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Advanced Counting Techniques

Advanced Counting Techniques. CSC-2259 Discrete Structures. Recurrence Relations. Sequence. Recurrence relation:. For any. Example:. Recurrence relation. Solutions to recurrence relation:. Example:. $10,000 bank deposit. %11 interest. :amount after years. Example:.

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Advanced Counting Techniques

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  1. Advanced Counting Techniques CSC-2259 Discrete Structures Konstantin Busch - LSU

  2. Recurrence Relations Sequence Recurrence relation: For any Konstantin Busch - LSU

  3. Example: Recurrence relation Solutions to recurrence relation: Konstantin Busch - LSU

  4. Example: $10,000 bank deposit %11 interest :amount after years Konstantin Busch - LSU

  5. Example: Fibonacci sequence Konstantin Busch - LSU

  6. Example: Towers of Hanoi bar1 bar2 bar3 discs Goal: move all discs to bar3 Rule: not allowed to put larger discs on top of smaller discs Konstantin Busch - LSU

  7. bar1 bar2 bar3 Step 1: move recursively discs to bar2 Konstantin Busch - LSU

  8. bar1 bar2 bar3 Step 2: move largest disc to bar3 Konstantin Busch - LSU

  9. bar1 bar2 bar3 Step 3: move recursively discs to bar3 Konstantin Busch - LSU

  10. :total disc moves 2 recursive calls with discs (steps 1&3) movement of largest disc (step 2) one move for one disc Konstantin Busch - LSU

  11. Konstantin Busch - LSU

  12. Solving Linear Recurrence Relations Linear homogeneous recurrence relation of degree : Constant coefficients: Konstantin Busch - LSU

  13. A sequence (solution) satisfying the relation is uniquely determined by the initial values: (these are different constants than the coefficients) Konstantin Busch - LSU

  14. Solution to recurrence relation: if an only if divide both sides with characteristic equation Konstantin Busch - LSU

  15. characteristic equation: factorize with roots characteristic roots: Multiple possible solutions: Konstantin Busch - LSU

  16. The solutions may not satisfy the initial conditions Konstantin Busch - LSU

  17. Theorem: Recurrence relation of degree 2 has unique solution where are solutions to the characteristic equation, and are constants that depend on initial conditions Konstantin Busch - LSU

  18. Proof: Characteristic Equation Roots: Konstantin Busch - LSU

  19. First compute from initial conditions Konstantin Busch - LSU

  20. Konstantin Busch - LSU

  21. Prove by induction that Basis cases: true for the specific choices of Konstantin Busch - LSU

  22. Inductive hypothesis: assume that for all Inductive step: prove that for Konstantin Busch - LSU

  23. By (strong) inductive hypothesis: Konstantin Busch - LSU

  24. By recurrence relation definition Inductive hypothesis characteristic equations End of Proof Konstantin Busch - LSU

  25. Example: Fibonacci sequence Has solution: Characteristic roots: Konstantin Busch - LSU

  26. Konstantin Busch - LSU

  27. Konstantin Busch - LSU

  28. Degree recurrence relation: has unique solution where are solutions to the characteristic equation, and are constants that depend on initial conditions Konstantin Busch - LSU

  29. Example: Solution: Characteristic equation: Roots: Konstantin Busch - LSU

  30. Final solution: Konstantin Busch - LSU

  31. Recurrence Relations for Divide and Conquer Algorithms Typical divide and conquer algorithm: • Input of size • Divide into sub-problems • each of size • Combine sub-problems with cost Konstantin Busch - LSU

  32. Divide an conquer recurrence relation: Cost of subproblem of size Konstantin Busch - LSU

  33. Examples: Binary search: Merge Sort: Fast Matrix Multiplication (Stassen’s Alg.): Konstantin Busch - LSU

  34. Theorem: if then Konstantin Busch - LSU

  35. Proof: Konstantin Busch - LSU

  36. Konstantin Busch - LSU

  37. End of Proof Konstantin Busch - LSU

  38. Theorem: if then Konstantin Busch - LSU

  39. Proof: From previous theorem Konstantin Busch - LSU

  40. Case: Konstantin Busch - LSU

  41. Case: End of Proof Konstantin Busch - LSU

  42. Example: Binary search: Konstantin Busch - LSU

  43. Master Theorem: if then Konstantin Busch - LSU

  44. Example: Merge Sort: Konstantin Busch - LSU

  45. Example: Fast Matrix Multiplication (Stassen’s Alg.): Konstantin Busch - LSU

  46. Generating Functions Find number of solutions for: Answer: Konstantin Busch - LSU

  47. Alternative solution choices for choices for choices for Konstantin Busch - LSU

  48. Alternative solution is the total number of solutions to equation Konstantin Busch - LSU

  49. Another problem: Find total number of solutions which satisfy: Konstantin Busch - LSU

  50. Alternative solution choices for choices for choices for Konstantin Busch - LSU

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