lecture8 - CS 473: Algorithms Chandra Chekuri...

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Unformatted text preview: CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, Urbana-Champaign Fall 2009 Chekuri CS473ug Problem Types Part I Problems and Terminology Chekuri CS473ug 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 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 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 Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness Part II Greedy Algorithms: Tools and Techniques Chekuri CS473ug Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness What is a Greedy Algorithm? Chekuri CS473ug Interval Scheduling Interval Partitioning Scheduling to Minimize Lateness What is a Greedy Algorithm? No real consensus on a universal definition. Chekuri CS473ug 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 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 Interval Scheduling Interval Partitioning...
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This note was uploaded on 01/22/2012 for the course CS 573 taught by Professor Chekuri,c during the Fall '08 term at University of Illinois, Urbana Champaign.

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lecture8 - CS 473: Algorithms Chandra Chekuri...

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