Transform

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Transform Conquer

• Replacing One Problem With Another

Saadiq Moolla
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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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Instance Simplification

• Reducing the Problem to a Simpler One
• Techniques
• Presorting
• Gaussian Elimination
• Search Trees

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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
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Presorting (Analysis)

• Old Idea, Many Different Ways
• Efficiency Dependant on Algorithm
• Compare Benefits vs Time Required
• Useful if Operation Repeated

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Gaussian Elimination
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Binary Search Trees
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Binary Search Trees (Analysis)

• Time Efficiency
• Slow
• Methods to Balance
• Special Trees
• AVL Trees
• B-trees

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Heaps
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• Type of Binary Tree
• Requirements
• Essentially Complete
• Parental Dominance

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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

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Heap Insertion
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Heaps (Analysis)

• Sorting
• Priority Queue
• O (n log n)

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Representation Change

• Change One Problem Into Another
• Steps
• Identify
• Transform
• Solve
• Mathematical Modeling

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Problem Reduction

• Reduce to a Known Problem
• Use Known Algorithms
• Dijkstras algorithm
• Horners Rule
• Karp Rabin algorithm
• etc.

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Horners Rule

• Used to Evaluate Polynomials
• 5x2 3x 8 (5x 3)x 8
• 7x3 x2 9x 2 ((7x 1)x 9)x 2
• Linear

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Horners 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
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Fast Exponentiation

• Evaluate xn
• 210 210102 28 22

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Fast Exponentiation Algorithm
Algorithm FastExp (x, n) // Returns xn term ?
x product ? 1 while n gt 0 do if n mod 2 1
then product ? product term term ? term
term n ? n/2 return product