Fast, Solvable Crosswords

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Fast, Solvable Crosswords

• James Connor, John Duchi, Bruce Lo

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Symmetry

• Heavily constrained puzzles often exhibit
  symmetry
• All five by five and six by six puzzles had
  totally symmetric solutions
• Many of the provided crosswords had symmetric
  solutions

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Symmetry Examples
The important one
4
Bit Arrays

• Allow for compact data representation in spite of
  empty space (20,000 words 2 MB)
• Bit-wise manipulations are very fast
• We feel cool for using things at such a low level

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Bit Arrays

• Call all the words of length 6 a class
• Each class is represented by a collection of bit
  arrays.
• Each letter ch at each position p gets a bit
  array
• These bit arrays represent the words with letter
  ch in position p.

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Bit Array Example

• a in position 1 001011
• The third, fifth and sixth words in the lexicon
  have an a in position 1.
• c in position 4 001000
• The third word in the lexicon has a c in
  position 4.

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Bit Array Example (contd)

• a 001011
• c 001000

ac 001000
Result The third word in the lexicon has an a
in the first position, a c in the fourth
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Timing Results

• Solved in an average of one-tenth of a second

13 x 13 crossword with 60 variables
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Easy vs. Hard

• Constraint Satisfaction is NP-Complete

Solve in around 6 seconds median
Never Solve!
10
New York Times Wednesday

• Proves very difficult to solve
• Solution rate of 60
• Average solution time (when successful) 180
  seconds

11
New York Times Thursday

• Significantly Easier
• 100 solution rate
• Average solution time 10 seconds

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Why Easier?

• Long words intersect with relatively few long
  words in the Thursday puzzle.
• Forward checking with conflict-directed
  backjumping can focus on a few words.
• Allows, basically, local search for the big
  words.
• Humans just are not good at big words.

13
The Hardest Puzzle