1. Introduction to Algorithms
Rabie A. Ramadan
rabieramadan.org 6
Ack : Carola Wenk nad Dr. Thomas Ottmann tutorials

2. 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.

3. 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 sequence of points on the convex hull polygon, in counter-clockwise order. 2 1 3 4 6 5

4. 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 )

5. 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

6. 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)

7. 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

8. Convex Hull: Runtime 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)

9. Convex Hull: Runtime T(n) = 2 T(n/2) + n Solves to T(n) = (n log n)
Runtime Recurrence:
T(n) = 2 T(n/2) + n
Solves to T(n) = (n log n)

10. Merging in O(n) time Find upper and lower tangents in O(n) time4
Find upper and lower tangents in O(n) time
Compute the convex hull of AB:
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 5 3 6 2 1 7 A B

11. Finding the lower tangent in O(n) time3
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 } while T not lower tangent to convex hull of B do{ b=b } } 4
4=b 2 3 5 5
a=2 6 1 7 1
can be checked in constant time
right turn or left turn?
Note that to determine if some line, T = AB, is a lower tangent to the left hull you only need to check the 2 points on either side of point A (A-1 and A+1). If both these points are above the line T then all the points in the left hull are above the line T. That result comes from the fact that the hulls are convex. Note that you have to be careful about accessing the A-1 and A+1 elements so you don’t go out of the array bounds. A B
T is lower tangent if all the points are above the line

12. Convex Hull – Divide & Conquer
Split set into two, compute convex hull of both, combine.
Convex Hull – Divide & Conquer

13. Convex Hull – Divide & Conquer
Split set into two, compute convex hull of both, combine.
Convex Hull – Divide & Conquer

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. Split set into two, compute convex hull of both, combine.

23. Merging two convex hulls.

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: (i) Find the lower 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: (ii) Find the upper tangent.

40. Merging two convex hulls: (iii) Eliminate non-hull edges.

41. Chapter 5 Decrease-and-Conquer
Copyright © 2007 Pearson Addison-Wesley. All rights reserved.

42. Decrease-and-Conquer
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

43. 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.

44. 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
Try to get the students involved in coming up with these:
Brute Force:
an= a*a*a*a*...*a n
Divide and conquer:
an= an/2 * an/2 (more accurately, an= an/2 * a n/2│)
Decrease by one:
an= an-1* a (one hopes a student will ask what is the difference with brute force here:
there is none in the resulting algorithm, of course, but you can arrive
at it in two different ways)
Decrease by constant factor:
an= (an/2) (again, if no student asks about it, be sure to point out the difference
with divide and conquer. Here there is a significant difference that leads to a
much more efficient algorithm – in divide and conquer we recompute an/2
T(n) = T(n/a) + a-1
= (a-1) n
= when a = 2

45. 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, |
4 6 |
| 8 5
| 5

46. Write a Pseudocode for Insertion Sort

47. 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