CSC 380: Design and Analysis of Algorithms Dr. Curry Guinn

Quick Info Dr. Curry Guinn CIS 2015 guinnc@uncw.edu www.uncw.edu/people/guinnc 962-7937 Office Hours: MWF: 10:00am-11:00m and by appointment

Today Divide-and-Conquer Transform-and-Conquer Multiplication of large integers Closest pair problem Transform-and-Conquer

Divide-and-Conquer Examples Sorting: mergesort and quicksort Binary tree traversals Multiplication of large integers Matrix multiplication: Strassen’s algorithm Closest-pair and convex-hull algorithms Binary search: decrease-by-half (or degenerate divide&conq.) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Multiplication of Large Integers Consider the problem of multiplying two (large) n-digit integers represented by arrays of their digits such as: A = 12345678901357986429 B = 87654321284820912836 The grade-school algorithm: a1 a2 … an b1 b2 … bn (d10) d11d12 … d1n (d20) d21d22 … d2n … … … … … … … (dn0) dn1dn2 … dnn Efficiency: n2 one-digit multiplications

Multiplying Integers x = 61,438,521 y = 94,736,407 Typical algorithm is N2 Use divide and conquer!

Multiplying Integers x = 61,438,521 y = 94,736,407 X = XL104+XR YL=9,473 YR=6,407 X = XL104+XR Y = YL104+YR XY = XLYL108 + (XLYR + XRYL)104 + XRYR T(N) = 4T(N/2) + O(N) = O(N2)

Multiplying Integers XY = XLYL108 + (XLYR + XRYL)104 + XRYR (XL – XR)(YR-YL)+XLYL+XRYR Use 1 multiplication plus 2 results already computed T(N) = 3T(N/2) + O(N) T(N) = O(Nlog23) = O(N1.59)

Example of Large-Integer Multiplication 2135  4014 A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Strassen’s Matrix Multiplication Strassen observed [1969] that the product of two matrices can be computed as follows: C00 C01 A00 A01 B00 B01 = * C10 C11 A10 A11 B10 B11 M1 + M4 - M5 + M7 M3 + M5 = M2 + M4 M1 + M3 - M2 + M6 A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Analysis of Strassen’s Algorithm Number of multiplications: M(n) = 7M(n/2), M(1) = 1 Solution: M(n) = 7log 2n = nlog 27 ≈ n2.807 vs. n3 of brute- force alg. Algorithms with better asymptotic efficiency are known but they are even more complex. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Closest-Points Problem p1 = (x1, y1) p2 = (x2, y2) Euclidean distance between p1 and p2 is [(x1-x2)2 + (y1-y2)2]1/2 Goal: Given N points, find closest pair Brute force algorithm?

Closest-Points Example

Closest-Points Problem Brute force algorithm? O(N2) Divide and Conquer similar to maximum subsequence sum sort by X coordinate find closest two in left half (dL) find closest two in right half (dR) find closest two where 1 is in left, 1 in right (dC) closest is minimum

Closest-Points Example dC dR dL

Closest-Points Problem find closest two where 1 is in left, 1 in right (dC) To ensure O(NlogN) solution, this step must be done in O(N) time d = min(dL, dR) Only need to look at points d from center Still cannot do brute force on these points – what would be running time? Observe: for each point, only need to consider points that are less than d away in y coordinate There are a maximum of 5 points to consider Points must be separated by at least d Don't need to look at points that have already been processed

Example: Worst Case

Efficiency of the Closest-Pair Algorithm Running time of the algorithm is described by T(n) = 2T(n/2) + M(n), where M(n)  O(n) By the Master Theorem (with a = 2, b = 2, d = 1) T(n)  O(n log n) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 5 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Transform and Conquer This group of techniques solves a problem by a transformation to a simpler/more convenient instance of the same problem (instance simplification) a different representation of the same instance (representation change) a different problem for which an algorithm is already available (problem reduction) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 6 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Instance simplification - Presorting Solve a problem’s instance by transforming it into another simpler/easier instance of the same problem Presorting Many problems involving lists are easier when list is sorted, e.g. searching computing the median (selection problem) checking if all elements are distinct (element uniqueness) Also: Topological sorting helps solving some problems for dags. Presorting is used in many geometric algorithms. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 6 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

How fast can we sort ? Efficiency of algorithms involving sorting depends on efficiency of sorting. Theorem (see Sec. 11.2): log2 n!  n log2 n comparisons are necessary in the worst case to sort a list of size n by any comparison-based algorithm. Note: About nlog2 n comparisons are also sufficient to sort array of size n (by mergesort). A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 6 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Searching with presorting Problem: Search for a given K in A[0..n-1] Presorting-based algorithm: Stage 1 Sort the array by an efficient sorting algorithm Stage 2 Apply binary search Efficiency: Θ(nlog n) + O(log n) = Θ(nlog n) Good or bad? Why do we have our dictionaries, telephone directories, etc. sorted? A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 6 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Element Uniqueness with presorting Presorting-based algorithm Stage 1: sort by efficient sorting algorithm (e.g. mergesort) Stage 2: scan array to check pairs of adjacent elements Efficiency: Θ(nlog n) + O(n) = Θ(nlog n) Brute force algorithm Compare all pairs of elements Efficiency: O(n2) Another algorithm? Hashing A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 6 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Taxonomy of Searching Algorithms List searching sequential search binary search interpolation search Tree searching binary search tree binary balanced trees: AVL trees, red-black trees multiway balanced trees: 2-3 trees, 2-3-4 trees, B trees Hashing open hashing (separate chaining) closed hashing (open addressing) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 6 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

For Next Class, Wednesday Homework 5 due Tuesday after Spring Break