1 / 33

Nifty assignments: Brute force and backtracking

Nifty assignments: Brute force and backtracking. Steve Weiss Department of Computer Science University of North Carolina at Chapel Hill weiss@cs.unc.edu. The puzzle. Brute force problem solving. Generate candidates. Filter. Solutions. Trash. Requirements. Candidate set must be finite.

garrett-duke
Download Presentation

Nifty assignments: Brute force and backtracking

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. Nifty assignments:Brute force and backtracking Steve Weiss Department of Computer Science University of North Carolina at Chapel Hill weiss@cs.unc.edu SIGCSE 2003

  2. The puzzle SIGCSE 2003

  3. SIGCSE 2003

  4. Brute force problem solving Generate candidates Filter Solutions Trash SIGCSE 2003

  5. Requirements • Candidate set must be finite. • Must be an “Oh yeah!” problem. SIGCSE 2003

  6. Example Combination lock 60*60*60 = 216,000 candidates SIGCSE 2003

  7. Example SIGCSE 2003

  8. Oh no! SIGCSE 2003

  9. Oh yeah! SIGCSE 2003

  10. Additional restrictions • Solution is a sequence s1, s2,…,sn • Solution length, n, is known (or at least bounded) in advance. • Each si is drawn from a finite pool T. SIGCSE 2003

  11. Sequence class • extend(x) Add x to the right end of the sequence. • retract() Remove the rightmost element. • size() How long is the sequence? • … SIGCSE 2003

  12. Generating the candidates Classic backtrack algorithm: At decision point, do something new (extend something that hasn’t been added to this sequence at this place before.) Fail: Backtrack: Undo most recent decision (retract). Fail: done SIGCSE 2003

  13. Recursive backtrack algorithm(pseudo Java) backtrack(Sequence s) { for each si in T { s.extend(si); if (s.size() == MAX) // Complete sequence display(s); else backtrack(s); s.retract(); } // End of for } // End of backtrack SIGCSE 2003

  14. Problem solver backtrack(Sequence s) { for each si in T { s.extend(si); if (s.size() == MAX) // Complete sequence if (s.solution()) display(s); else backtrack(s); s.retract(); } // End of for } // End of backtrack SIGCSE 2003

  15. Problems • Too slow, even on very fast machines. • Case study: 8-queens • Example: 8-queens has more than 281,474,976,711,000 candidates. SIGCSE 2003

  16. 8-queens • How can you place 8 queens on a chessboard so that no queen threatens any of the others. • Queens can move left, right, up, down, and along both diagonals. SIGCSE 2003

  17. Problems • Too slow, even on very fast machines. • Case study: 8-queens • Example: 8-queens has more than 281,474,976,711,000 candidates. SIGCSE 2003

  18. Faster! • Reduce size of candidate set. • Example: 8-queens, one per row, has only 16,777,216 candidates. SIGCSE 2003

  19. Faster still! • Prune: reject nonviable candidates early, not just when sequence is complete. • Example: 8-queens with pruning looks at about 16,000 partial and complete candidates. SIGCSE 2003

  20. Backtrack with pruning backtrack(Sequence s) { for each si in T if (s.okToAdd(si)) // Pruning { s.extend(si); if (s.size() == MAX) // Complete solution display(s); else backtrack(s); s.retract(); } // End of if } // End of backtrack SIGCSE 2003

  21. Nifty assignments • Map coloring: Given a map with n regions, and a palate of c colors, how many ways can you color the map so that no regions that share a border are the same color? SIGCSE 2003

  22. Nifty assignments • Solution is a sequence of known length (n) where each element is one of the colors. 1 2 4 3 SIGCSE 2003

  23. Nifty assignments • 2. Running a maze: How can you get from start to finish legally in a maze? 20 x 20 grid SIGCSE 2003

  24. Nifty assignments • Solution is a sequence of unknown length, but bounded by 400, where each element is S, L, or R. SIGCSE 2003

  25. Nifty assignments • 3. Making change. • How many ways are there to make $1.00 with coins. Don’t forget Sacagawea. SIGCSE 2003

  26. Nifty assignments • 3. Making change. • Have the coin set be variable. • Exclude the penny. SIGCSE 2003

  27. Nifty assignments 4. Unscrambling a word “ptos” == “stop”, “post”, “pots”, ”spot” SIGCSE 2003

  28. Nifty assignments 4. Unscrambling a word Requires a dictionary Data structures and efficient search Permutations SIGCSE 2003

  29. Nifty assignments • 5. Solving the 9 square problem. • Solution is sequence of length 9 where each element is a different puzzle piece and where the touching edges sum to zero. SIGCSE 2003

  30. The puzzle SIGCSE 2003

  31. Nifty assignments Challenges: Data structures to store the pieces and the 3 x 3 board. Canonical representation so that solutions don’t appear four times. Pruning nonviable sequences: puzzle piece used more than once. edge rule violation not canonical SIGCSE 2003

  32. Nifty assignments Challenges: Algorithm analysis: instrumenting the program to keep track of running time and number of calls to the filter and to the backtrack method. SIGCSE 2003

  33. SIGCSE 2003

More Related