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Divide and Conquer Paradigm
By: Melissa Manley
How does it work?
Divide: the original problem into two or more sub-problems
Recursively solves the sub-problems
Conquer: by then combining the solutions to these sub-problems
Divide and Conquer Paradigm By: Melissa Manley How does it work? - - PowerPoint PPT Presentation
Divide and Conquer Paradigm By: Melissa Manley How does it work? - - PowerPoint PPT Presentation
Divide and Conquer Paradigm By: Melissa Manley How does it work? Divide : the original problem into two or more sub-problems Recursively solves the sub-problems Conquer : by then combining the solutions to these sub-problems
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Karatsuba Karatsuba
& Gauss
& Gauss
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Advantages
Allows us to solve difficult problems Helps find other efficient algorithms Effectively uses Memory Caches Sometimes will produce more precise
- utcomes.
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Disadvantages
Recursion is slow
Occasionally more complicated than an iterative
approach
Sub-problems can occur more than once
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Binary Search
Referred to as the “ultimate divide and conquer algorithm”
The main concept: decides to use one half
- f the data set, or the other
For example, if trying to find a key k in a set
- f keys containing data z[0,1,…,n-1] :
Compare k to n/2 Based on this result, use either the 1st or 2nd half of the data
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Binary Search Tree
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Mergesort Algorithm
The merge sort divides the data into two halves
It then recursively solves each half of the data
Then merges the data sets back together after they have been sorted
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Quicksort Algorithm
A Random sorting algorithm
Sorts by applying the divide and conquer strategy.
How it works:
Pick a random element from the set Divide the data into 3 groups Recursively sort the sub-list of lesser elements
and the sub-list of greater elements.
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Cooley Tukey FFT Algorithm
most common Fast
Fourier Transform
Originally used by
Carl Friedrich Gauss
Been rediscovered
many times and popularized by J. W. Cooley and J. W. Tukey in 1965
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Cooley-Tukey
It is a D & C algorithm Recursively breaks down a Discrete Fourier
Transform of size N = N1N2 into smaller DFT’s of size N1 and N2.
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Running Time
Based on three criteria:
The # of sub instances the problem is split into
The ratio of the original problem size to the sub problem size
The number of steps required to divide and conquer
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Comparing different sorting algorithm Running Times
O(nlog(n)) (Fastest) Quicksort O(nlog(n)) (Fast) Mergesort O(n^2) (Slow) Insertion Sort O(n^2) (Slow) Selection Sort Running Times Algorithm
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