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Greedy Algorithms Be greedy! always make the choice that looks best at the moment. Local optimization. Not always yielding a globally optimal solution. Applicable to a wide range of problems (can often come close to the optimal even if it is not guaranteed). Notable greedy algorithms in this course: Minimum spanning tree algorithms Dijkstra’s single source shortest path algorithm Optimization often goes through a sequence of steps.

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Activity Selection Set S = {1, 2, …, n } of activities. time compatible (no overlap) Problem: Find the largest set A of compatible events. s s f f ii j j incompatible events (overlap) i j k

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Overlapping Subproblems Recursively try all possible compatible subsets. 1 A ? S S'S' S' = {i S | S f } i 1 S–{1} yes no 2 A ? yesno 2 A ? S–{1,2} S'' yesno S'' S'–{2} S'' = {i S | S f } i 2

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A Greedy Solution // The input activities are in order by increasing finish time: f f … f 1 2 n // // Otherwise, sort them first. Greedy-Activity-Selector(s, f ) // (n) without the sorting n = length[s]; A = {1} j = 1 // last activity scheduled (current activity) for i = 2 to n do if s f // next activity starts after current one finishes then A = A + {i} j = i return A i j // update the last scheduled activity

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An Example 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 3 5 6 Step 1 1 Step 2 1 Step 3 1 4 Step 4 1 4 Step 5 2 incompatible – discard the event 4 compatible – schedule the event 14 6 A = {1, 4, 6} schedule

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Greedy Choice Claim 1 There exists an optimal schedule A S such that activity 1 is in A. Proof Suppose B S is an optimal schedule. There are two cases: (1) If 1 is in B then let A = B. (2) Otherwise, let k be the first activity in B... k Let A = B {k} + {1}: Since f f, activities in A are compatible. Thus A is also optimal. 1 k... 1

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Optimal Substructure Claim 2 Let A be an optimal schedule, then A {1} is an optimal schedule for S' = { i in S | s f } i 1 Proof Suppose not true. Then there exists an optimal schedule B for S' with |B| > | A – {1} | = |A| – 1. Then the following solution to S has more activities than A....B:B: A – {1}:... 1 B + {1}: Contradiction.

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Correctness of Greedy Algorithm Combine Claims 1 and 2 and induct on the number of choices: Theorem Algorithm Greedy-Activity-Selector produces solutions of maximum size for the activity-selection problem. Local optimal (greedy) choice → globally optimal solution In activity-selection

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Greedy Algorithm vs Dynamic Programming Dynamic programming solves subproblems first, then makes a decision. Greedy algorithm makes decision first, then solve subproblems. (Greedy-choice property gains efficiency.) Both techniques rely on the presence of optimal substructure. The optimal solution contains the optimal solutions to subproblems.

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Greedy Algorithms Many of the slides are from Prof. Plaisted’s resources at University of North Carolina at Chapel Hill.

Greedy Algorithms Many of the slides are from Prof. Plaisted’s resources at University of North Carolina at Chapel Hill.

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