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
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The puzzle
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Brute force problem solving
Generate
candidates
Filter
Solutions
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
2
4
3
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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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