Introduction to Algorithms
Rabie A. Ramadan
[email protected]
http://www. rabieramadan.org
6
Ack : Carola Wenk nad Dr. Thomas Ottmann tutorials
Chapter 4
Divide-and-Conquer
Acknowledgment :
Some of the slides are based
on a tutorial made by
Kruse and Ryba
Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
Convex Hull
 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
4
5
3
6
2
1
0
sequence of points on the convex hull
polygon, in counter-clockwise order.
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Brute force Solution
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 )
4
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
Pmax
P2
P1
QuickHull Algorithm

How to find the Pmax 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)
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Convex Hull: Divide & Conquer
 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
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Convex Hull: Runtime
Preprocessing: sort the points by xcoordinate
 Divide the set of points into two sets A and
B:
O(n log n)
just once
O(1)
 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
T(n/2)
T(n/2)
O(n)
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Convex Hull: Runtime
 Runtime Recurrence:
T(n) = 2 T(n/2) + n
 Solves to T(n) = (n log n)
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Merging in O(n) time
 Find upper and lower tangents in O(n) time
 Compute the convex hull of AB:
 walk clockwise around the convex hull of A,
4
5
3
starting with left endpoint of lower tangent
 when hitting the left endpoint of the upper
tangent, cross over to the convex hull of B
6
2
 walk counterclockwise around the convex
hull of B
 when hitting right endpoint of the lower
7
1
A
B
tangent we’re done
This takes O(n) time
10
Finding the lower tangent in O(n) time
3
a = rightmost point of A
4=b
4
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
}
can be checked
3
5
2
5
a=2
6
7
1
1
0
0
A
in constant time
B
right turn or
left turn?
T is lower tangent if all the points are above the line
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
Split set into two, compute convex hull of
both, combine.
Convex Hull – Divide & Conquer
12

Split set into two, compute convex hull of
both, combine.
Convex Hull – Divide & Conquer
13

Split set into two, compute convex hull of
both, combine.
14

Split set into two, compute convex hull of
both, combine.
15

Split set into two, compute convex hull of
both, combine.
16

Split set into two, compute convex hull of
both, combine.
17

Split set into two, compute convex hull of
both, combine.
18

Split set into two, compute convex hull of
both, combine.
19

Split set into two, compute convex hull of
both, combine.
20
Split set into two, compute convex hull of
both, combine.
21

Split set into two, compute convex hull of
both, combine.
22
Merging two convex hulls.
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
Merging two convex hulls: (i) Find the lower
tangent.
24

Merging two convex hulls: (i) Find the lower
tangent.
25

Merging two convex hulls: (i) Find the lower
tangent.
26

Merging two convex hulls: (i) Find the lower
tangent.
27

Merging two convex hulls: (i) Find the lower
tangent.
28

Merging two convex hulls: (i) Find the lower
tangent.
29
Merging two convex hulls: (i) Find the lower
tangent.
30
Merging two convex hulls: (i) Find the lower
tangent.
31
Merging two convex hulls: (i) Find the lower
tangent.
32
Merging two convex hulls: (ii) Find the upper
tangent.
33
Merging two convex hulls: (ii) Find the upper
tangent.
34

Merging two convex hulls: (ii) Find the
upper tangent.
35

Merging two convex hulls: (ii) Find the
upper tangent.
36
Merging two convex hulls: (ii) Find the upper
tangent.
37
Merging two convex hulls: (ii) Find the upper
tangent.
38
Merging two convex hulls: (ii) Find the upper
tangent.
39

Merging two convex hulls: (iii) Eliminate
non-hull edges.
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Chapter 5
Decrease-and-Conquer
Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
Decrease-and-Conquer
1.
2.
3.

Reduce problem instance to smaller instance of
the same problem
Solve smaller instance
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 (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) log a n
= log 2 n
when a = 2
Insertion Sort
To 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