1 / 26

Brute Force Approaches

Brute Force Approaches. Techniques for finding optimal solutions to hard problems “relatively” quickly Backtracking Branch and Bound Combining efficient solutions with brute force approaches Other issues. The Knapsack Problem. Input Capacity K n items with weights w i and values v i

Anita
Download Presentation

Brute Force Approaches

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Brute Force Approaches • Techniques for finding optimal solutions to hard problems “relatively” quickly • Backtracking • Branch and Bound • Combining efficient solutions with brute force approaches • Other issues

  2. The Knapsack Problem • Input • Capacity K • n items with weights wi and values vi • Goal • Output a set of items S such that • the sum of weights of items in S is at most K • and the sum of values of items in S is maximized

  3. Greedy Does Not Work • What are possible greedy strategies? • Highest Density First • Highest Value First • Lowest Weight First • Prove these are not optimal for the 0-1 knapsack problem.

  4. Multi-constraint Knapsack • Input • Capacity K1 and K2 • n items with integer weights wi, size si, and values vi • Goal • Output a set of items S such that • the sum of weights of items in S is at most K1 • the sum of sizes of items in S is at most K2 • and the sum of values of items in S is maximized • Can we use dynamic programming?

  5. Situation where dynamic programming does not work What if our problem can’t be described with integers? wA = 2 vA = $40 wB =  vB = $50 wC = 1.98 vC = $100 wD = 5 vD = $95 wE = 3 vE = $30 We have to resort to brute force….

  6. Brute Force • Generate all possible solutions • With n items, there are 2n solutions to be generated • Check each to see if they satisfy the constraint • Save maximum solution that satisfies constraint • Can be represented as a tree

  7. C E D A C D E E D E E D E E C D E E D E E E C E E D E E D B B Brute Force: Branching In Out Weight = 15.12 Value = $315 Weight = 8.98 Value = $235 Weight = 9.98 Value = $225

  8. Backtracking • In the tree representation, we can think of the previous algorithm doing a DFS in the tree • If we reach a point where a solution no longer is feasible, there is no need to continue exploring • We can “backtrack” from this point and potentially cut off much of the tree and many of the solutions • In the given example, backtracking would be much more effective if we had even more items or a smaller knapsack capacity

  9. C C E A E C D E D D E C D E E E D E D D E E E E D E 2, $40 , $50 1.98, $100 5, $95 3, $30 Backtracking In Out B B E E E Weight = 8.98 Value = $235 Weight = 9.98 Value = $225

  10. Branch and Bound • We can backtrack if we know the best possible solution in current subtree is worse than current best solution obtained so far. • Estimates for improvement given current ordering • A down can give $315 • B down -> $275 • C down -> $225 • D down -> $125 • E down -> $30

  11. A D C D C E D E D C 2, $40 , $50 1.98, $100 5, $95 3, $30 Branch and Bound In Out B B C D D E E E E E E Weight = 8.98 Value = $235 Weight = 7.12 Value = $190

  12. Generating a good bound • The key to branch and bound is having a good initial bound. • How might we generate a good initial bound?

  13. 2, $40 , $50 1.98, $100 5, $95 3, $30 Order of items • Since there is no fixed ordering of items, is there a better way to order the input items? • Highest weight first • Generate infeasible solutions quickly • Highest density first • Generate good solution quickly • Which is better depends on input

  14. C D C E A A C C E A D: 5, $95 B:, $50 E: 3, $30 A: 2, $40 C: 1.98, $100 Heaviest on Top In Out B B (Max remaining = $220) A C C Weight = 8.14 Value = $145 Weight = 10.0 Value = $165 Weight = 9.98 Value = $225 Weight = 8.98 Value = $235

  15. D C B D B E C: 1.98, $100 A: 2, $40 D: 5, $95 B:, $50 E: 3, $30 Best Density on Top In Out A A B B E E E Weight = 8.98 Value = $235

  16. Greedy and Brute Force • Suppose half the inputs to our knapsack problem all have the same weight. • If all inputs had the same weight, we could implement a greedy solution • However, since half do not, we cannot use greedy alone to find optimal solution • Combine brute-force approach with greedy to find optimal solution more quickly.

  17. F1 F4 F4 F4 The Breakdown... In Out F2 F2 F3 F3 F3 F3 F4 F4 F4 F4 F4 F5 F5 F5 F5 F5 F5 F5 F5 F5 F5 Greedy on half!

  18. Algorithm • Backtrack on half the inputs • At leafs, apply greedy strategy on the other half of the inputs • Comparison • Pure brute force: O(2n) • Combination: O(n2n/2)

  19. How much better? • Assumptions • Suppose n=50 • We can test 1,000,000 solutions/second. • 250 would take over 35 years • 225 can be generated in half an hour • Plus marginal time to generate greedy solution for each of the 225 solutions

  20. Another combination • Suppose half the inputs to our knapsack problem have small integral weights/values while the other half have real weights/values. • How can we combine approaches to solve this efficiently?

  21. The n-Queens Problem • Input • Positive integer n • Task • Place n queens on an n by n chessboard so that no two queens attack each other (on same row, column, diagonal), or report that this is impossible • Solve the n-queens problem for n = 1, 2, 3

  22. n=4 • Pure brute force search • 16 squares on a 4 by 4 chessboard • Leads to 16 * 15 * 14 * 13 = 43,680 possible solutions • Improvements • At most one queen per row: 44 = 256 possible solutions • Backtracking: If two queens already attack before final queen placed, backtrack

  23. Larger values of n • n=8 • At most one queen per row: 88 = 16,777,216 • Early backtracking: 2057 nodes total • Time to find first solution: 114 nodes • n=12 • At most one queen per row: 1212 • Early backtracking: 856,189 nodes total • Time to find first solution: 262 nodes

  24. The Problem • In the U.S. navy, the SEALS are each specially trained in a wide variety of skills so that small teams can handle a multitude of missions. • If there are k different skills needed for a mission, and n SEAL members that can be assigned to the team, find the smallest team that will cover all of the required skills. • Andersen knows hand-to-hand, first aid, and camouflage • Butler knows hand-to-hand and snares • Cunningham knows hand-to-hand • Douglas knows hand-to-hand, sniping, diplomacy, and snares • Eckers knows first-aid, sniping, and diplomacy

  25. Greedy Algorithm • What is the obvious greedy algorithm for this problem? • Find a counter-example to the optimality of greedy for this problem.

  26. Brute Force Approach • What is the brute-force approach? • How can we simplify the problem as far as possible in polynomial time? • How can we set bounds on the solutions? • When will we need to do backtracking? • What order should we test the potential members in when branching?

More Related