sigcse 20031 nifty assignments: brute force and backtracking steve weiss department of computer...
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SIGCSE 2003 1
Nifty assignments:Brute force and backtracking
Steve Weiss
Department of Computer Science
University of North Carolina at Chapel Hill
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The puzzle
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Brute force problem solving
Generate candidates
FilterSolutions
Trash
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Requirements
• Candidate set must be finite.
• Must be an “Oh yeah!” problem.
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Example
Combination lock
60*60*60 = 216,000
candidates
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Example
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Oh no!
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Oh yeah!
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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.
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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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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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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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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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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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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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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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Faster!
• Reduce size of candidate set.
• Example: 8-queens, one per row, has only 16,777,216 candidates.
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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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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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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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Nifty assignments
Solution is a sequence of known length (n) where each element is one of the colors.
1
43
2
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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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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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Nifty assignments
3. Making change.
How many ways are there to make $1.00 with coins. Don’t forget Sacagawea.
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Nifty assignments
3. Making change.
Have the coin set be variable.
Exclude the penny.
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Nifty assignments
4. Unscrambling a word
“ptos” == “stop”, “post”, “pots”, ”spot”
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Nifty assignments
4. Unscrambling a word
Requires a dictionary
Data structures and efficient search
Permutations
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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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The puzzle
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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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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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