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Introduction to Algorithms
Rabie A. Ramadanrabie@rabieramadan.org
http://www. rabieramadan.org
6
Ack : Carola Wenk nad Dr. Thomas Ottmann tutorials
Chapter 4Chapter 4
Divide-and-ConquerDivide-and-Conquer
Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
Acknowledgment :Some of the slides are based on a tutorial made by Kruse and Ryba
Convex Hull
3
Given a set of pins on a pinboard
And a rubber band around them
How does the rubber band look when it snaps tight?
We represent the convex hull as the sequence of points on the convex hull polygon, in counter-clockwise order.
0
2
1
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Brute force Solution
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Based on the following observation:
•A line segment connecting two points Pi and Pj of a set on n points is a part of the convex hull’s boundary if and only if all the other points of the set lie on the same side of the straight line through these two points.
•We need to try every pair of points O(n3 )
Quickhull Algorithm
Convex hull: smallest convex set that includes given points.
Assume points are sorted by x-coordinate values Identify extreme points P1 and P2 (leftmost and rightmost) Compute upper hull recursively:
• find point Pmax that is farthest away from line P1P2
• compute the upper hull of the points to the left of line P1Pmax
• compute the upper hull of the points to the left of line PmaxP2
Compute lower hull in a similar manner
P1
P2
Pmax
QuickHull Algorithm
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How to find the How to find the PPmaxmax point point
• Pmax maximizes the area of the triangle PmaxP1P2
• if tie, select the Pmax that maximizes the angle PmaxP1P2
The points inside triangle PmaxP1P2 can be excluded from further consideration
Worst case (almost like quick sort) : O(n2)
Convex Hull: Divide & Conquer
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Preprocessing: sort the points by x-coordinate
Divide the set of points into two sets A and B:
A contains the left n/2 points,
B contains the right n/2 points
Recursively compute the convex hull of A
Recursively compute the convex hull of B
Merge the two convex hulls
A B
Convex Hull: Runtime
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Preprocessing: sort the points by x-coordinate
Divide the set of points into two sets A and B:
A contains the left n/2 points,
B contains the right n/2 points
Recursively compute the convex hull of A
Recursively compute the convex hull of B
Merge the two convex hulls
O(n log n) just once
O(1)
T(n/2)
T(n/2)
O(n)
Convex Hull: Runtime
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Runtime Recurrence:
T(n) = 2 T(n/2) + n
Solves to T(n) = (n log n)
Merging in O(n) time
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Find upper and lower tangents in O(n) time
Compute the convex hull of AB:
walk clockwise around the convex hull of A, starting with left endpoint of lower tangent
when hitting the left endpoint of the upper tangent, cross over to the convex hull of B
walk counterclockwise around the convex hull of B
when hitting right endpoint of the lower tangent we’re done
This takes O(n) time
A B
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2
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45
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Finding the lower tangent in O(n) time
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can be checked in constant time
right turn or left turn?
a = rightmost point of A
b = leftmost point of B
while T=ab not lower tangent to both convex hulls of A and B do{
while T not lower tangent to convex hull of A do{ a=a-1 } while T not lower tangent to convex hull of B do{ b=b+1 } }
A B
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a=2
1
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0
1
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4=b
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T is lower tangent if all the points are above the line
Convex Hull – Divide & Conquer
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Split set into two, compute convex hull of both, combine.
Convex Hull – Divide & Conquer
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Split set into two, compute convex hull of both, combine.
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Merging two convex hulls.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (i) Find the lower tangent.
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Merging two convex hulls: (ii) Find the upper tangent.
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Merging two convex hulls: (ii) Find the upper tangent.
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Merging two convex hulls: (ii) Find the upper tangent.
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Merging two convex hulls: (ii) Find the upper tangent.
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Merging two convex hulls: (ii) Find the upper tangent.
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Merging two convex hulls: (ii) Find the upper tangent.
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Merging two convex hulls: (ii) Find the upper tangent.
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Merging two convex hulls: (iii) Eliminate non-hull edges.
Chapter 5 Chapter 5
Decrease-and-ConquerDecrease-and-Conquer
Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
Decrease-and-Conquer
1. Reduce problem instance to smaller instance of the same problem
2. Solve smaller instance
3. Extend solution of smaller instance to obtain solution to original instance
Also referred to as inductive or incremental approach
3 Types of Decrease and Conquer
Decrease by a constant Decrease by a constant (usually by 1):(usually by 1):• insertion sort• graph traversal algorithms (DFS and BFS)• topological sorting• algorithms for generating permutations, subsets
Decrease by a constant factor (usually by half)• binary search and bisection method• exponentiation by squaring• multiplication à la russe
Variable-size decrease• Euclid’s algorithm• selection by partition• Nim-like games
This usually results in a recursive algorithm.
What is the difference?
Consider the problem of exponentiation: Compute xn
Brute Force:
Divide and conquer:
Decrease by one:
Decrease by constant factor:
n-1 multiplications
T(n) = 2*T(n/2) + 1 = n-1
T(n) = T(n-1) + 1 = n-1
T(n) = T(n/a) + a-1
= (a-1) n
= when a = 2
alogn2log
Insertion SortTo sort array A[0..n-1], sort A[0..n-2] recursively and then insert A[n-1]
in its proper place among the sorted A[0..n-2]
Usually implemented bottom up (nonrecursively) (Video)
Example: Sort 6, 4, 1, 8, 5
6 | 4 1 8 5 4 6 | 1 8 5 1 4 6 | 8 5 1 4 6 8 | 5 1 4 5 6 8
Write a Pseudocode for Insertion Sort
Analysis of Insertion Sort Time efficiency
Cworst(n) = n(n-1)/2 Θ(n2)
Cbest(n) = n - 1 Θ(n) (also fast on almost sorted arrays)
Space efficiency: in-place
Best elementary sorting algorithm overall
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