Introduction to Algorithms, Second Edition

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The University of Texas at Austin Lecture 11 Department of Computer Sciences Professor Vijaya Ramachandran Greedy, MST CS357: ALGORITHMS, Spring 2006 1 The Greedy Framework Rcall that an optimization problem is one for which an input has a collection of feasible solutions, each with an associated cost, and we need to find a feasible solution that optimizes the cost; here ‘optimizes’ would mean either minimizes or maximizes , depending on the nature of the problem. Such a solution is called an optimal solution. A greedy strategy for an optimization problem constructs an optimal solution (which is typ- ically a set of elements) incrementally by making a locally optimal choice at each step to decide the next element to be added to the solution. If this strategy generates an optimal solution, then this represents a greedy algorithm . 2 A Toy Problem Activity Selection Problem Given a collection of activities S = { ( s i ,f i ), 1 i n } ,where s i and f i are the start and finish times of the i th activity, and f 1 f 2 ≤ ··· ≤ f n ,a feasible solution is a subset of activities whose durations do not overlap. A feasible solution with the maximum number of activities is an optimal solution . Algorithm Greedy Activity Selection Start with an initially empty solution A .Le t T := S repeat Select an activity α in T with smallest finish time and set A := A ∪{ α } . Delete from T those activities whose durations overlap with the selected activity. until T = φ Correctness Lemma 1 (Greedy Choice) :L e t X be a subset of an optimal solution for the activity selection problem on S .L e t S 0 be the set of activities in S that do not overlap with any activity in X ,andl e t α =( s, f ) be the activity with smallest finish time in S 0 . Then, X 0 =
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Proof. Clearly X 0 is a feasible solution since none of the activities in X 0 overlap with one another. Consider an optimal solution A for S that contains X . (We are given that such an optimal solution exists.) If A contains α then we are done. Otherwise let β =( s 0 ,f 0 ) be the activity with smallest finish time in A - X . Then, f 0 >f since ( s, f ) is the activity with smallest finish time that does not overlap with any activity in X .S in c e α is in S 0 , it does not overlap with any activity in X . Also, α does not overlap with any activity in A - X -{ β } since any activity in A - X -{ β } must have a start time greater than f 0 and the finish time f of α is smaller than f 0 . Hence α does not overlap with any activity in A -{ β } . Now consider the set A 0 = A -{ β }∪{ α } . This is a feasible set since none of the activities in A 0 over lapw itheachother .But | A 0 | = | A | , and since A is an optimal solution, A 0 is also
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Lecture 11-12 - The University of Texas at Austin...

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