tailieunhanh - Greedy Algorithms

Simple recursive algorithms Backtracking algorithms Divide and conquer algorithms Dynamic programming algorithms Greedy | Greedy Algorithms A short list of categories Algorithm types we will consider include: Simple recursive algorithms Backtracking algorithms Divide and conquer algorithms Dynamic programming algorithms Greedy algorithms Branch and bound algorithms Brute force algorithms Randomized algorithms Optimization problems An optimization problem is one in which you want to find, not just a solution, but the best solution A “greedy algorithm” sometimes works well for optimization problems A greedy algorithm works in phases. At each phase: You take the best you can get right now, without regard for future consequences You hope that by choosing a local optimum at each step, you will end up at a global optimum 2 Example: Counting money Suppose you want to count out a certain amount of money, using the fewest possible bills and coins A greedy algorithm would do this would be: At each step, take the largest possible bill or coin that does not overshoot Example: To make $, you can choose: a | Greedy Algorithms A short list of categories Algorithm types we will consider include: Simple recursive algorithms Backtracking algorithms Divide and conquer algorithms Dynamic programming algorithms Greedy algorithms Branch and bound algorithms Brute force algorithms Randomized algorithms Optimization problems An optimization problem is one in which you want to find, not just a solution, but the best solution A “greedy algorithm” sometimes works well for optimization problems A greedy algorithm works in phases. At each phase: You take the best you can get right now, without regard for future consequences You hope that by choosing a local optimum at each step, you will end up at a global optimum 2 Example: Counting money Suppose you want to count out a certain amount of money, using the fewest possible bills and coins A greedy algorithm would do this would be: At each step, take the largest possible bill or coin that does not overshoot Example: To make $, you can choose: a $5 bill a $1 bill, to make $6 a 25¢ coin, to make $ A 10¢ coin, to make $ four 1¢ coins, to make $ For US money, the greedy algorithm always gives the optimum solution 3 A failure of the greedy algorithm In some (fictional) monetary system, “krons” come in 1 kron, 7 kron, and 10 kron coins Using a greedy algorithm to count out 15 krons, you would get A 10 kron piece Five 1 kron pieces, for a total of 15 krons This requires six coins A better solution would be to use two 7 kron pieces and one 1 kron piece This only requires three coins The greedy algorithm results in a solution, but not in an optimal solution 4 A scheduling problem You have to run nine jobs, with running times of 3, 5, 6, 10, 11, 14, 15, 18, and 20 minutes You have three processors on which you can run these jobs You decide to do the longest-running jobs first, on whatever processor is available 20 18 15 14 11 10 6 5 3 P1 P2 P3 Time to completion: 18 + 11 + 6 = 35 minutes This solution isn’t bad, .