC problems: have no known algorithm that runs in polynomial time There is no proof that such an algorithm doesn’t exist Examples: Hamiltonian path: path that traverses all nodes exactly once 3-SAT: assign values to Boolean variables that satisfy a set of clauses Graph coloring: assign colors to graphs, adjacent nodes have different colors Cook’s theorem (paraphrased): if the satisfiability problem can be solved in polynomial time, any problem in NP can be solved in polynomial time
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7 NP complete Why is it useful to know a problem is NP complete? So you can stop looking for trivial answers Focus on heuristics or local optimality Evaluate pseudo-polynomial options
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8 NPC Proof
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9 Clique
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10 Clique – NPC proof
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11 Clique – NPC proof
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12 Hamiltonian Path
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13 Hamiltonian Path – NPC proof
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14 Hamiltonian Path – NPC proof
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15 Traveling Salesman Problem
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16 Traveling Salesman Problem
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17 Traveling Salesman Problem
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18 Subset Sum
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