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

Chapter 4 Divide-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

Brute force Solution 4 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(n 3 )

Quickhull Algorithm Convex hull: smallest convex set that includes given points. Assume points are sorted by x-coordinate values Identify extreme points P 1 and P 2 (leftmost and rightmost) Compute upper hull recursively: find point P max that is farthest away from line P 1 P 2 compute the upper hull of the points to the left of line P 1 P max compute the upper hull of the points to the left of line P max P 2 Compute lower hull in a similar manner P1P1 P2P2 P max

QuickHull Algorithm 6 How to find the P max point How to find the P max point P max maximizes the area of the triangle P max P 1 P 2 if tie, select the P max that maximizes the angle P max P 1 P 2 The points inside triangle P max P 1 P 2 can be excluded from further consideration Worst case (almost like quick sort) : O(n 2 )

Convex Hull: Divide & Conquer 7  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 8  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) O(n)

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

Merging in O(n) time 10  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 A B

Finding the lower tangent in O(n) time 11 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 0 a= =b T is lower tangent if all the points are above the line

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 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.

Chapter 5 Decrease-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 (usually by 1): 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 x n 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

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)Video Example: Sort 6, 4, 1, 8, 5 6 | | | |

Write a Pseudocode for Insertion Sort

Analysis of Insertion Sort Time efficiency C worst (n) = n(n-1)/2  Θ(n 2 ) C best (n) = n - 1  Θ(n) (also fast on almost sorted arrays) Space efficiency: in-place Best elementary sorting algorithm overall