lecture14GreedyDesign2

lecture14GreedyDesign2 - 1 Greedy Algorithms • Algorithm...

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Unformatted text preview: 1 Greedy Algorithms • Algorithm Design Techniques • Several greedy algorithms 2 The point is, ladies and gentleman, that greed-- for lack of a better word -- is good . Greed is right. Greed works. Greed clarifies, cuts through, and captures the essence of the evolutionary spirit. Greed, in all of its forms -- greed for life, for money, for love, knowledge -- has marked the upward surge of mankind. Gordon Gecko in the movie “Wall Street” 3 Greedy Algorithms • Used to solve optimization problems – Multiple solutions exist, need to find the ‘best’ one • Make the choice that looks best at the moment • A locally optimal choice at each step in expectation of achieving global optimum • Does not always yield optimal solutions • Usually simple and fast 4 Making Change • You need to make $1.34 in change – 1 dollar – 1 quarter – 1 nickel – 4 pennies 5 Making Change • You need to make $1.34 in change Goal: minimize the number of coins – 1 dollar – 1 quarter – 1 nickel – 4 pennies • Greedy algorithm: – As much as you can from largest available denomination 6 Greedy vs. Non-greedy • Greedy – 1.00 + 0.25 + 0.05 + 4 * 0.01 = 7 coins • Non-greedy – 4 * 0.25 + 3 * 0.10 + 4 * 0.01 = 11 coins 7 Is this “optimal”? • Define optimal • Is this algorithm optimal for… – US currency? – Any possible currency? 8 Proving not optimal • Let a currency exist with the following coin values: – 5, 4, 1 • Assertion: Algorithm is not optimal for this currency How to prove this? 9 Proving not optimal • Let a currency exist with the following coin values: – 5, 4, 1 • What’s a change request where the greedy algorithm fails? You can disprove with an example. 10 Proving optimal • Assume currency values of: – 25, 10, 5, 1 – Let’s prove the algorithm does work • Much harder to prove than disprove 11 Assertion 1 • If x is the largest coin such that x ≤ n, then there exists an optimum solution containing x. – This is a greedy choice property • Proof: Let A be an optimal solution – If n < 5, then x = 1. Clearly only one solution – If 5 ≤ n < 10 then x = 5. If A does not contain the greedy choice 5, then it contains 5 1’s, which can be replaced by 1 5, so it’s not an optimal solution. – Similar argument applies for 10 ≤ n < 25 – Continued… 12 Cont… • If 25 ≤ n then x=25. Assume A does not contain a 25. – If A contains at least 3 10’s, they can be traded for 25 + 5 – Otherwise, A consists of 10’s, 5’s, and 1’s and must contain some subset = 25 (why?), which we can replace with a 25 – The assertion is proven for all cases. 13 Assertion 2 • The greedy choice x, followed by an optimal solution for n-x, is optimal – Let B be an optimal solution for n-x, Assume B ∪ {x} is not optimal. Let C ∪ {x} be a better solution, which means that |C|<|B|, but C can’t be smaller than B, so we are done....
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This note was uploaded on 07/25/2008 for the course CSE 331 taught by Professor M.mccullen during the Spring '08 term at Michigan State University.

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lecture14GreedyDesign2 - 1 Greedy Algorithms • Algorithm...

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