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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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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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