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lecture8 - CS 473 Algorithms Chandra Chekuri...

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CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, Urbana-Champaign Fall 2009 Chekuri CS473ug
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Problem Types Part I Problems and Terminology Chekuri CS473ug
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Problem Types Problem Types Decision Problem : Is the input a YES or NO input? Example: Given graph G , nodes s , t , is there a path from s to t in G ? Search Problem: Find a solution if input is a YES input. Example: Given graph G , nodes s , t , find an s - t path. Optimization Problem: Find a best solution among all solutions for the input. Example: Given graph G , nodes s , t , find a shortest s - t path. Chekuri CS473ug
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Problem Types Terminology A problem Π consists of an infinite collection of inputs { I 1 , I 2 , . . . , } . Each input is referred to as an instance . The size of an instance I is the number of bits in its representation. For an instance I , sol ( I ) is a set of feasible solutions to I . Typical implicit assumption: given instance I and y Σ * , there is an way to check if y sol ( I ). In other words, problem is in NP. For optimization problems each solution s sol ( I ) has an associated value . Typical implicit assumption: given s , can compute value efficiently. Chekuri CS473ug
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Problem Types Problem Types Decision Problem : Given I output whether sol ( I ) = or not. Search Problem: Given I , find a solution s sol ( I ) if sol ( I ) 6 = . Optimization Problem: Given I , Minimization problem. Find a solution s sol ( I ) of minimum value Maximization problem. Find a solution s sol ( I ) of maximum value Notation: opt ( I ) : interchangeably (when there is no confusion) used to denote the value of an optimum solution or some fixed optimum solution. Chekuri CS473ug
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Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness Part II Greedy Algorithms: Tools and Techniques Chekuri CS473ug
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Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness What is a Greedy Algorithm? Chekuri CS473ug
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Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness What is a Greedy Algorithm? No real consensus on a universal definition. Chekuri CS473ug
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Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness What is a Greedy Algorithm? No real consensus on a universal definition. Greedy algorithms: make decision incrementally in small steps without backtracking decision at each step is based on improving local or current state in a myopic fashion without paying attention to the global situation decisions often based on some fixed and simple priority rules Chekuri CS473ug
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Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness Pros and Cons of Greedy Algorithms Pros: Usually (too) easy to design greedy algorithms Easy to implement and often run fast since they are simple Several important cases where they are effective/optimal Lead to a first-cut heuristic when problem not well understood Chekuri CS473ug
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Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness Pros and Cons of Greedy Algorithms Pros: Usually (too) easy to design greedy algorithms
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