Fast, Solvable Crosswords

1. Fast, Solvable Crosswords James Connor, John Duchi, Bruce Lo

2. 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

3. 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

5. 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.

6. 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.

7. Bit Array Example (cont’d) ‘a’: 001011 ‘c’: 001000 *a**c*: 001000 Result: The third word in the lexicon has an ‘a’ in the first position, a ‘c’ in the fourth

8. Timing Results • Solved in an average of one-tenth of a second 13 x 13 crossword with 60 variables

9. 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

12. 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