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SIGCSE 20031 Nifty assignments: Brute force and backtracking Steve Weiss Department of Computer Science University of North Carolina at Chapel Hill weiss@cs.unc.edu

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SIGCSE 20032 The puzzle

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SIGCSE 20033

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4 Brute force problem solving Generate candidates Filter Solutions Trash

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SIGCSE 20035 Requirements Candidate set must be finite. Must be an “Oh yeah!” problem.

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SIGCSE 20036 Example Combination lock 60*60*60 = 216,000 candidates

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SIGCSE 20037 Example

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SIGCSE 20038 Oh no!

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SIGCSE 20039 Oh yeah!

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SIGCSE 200310 Additional restrictions Solution is a sequence s 1, s 2,…,s n Solution length, n, is known (or at least bounded) in advance. Each s i is drawn from a finite pool T.

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SIGCSE 200311 Sequence class extend(x)Add x to the right end of the sequence. retract()Remove the rightmost element. size()How long is the sequence? …

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SIGCSE 200312 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

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SIGCSE 200313 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

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SIGCSE 200314 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

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SIGCSE 200315 Problems Too slow, even on very fast machines. Case study: 8-queens Example: 8-queens has more than 281,474,976,711,000 candidates.

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SIGCSE 200316 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.

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SIGCSE 200317 Problems Too slow, even on very fast machines. Case study: 8-queens Example: 8-queens has more than 281,474,976,711,000 candidates.

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SIGCSE 200318 Faster! Reduce size of candidate set. Example: 8-queens, one per row, has only 16,777,216 candidates.

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SIGCSE 200319 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.

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SIGCSE 200320 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

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SIGCSE 200321 Nifty assignments 1.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?

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SIGCSE 200322 Nifty assignments Solution is a sequence of known length (n) where each element is one of the colors. 1 4 3 2

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SIGCSE 200323 Nifty assignments 2. Running a maze: How can you get from start to finish legally in a maze? 20 x 20 grid

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SIGCSE 200324 Nifty assignments Solution is a sequence of unknown length, but bounded by 400, where each element is S, L, or R.

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SIGCSE 200325 Nifty assignments 3. Making change. How many ways are there to make $1.00 with coins. Don’t forget Sacagawea.

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SIGCSE 200326 Nifty assignments 3. Making change. Have the coin set be variable. Exclude the penny.

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SIGCSE 200327 Nifty assignments 4. Unscrambling a word “ptos” == “stop”, “post”, “pots”, ”spot”

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SIGCSE 200328 Nifty assignments 4. Unscrambling a word Requires a dictionary Data structures and efficient search Permutations

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SIGCSE 200329 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.

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SIGCSE 200330 The puzzle

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SIGCSE 200331 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

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SIGCSE 200332 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.

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SIGCSE 200333

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Department of Computer Science 1 Recursion & Backtracking 1.The game of NIM 2.Getting out of a maze 3.The 8 Queen’s Problem 4.Sudoku.

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