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Transform & Conquer

Transform & Conquer. Replacing One Problem With Another . Saadiq Moolla. Introduction. Initial Problem. Two Stage Solution Transform into Another Problem Solve New Problem Three Variations Instance Simplification Representation Change Problem Reduction. New Representation. Solution.

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Transform & Conquer

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  1. Transform & Conquer Replacing One Problem With Another Saadiq Moolla

  2. Introduction Initial Problem • Two Stage Solution • Transform into Another Problem • Solve New Problem • Three Variations • Instance Simplification • Representation Change • Problem Reduction New Representation Solution

  3. Instance Simplification • Reducing the Problem to a Simpler One • Techniques: • Presorting • Gaussian Elimination • Search Trees

  4. Presorting • Example: Given a random list of numbers, determine if there are any duplicates. • Brute Force: Compare Every Pair • Transform and Conquer: • Sort the List • Compare A [i] with A [i + 1] i A

  5. Presorting (Analysis) • Old Idea, Many Different Ways • Efficiency Dependant on Algorithm • Compare Benefits vs Time Required • Useful if Operation Repeated

  6. Gaussian Elimination a11x + a12x + … + a1nx = b1 a’11x + a’12x + … + a’1nx = b’1 a21x + a22x + … + a2nx = b2 a’22x + … + a’2nx = b2 … … an1x + an2x + … + annx = b1 a’nnx = b1

  7. Binary Search Trees 8 4 15 1 7 9 5

  8. Binary Search Trees (Analysis) • Time Efficiency • Slow • Methods to Balance • Special Trees: • AVL Trees • B-trees

  9. Heaps 9 • Type of Binary Tree • Requirements: • Essentially Complete • Parental Dominance 5 7 4 2 1

  10. Properties of Heaps • Height is log2n • Root is largest element • Node + Descendents = Heap • Stored in Array: • Children of A [i] are A [2i] and A [2i + 1]

  11. Heap Insertion 9 5 7 9 8 4 2 1 5 8 7 4 2 1

  12. Heaps (Analysis) • Sorting • Priority Queue • O (n log n)

  13. Representation Change • Change One Problem Into Another • Steps: • Identify • Transform • Solve • Mathematical Modeling

  14. Problem Reduction • Reduce to a Known Problem • Use Known Algorithms: • Dijkstra’s algorithm • Horner’s Rule • Karp – Rabin algorithm • etc.

  15. Horner’s Rule • Used to Evaluate Polynomials • 5x^2 – 3x + 8 » (5x + 3)x + 8 • 7x^3 + x^2 – 9x – 2 » ((7x+ 1)x – 9)x – 2 • Linear

  16. Horner’s Algorithm algorithm Horner (x, P []) // Evaluates a polynomial with coefficients P [] at x for i ← n – 1 downto 0 do v← x * v + P [i] return v

  17. Fast Exponentiation • Evaluate x^n • 2^10 = 2^10102 = 2^8 * 2^2

  18. Fast Exponentiation Algorithm Algorithm FastExp (x, n) // Returns x^n term ← x product ← 1 while n > 0 do if n mod 2 = 1 then product ← product * term term ← term * term n ← └ n/2 ┘ return product

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