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Greedy Algorithms

Greedy Algorithms. Basic idea Connection to dynamic programming Proof Techniques. Making Change. Input Positive integer n Task Compute a minimal multiset of coins from C = {d 1 , d 2 , d 3 , …, d k } such that the sum of all coins chosen equals n Example n = 73, C = {1, 3, 6, 12, 24}

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Greedy Algorithms

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  1. Greedy Algorithms • Basic idea • Connection to dynamic programming • Proof Techniques

  2. Making Change • Input • Positive integer n • Task • Compute a minimal multiset of coins from C = {d1, d2, d3, …, dk} such that the sum of all coins chosen equals n • Example • n = 73, C = {1, 3, 6, 12, 24} • Solution: 3 coins of size 24, 1 coin of size 1

  3. Dynamic programming solution • Subsolutions: T(k) for 0 <= k <= n • Recurrence relation • T(n) = mini(T(n-di) + 1) • T(di) = 1 • Linear array of values to compute • Compute T(k) starting at T(1) skipping bas case values

  4. Greedy Solution • T(n) = mini(T(n-di) + 1) • Key observation • For many (but not all) sets of coins, the optimal di will always be the maximum di • That is, T(n) = T(n-dmax) + 1 where dmax is the largest di <= n • Algorithm • Choose largest di smaller than n and recurse

  5. Greedy Technique • When trying to solve a problem, make a local greedy choice that optimizes progress towards global solution and recurse • Running time analysis is typically straightforward • Proof of optimality is harder • Based on structure of optimal solution • “Swapping” is common technique • Must consider that greedy is not optimal and consider counter-examples

  6. Proof of Greedy Optimality • Example: Any n, C = {1, 3, 9, 27, 81} • Let S be an optimal solution and G be the greedy solution • Let Ak denote the number of coins of size k in solution A • Let kdiff be the largest value of k s.t. Gk != Sk • Claim 1: Gkdiff >= Skdiff. Why? • Claim 2: Sk must contain at least 3 coins of size di for some di < dkdiff. Why? • Claim 3: We can create a better solution than Sk. How? • These three claims imply kdiff does not exist and Gk is optimal.

  7. Proof that Greedy is NOT optimal • Consider the following coin set • C = {1, 3, 6, 12, 24, 30} • Prove that greedy will not produce an optimal solution for all n • What about the following coin set? • C = {1, 5, 10, 25, 50}

  8. Activity selection problem • Input • Set of n intervals (si, fi) where • fi > si >= 0 for all intervals n • Task • Identify a maximal set of intervals that do not overlap • Example • {(0,2), (1,3), (5,7), (6, 11), (8,10)} • Solution: {(1,3), (5,7), (8,10)}

  9. Possible Greedy Strategies • Choose the shortest interval and recurse • Choose the interval that starts earliest and recurse • Choose the interval that ends earliest and recurse • Choose the interval that starts latest and recurse • Choose the interval that ends latest and recurse • Do any of these strategies work?

  10. Earliest end time Algorithm • Sort intervals by end time • A: Choose the interval with the earliest end time breaking ties arbitrarily • Prune intervals that overlap with this interval and goto A • Running time?

  11. Proof of Optimality • For any instance of the problem, there exists an optimal solution that includes an interval with earliest end time • Let S be an optimal solution • Suppose S does not include an interval with the earliest end time • We can swap the interval with earliest end time in S with an interval with earliest overall end time to obtain feasible solution S’ • S’ has at least as many intervals as S and the result follows • We recursively apply this observation to the subproblem induced by selecting an interval with earliest end time to see that greedy produces an optimal schedule

  12. Shortest Interval Algorithm • Sort intervals by interval length • A: Choose the interval with shortest length breaking ties arbitrarily • Prune intervals that overlap with this interval and goto A • Running time?

  13. Proof of Optimality? • For any instance of the problem, there exists an optimal solution that includes an interval with earliest end time • Let S be an optimal solution • Suppose S does not include a shortest interval • Can we produce an equivalent solution S’ from S that includes a shortest interval? • If not, how does this lead to a counterexample?

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