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DCS Lecture how to solve it

DCS Lecture how to solve it. Patrick Prosser. Your Challenge. Put a different number in each circle (1 to 8) such that adjacent circles cannot take consecutive numbers. That’s illegal, okay?. 6. 5. Put a different number in each circle (1 to 8) such

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DCS Lecture how to solve it

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  1. DCS Lecture how to solve it Patrick Prosser

  2. Your Challenge Put a different number in each circle (1 to 8) such that adjacent circles cannot take consecutive numbers

  3. That’s illegal, okay? 6 5 Put a different number in each circle (1 to 8) such that adjacent circles cannot take consecutive numbers

  4. That’s illegal, okay? 3 3 Put a different number in each circle (1 to 8) such that adjacent circles cannot take consecutive numbers

  5. The Puzzle • Place numbers 1 through 8 on nodes • Each number appears exactly once • No connected nodes have consecutive numbers ? ? ? ? ? ? ? ? You have 4 minutes!

  6. Bill Gates asks … how do we solve it? How do we solve it?

  7. Heuristic Search Which nodes are hardest to number? ? ? ? ? ? ? ? ? Heuristic: a rule of thumb

  8. Heuristic Search ? ? ? ? ? ? ? ?

  9. Heuristic Search Which are the least constraining values to use? ? ? ? ? ? ? ? ?

  10. Heuristic Search Values 1 and 8 ? ? ? 1 8 ? ? ?

  11. Heuristic Search Values 1 and 8 ? ? ? 1 8 ? ? ? Symmetry means we don’t need to consider: 8 1

  12. ? ? ? 1 8 ? ? ? Inference/propagation We can now eliminate many values for other nodes Inference/propagation: reasoning

  13. Inference/propagation {1,2,3,4,5,6,7,8} ? ? ? 1 8 ? ? ?

  14. Inference/propagation {2,3,4,5,6,7} ? ? ? 1 8 ? ? ?

  15. Inference/propagation {3,4,5,6} ? ? ? 1 8 ? ? ?

  16. Inference/propagation {3,4,5,6} ? ? ? 1 8 ? ? ? {3,4,5,6} By symmetry

  17. Inference/propagation {3,4,5,6} {1,2,3,4,5,6,7,8} ? ? ? 1 8 ? ? ? {3,4,5,6}

  18. Inference/propagation {3,4,5,6} {2,3,4,5,6,7} ? ? ? 1 8 ? ? ? {3,4,5,6}

  19. Inference/propagation {3,4,5,6} {3,4,5,6} ? ? ? 1 8 ? ? ? {3,4,5,6}

  20. Inference/propagation {3,4,5,6} {3,4,5,6} ? ? ? 1 8 ? ? ? {3,4,5,6} {3,4,5,6} By symmetry

  21. Inference/propagation {3,4,5,6} {3,4,5,6} ? ? ? 1 8 ? {2,3,4,5,6} {3,4,5,6,7} ? ? {3,4,5,6} {3,4,5,6}

  22. Inference/propagation {3,4,5,6} {3,4,5,6} ? ? ? 1 8 ? {2,3,4,5,6} {3,4,5,6,7} ? ? {3,4,5,6} {3,4,5,6} Value 2 and 7 are left in just one node’s domain

  23. Inference/propagation {3,4,5,6} {3,4,5,6} ? ? 7 1 8 2 {2,3,4,5,6} {3,4,5,6,7} ? ? {3,4,5,6} {3,4,5,6} And propagate …

  24. Inference/propagation {3,4,5} {3,4,5,6} ? ? 7 1 8 2 {2,3,4,5,6} {3,4,5,6,7} ? ? {3,4,5} {3,4,5,6} And propagate …

  25. Inference/propagation {3,4,5} {4,5,6} ? ? 7 1 8 2 {2,3,4,5,6} {3,4,5,6,7} ? ? {3,4,5} {4,5,6} And propagate …

  26. Inference/propagation {3,4,5} {4,5,6} ? ? 7 1 8 2 ? ? {3,4,5} {4,5,6} Guess a value, but be prepared to backtrack … Backtrack?

  27. Inference/propagation {3,4,5} {4,5,6} 3 ? 7 1 8 2 ? ? {3,4,5} {4,5,6} Guess a value, but be prepared to backtrack …

  28. Inference/propagation {3,4,5} {4,5,6} 3 ? 7 1 8 2 ? ? {3,4,5} {4,5,6} And propagate …

  29. Inference/propagation {5,6} 3 ? 7 1 8 2 ? ? {4,5} {4,5,6} And propagate …

  30. Inference/propagation {5,6} 3 ? 7 1 8 2 ? ? {4,5} {4,5,6} Guess another value …

  31. Inference/propagation 3 5 7 1 8 2 ? ? {4,5} {4,5,6} Guess another value …

  32. Inference/propagation 3 5 7 1 8 2 ? ? {4,5} {4,5,6} And propagate …

  33. Inference/propagation 3 5 7 1 8 2 ? ? {4} {4,6} And propagate …

  34. Inference/propagation 3 5 7 1 8 2 4 ? {4} {4,6} One node has only a single value left …

  35. Inference/propagation 3 5 7 1 8 2 4 6 {6}

  36. Solution! 3 5 7 1 8 2 4 6

  37. Bill Gates says … how does a computer solve it? How does a computer solve it?

  38. A Constraint Satisfaction Problem ? ? ? ? ? ? ? ? • Variable,vi for each node • Domain of {1, …, 8} • Constraints • All values used Alldifferent(v1 v2 v3 v4 v5 v6 v7 v8) • No consecutive numbers for adjoining nodes |v1 - v2 | > 1 |v1 - v3 | > 1 …

  39. How we might input the problem to a program Viewing the problem as a “graph” with 8 “vertices” and 17 “edges”

  40. Graph Theory?

  41. Our Problem as a Graph vertex 0 is adjacent to vertex 1 vertex 3 is adjacent to vertex 7 8 vertices, 17 edges 1 2 0 6 7 3 5 4

  42. By the way, Bill Gates says … Computer scientists count from zero 

  43. A Java (Constraint) Program to solve our problem

  44. Read in the name of the input file

  45. Make a “Problem” and attach “variables” to it Note: variables represent our vertices

  46. Constrain all variables take different values

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