CS473  Algorithms I


 Wilfred Armstrong
 3 years ago
 Views:
Transcription
1 CS473  Algorithms I Lecture 4 The DivideandConquer Design Paradigm View in slideshow mode 1
2 Reminder: Merge Sort Input array A sort this half sort this half Divide Conquer merge two sorted halves Combine 2
3 The DivideandConquer Design Paradigm 1. Divide the problem (instance) into subproblems. 2. Conquer the subproblems by solving them recursively. 3. Combine subproblem solutions. CS473 Lecture 4 3
4 Example: Merge Sort 1. Divide: Trivial. 2. Conquer: Recursively sort 2 subarrays. 3. Combine: Linear time merge. T(n) = 2 T(n/2) + Θ(n) # subproblems subproblem size work dividing and combining CS473 Lecture 4 4
5 Master Theorem: Reminder T(n) = at(n/b) + f(n) Case 1: T(n) = Case 2: T(n) = Case 3: and a f (n/b) c f (n) for c < 1 T(n) = 5
6 Merge Sort: Solving the Recurrence T(n) = 2 T(n/2) + Θ(n) a = 2, b = 2, f(n) = Θ(n), n log ba = n Case 2: T(n) = holds for k = 0 T(n) = Θ (nlgn) 6
7 Binary Search Find an element in a sorted array: 1. Divide: Check middle element. 2. Conquer: Recursively search 1 subarray. 3. Combine: Trivial. Example: Find CS473 Lecture 4 7
8 Binary Search Find an element in a sorted array: 1. Divide: Check middle element. 2. Conquer: Recursively search 1 subarray. 3. Combine: Trivial. Example: Find CS473 Lecture 4 8
9 Binary Search Find an element in a sorted array: 1. Divide: Check middle element. 2. Conquer: Recursively search 1 subarray. 3. Combine: Trivial. Example: Find CS473 Lecture 4 9
10 Binary Search Find an element in a sorted array: 1. Divide: Check middle element. 2. Conquer: Recursively search 1 subarray. 3. Combine: Trivial. Example: Find CS473 Lecture 4 10
11 Binary Search Find an element in a sorted array: 1. Divide: Check middle element. 2. Conquer: Recursively search 1 subarray. 3. Combine: Trivial. Example: Find CS473 Lecture 4 11
12 Binary Search Find an element in a sorted array: 1. Divide: Check middle element. 2. Conquer: Recursively search 1 subarray. 3. Combine: Trivial. Example: Find CS473 Lecture 4 12
13 Recurrence for Binary Search T(n) = 1 T(n/2) + Ө(1) # subproblems subproblem size work dividing and combining CS473 Lecture 4 13
14 Binary Search: Solving the Recurrence T(n) = T(n/2) + Θ(1) a = 1, b = 2, f(n) = Θ(1), n log ba = n0 = 1 Case 2: T(n) = holds for k = 0 T(n) = Θ (lgn) 14
15 Powering a Number Problem: Compute a n, where n is a natural number NaivePower (a, n) powerval 1 for i 1 to n powerval powerval. a return powerval What is the complexity? T(n) = Θ (n) 15
16 Powering a Number: Divide & Conquer Basic idea: a n = a n/2. a n/2 even a (n1)/2. a (n1)/2. a if n is if n is odd 16
17 Powering a Number: Divide & Conquer POWER (a, n) if n = 0 then return 1 else if n is even then val POWER (a, n/2) return val * val else if n is odd then val POWER (a, (n1)/2) return val * val * a 17
18 Powering a Number: Solving the Recurrence T(n) = T(n/2) + Θ(1) a = 1, b = 2, f(n) = Θ(1), n log ba = n0 = 1 Case 2: T(n) = holds for k = 0 T(n) = Θ (lgn) 18
19 Matrix Multiplication Input : A = [a ij ], B = [b ij ]. Output: C = [c ij ] = A. B. i, j = 1, 2,..., n. c 11 c c 1n a 11 a a 1n b 11 b b 1n c 21 c c 2n = a 21 a a 2n. b 21 b b 2n c n1 c n2... c nn a n1 a n2... a nn b n1 b n2... b nn c ij = 1 k n a ik.b kj CS473 Lecture 4 19
20 Standard Algorithm for i 1 to n do for j 1 to n do c ij 0 for k 1 to n do c ij c ij + a ik. b kj Running time = Θ(n 3 ) CS473 Lecture 4 20
21 Matrix Multiplication: Divide & Conquer IDEA: Divide the n x n matrix into 2x2 matrix of (n/2)x(n/2) submatrices C A B c 11 =. c 12 a 11 a 12 b 11 b 12 c 21 c 22 a 21 a 22 b 21 b 22 c 11 = a 11 b 11 + a 12 b 21 21
22 Matrix Multiplication: Divide & Conquer IDEA: Divide the n x n matrix into 2x2 matrix of (n/2)x(n/2) submatrices C A B c 11 b 11 =. c 12 a 11 a 12 b 12 c 21 c 22 a 21 a 22 b 21 b 22 c 12 = a 11 b 12 + a 12 b 22 22
23 Matrix Multiplication: Divide & Conquer IDEA: Divide the n x n matrix into 2x2 matrix of (n/2)x(n/2) submatrices C A B c 11 a 11 a 12 =. c 12 b 11 b 12 c 21 c 22 a 21 a 22 b 21 b 22 c 21 = a 21 b 11 + a 22 b 21 23
24 Matrix Multiplication: Divide & Conquer IDEA: Divide the n x n matrix into 2x2 matrix of (n/2)x(n/2) submatrices C A B c 11 c 12 a 11 b 11 =. a 12 b 12 c 21 c 22 a 21 a 22 b 21 b 22 c 22 = a 21 b 12 + a 22 b 22 24
25 Matrix Multiplication: Divide & Conquer C A B c 11 c 12 a 11 a 12 b =. 11 b 12 c 21 c 22 a 21 a 22 b 21 b 22 c 11 = a 11 b 11 + a 12 b 21 c 12 = a 11 b 12 + a 12 b 22 c 21 = a 21 b 11 + a 22 b 21 8 mults of (n/2)x(n/2) submatrices 4 adds of (n/2)x(n/2) submatrices c 22 = a 21 b 12 + a 22 b 22 25
26 Matrix Multiplication: Divide & Conquer MATRIXMULTIPLY (A, B) // Assuming that both A and B are nxn matrices if n = 1 then return A * B else partition A, B, and C as shown before c 11 = MATRIXMULTIPLY (a 11, b 11 ) + MATRIXMULTIPLY (a 12, b 21 ) c 12 = MATRIXMULTIPLY (a 11, b 12 ) + MATRIXMULTIPLY (a 12, b 22 ) c 21 = MATRIXMULTIPLY (a 21, b 11 ) + MATRIXMULTIPLY (a 22, b 21 ) c 22 = MATRIXMULTIPLY (a 21, b 12 ) + MATRIXMULTIPLY (a 22, b 22 ) return C 26
27 Matrix Multiplication: Divide & Conquer Analysis T(n) = 8 T(n/2) + Θ(n 2 ) 8 recursive calls each subproblem has size n/2 submatrix addition 27
28 Matrix Multiplication: Solving the Recurrence T(n) = 8 T(n/2) + Θ(n 2 ) a = 8, b = 2, f(n) = Θ(n 2 ), n log ba = n 3 Case 1: T(n) = T(n) = Θ (n 3 ) No better than the ordinary algorithm! 28
29 Matrix Multiplication: Strassen s Idea C A B c 11 c 12 a 11 a 12 b =. 11 b 12 c 21 c 22 a 21 a 22 b 21 b 22 Compute c 11, c 12, c 21, and c 22 using 7 recursive multiplications 29
30 Matrix Multiplication: Strassen s Idea P 1 = a 11 x (b 12  b 22 ) P 2 = (a 11 + a 12 ) x b 22 P 3 = (a 21 + a 22 ) x b 11 P 4 = a 22 x (b 21  b 11 ) P 5 = (a 11 + a 22 ) x (b 11 + b 22 ) P 6 = (a 11  a 22 ) x (b 21 + b 22 ) P 7 = ( a 11  a 21 ) x (b 11 + b 12 ) Reminder: Each submatrix is of size (n/2)x(n/2) Each add/sub operation takes Θ(n 2 ) time Compute P 1..P 7 using 7 recursive calls to matrixmultiply How to compute c ij using P 1.. P 7? 30
31 Matrix Multiplication: Strassen s Idea P 1 = a 11 x (b 12  b 22 ) P 2 = (a 11 + a 12 ) x b 22 P 3 = (a 21 + a 22 ) x b 11 P 4 = a 22 x (b 21  b 11 ) P 5 = (a 11 + a 22 ) x (b 11 + b 22 ) P 6 = (a 11  a 22 ) x (b 21 + b 22 ) P 7 = ( a 11  a 21 ) x (b 11 + b 12 ) c 11 = P 5 + P 4 P 2 + P 6 c 12 = P 1 + P 2 c 21 = P 3 + P 4 c 22 = P 5 + P 1 P 3 P 7 7 recursive multiply calls 18 add/sub operations Does not rely on commutativity of multiplication 31
32 Matrix Multiplication: Strassen s Idea P 1 = a 11 x (b 12  b 22 ) P 2 = (a 11 + a 12 ) x b 22 P 3 = (a 21 + a 22 ) x b 11 P 4 = a 22 x (b 21  b 11 ) P 5 = (a 11 + a 22 ) x (b 11 + b 22 ) P 6 = (a 11  a 22 ) x (b 21 + b 22 ) P 7 = ( a 11  a 21 ) x (b 11 + b 12 ) e.g. Show that c 12 = P 1 +P 2 c 12 = P 1 + P 2 = a 11 (b 12 b 22 )+(a 11 +a 12 )b 22 = a 11 b 12 a 11 b 22 +a 11 b 22 +a 12 b 22 = a 11 b 12 +a 12 b 22 32
33 Strassen s Algorithm 1. Divide: Partition A and B into (n/2) x (n/2) submatrices. Form terms to be multiplied using + and. 2. Conquer: Perform 7 multiplications of (n/2) x (n/2) submatrices recursively. 3. Combine: Form C using + and on (n/2) x (n/2) submatrices. Recurrence: T(n) = 7 T(n/2) + Ө(n 2 ) 33
34 Strassen s Algorithm: Solving the Recurrence T(n) = 7 T(n/2) + Θ(n 2 ) a = 7, b = 2, f(n) = Θ(n 2 ), n log ba = n lg7 Case 1: T(n) = T(n) = Θ (n lg7 ) Note: lg
35 Strassen s Algorithm The number 2.81 may not seem much smaller than 3 But, it is significant because the difference is in the exponent. Strassen s algorithm beats the ordinary algorithm on today s machines for n 30 or so. Best to date: Θ(n ) (of theoretical interest only) 35
36 VLSI Layout: Binary Tree Embedding Problem: Embed a complete binary tree with n leaves into a 2D grid with minimum area. Example: 36
37 Binary Tree Embedding Use divide and conquer root LEFT SUBTREE RIGHT SUBTREE 1. Embed the root node 2. Embed the left subtree 3. Embed the right subtree What is the minarea required for n leaves? 37
38 Binary Tree Embedding root H(n/2) EMBED LEFT SUBTREE HERE EMBED RIGHT SUBTREE HERE H(n) = H(n/2) + 1 W(n/2) W(n/2) W(n) = 2W(n/2)
39 Binary Tree Embedding Solve the recurrences: W(n) = 2W(n/2) + 1 H(n) = H(n/2) + 1 W(n) = Ө(n) H(n) = Ө(lgn) Area(n) = Ө(nlgn) 39
40 Binary Tree Embedding Example: W(n) H(n) 40
41 Binary Tree Embedding: HTree Use a different divide and conquer method root left subtree 1 subtree 2 subtree 3 right subtree 4 1. Embed root, left, right nodes 2. Embed subtree 1 3. Embed subtree 2 4. Embed subtree 3 5. Embed subtree 4 What is the minarea required for n leaves? 41
42 Binary Tree Embedding: HTree H(n/4) H(n/4) SUBTREE 1 SUBTREE 2 SUBTREE 3 SUBTREE 4 W(n) = 2W(n/4) + 1 H(n) = 2H(n/4) + 1 W(n/4) W(n/4) 42
43 Binary Tree Embedding: HTree Solve the recurrences: W(n) = 2W(n/4) + 1 H(n) = 2H(n/4) + 1 W(n) = Ө( ) H(n) = Ө( ) Area(n) = Ө(n) 43
44 Binary Tree Embedding: HTree Example: W(n) H(n) 44
45 Correctness Proofs Proof by induction commonly used for D&C algorithms Base case: Show that the algorithm is correct when the recursion bottoms out (i.e., for sufficiently small n) Inductive hypothesis: Assume the alg. is correct for any recursive call on any smaller subproblem of size k (k < n) General case: Based on the inductive hypothesis, prove that the alg. is correct for any input of size n 45
46 Example Correctness Proof: Powering a Number POWER (a, n) if n = 0 then return 1 else if n is even then val POWER (a, n/2) return val * val else if n is odd then val POWER (a, (n1)/2) return val * val * a 46
47 Example Correctness Proof: Powering a Number Base case: POWER (a, 0) is correct, because it returns 1 Ind. hyp: Assume POWER (a, k) is correct for any k < n General case: In POWER (a, n) function: If n is even: val = a n/2 (due to ind. hyp.) it returns val. val = a n If n is odd: val = a (n1)/2 (due to ind. hyp.) it returns val. val. a = a n The correctness proof is complete 47
48 Maximum Subarray Problem Input: An array of values Output: The contiguous subarray that has the largest sum of elements Input array: the maximum contiguous subarray 48
49 Maximum Subarray Problem: Divide & Conquer Basic idea: Divide the input array into 2 from the middle Pick the best solution among the following: 1. The max subarray of the left half 2. The max subarray of the right half 3. The max subarray crossing the midpoint Crosses the midpoint A Entirely in the left half Entirely in the right half 49
50 Maximum Subarray Problem: Divide & Conquer Divide: Trivial (divide the array from the middle) Conquer: Recursively compute the max subarrays of the left and right halves Combine: Compute the maxsubarray crossing the midpoint (can be done in Θ(n) time). Return the max among the following: 1. the max subarray of the left subarray 2. the max subarray of the right subarray 3. the max subarray crossing the midpoint See textbook for the detailed solution. 50
51 Conclusion Divide and conquer is just one of several powerful techniques for algorithm design. Divideandconquer algorithms can be analyzed using recurrences and the master method (so practice this math). Can lead to more efficient algorithms CS473 Lecture 4 51
Divide And Conquer Algorithms
CSE341T/CSE549T 09/10/2014 Lecture 5 Divide And Conquer Algorithms Recall in last lecture, we looked at one way of parallelizing matrix multiplication. At the end of the lecture, we saw the reduce SUM
More informationCS473  Algorithms I
CS473  Algorithms I Lecture 9 Sorting in Linear Time View in slideshow mode 1 How Fast Can We Sort? The algorithms we have seen so far: Based on comparison of elements We only care about the relative
More informationMany algorithms, particularly divide and conquer algorithms, have time complexities which are naturally
Recurrence Relations Many algorithms, particularly divide and conquer algorithms, have time complexities which are naturally modeled by recurrence relations. A recurrence relation is an equation which
More informationBinary Search. Search for x in a sorted array A.
Divide and Conquer A general paradigm for algorithm design; inspired by emperors and colonizers. Threestep process: 1. Divide the problem into smaller problems. 2. Conquer by solving these problems. 3.
More informationIntro. to the DivideandConquer Strategy via Merge Sort CMPSC 465 CLRS Sections 2.3, Intro. to and various parts of Chapter 4
Intro. to the DivideandConquer Strategy via Merge Sort CMPSC 465 CLRS Sections 2.3, Intro. to and various parts of Chapter 4 I. Algorithm Design and DivideandConquer There are various strategies we
More informationThe PRAM Model and Algorithms. Advanced Topics Spring 2008 Prof. Robert van Engelen
The PRAM Model and Algorithms Advanced Topics Spring 2008 Prof. Robert van Engelen Overview The PRAM model of parallel computation Simulations between PRAM models Worktime presentation framework of parallel
More informationMerge Sort. 2004 Goodrich, Tamassia. Merge Sort 1
Merge Sort 7 2 9 4 2 4 7 9 7 2 2 7 9 4 4 9 7 7 2 2 9 9 4 4 Merge Sort 1 DivideandConquer Divideand conquer is a general algorithm design paradigm: Divide: divide the input data S in two disjoint subsets
More informationMatrixChain Multiplication
MatrixChain Multiplication Let A be an n by m matrix, let B be an m by p matrix, then C = AB is an n by p matrix. C = AB can be computed in O(nmp) time, using traditional matrix multiplication. Suppose
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
More informationFormat of DivideandConquer algorithms:
CS 360: Data Structures and Algorithms DivideandConquer (part 1) Format of DivideandConquer algorithms: Divide: Split the array or list into smaller pieces Conquer: Solve the same problem recursively
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationSorting revisited. Build the binary search tree: O(n^2) Traverse the binary tree: O(n) Total: O(n^2) + O(n) = O(n^2)
Sorting revisited How did we use a binary search tree to sort an array of elements? Tree Sort Algorithm Given: An array of elements to sort 1. Build a binary search tree out of the elements 2. Traverse
More informationCSC148 Lecture 8. Algorithm Analysis Binary Search Sorting
CSC148 Lecture 8 Algorithm Analysis Binary Search Sorting Algorithm Analysis Recall definition of Big Oh: We say a function f(n) is O(g(n)) if there exists positive constants c,b such that f(n)
More informationFUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM
International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 3448 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT
More informationDivide and Conquer. Textbook Reading Chapters 4, 7 & 33.4
Divide d Conquer Textook Reading Chapters 4, 7 & 33.4 Overview Design principle Divide d conquer Proof technique Induction, induction, induction Analysis technique Recurrence relations Prolems Sorting
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
More informationMathematical Induction. Lecture 1011
Mathematical Induction Lecture 1011 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationThe Tower of Hanoi. Recursion Solution. Recursive Function. Time Complexity. Recursive Thinking. Why Recursion? n! = n* (n1)!
The Tower of Hanoi Recursion Solution recursion recursion recursion Recursive Thinking: ignore everything but the bottom disk. 1 2 Recursive Function Time Complexity Hanoi (n, src, dest, temp): If (n >
More informationParallel Random Access Machine (PRAM) PRAM Algorithms. Shared Memory Access Conflicts. A Basic PRAM Algorithm. Time Optimality.
Parallel Random Access Machine (PRAM) PRAM Algorithms Arvind Krishnamurthy Fall 2 Collection of numbered processors Accessing shared memory cells Each processor could have local memory (registers) Each
More informationOPTIMUM PARTITION PARAMETER OF DIVIDEANDCONQUER ALGORITHM FOR SOLVING CLOSESTPAIR PROBLEM
OPTIMUM PARTITION PARAMETER OF DIVIDEANDCONQUER ALGORITHM FOR SOLVING CLOSESTPAIR PROBLEM Mohammad Zaidul Karim 1 and Nargis Akter 2 1 Department of Computer Science and Engineering, Daffodil International
More informationMergesort and Quicksort
Mergesort Mergesort and Quicksort Basic plan: Divide array into two halves. Recursively sort each half. Merge two halves to make sorted whole. mergesort mergesort analysis quicksort quicksort analysis
More informationClosest Pair Problem
Closest Pair Problem Given n points in ddimensions, find two whose mutual distance is smallest. Fundamental problem in many applications as well as a key step in many algorithms. p q A naive algorithm
More informationAnalysis of Algorithms I: Optimal Binary Search Trees
Analysis of Algorithms I: Optimal Binary Search Trees Xi Chen Columbia University Given a set of n keys K = {k 1,..., k n } in sorted order: k 1 < k 2 < < k n we wish to build an optimal binary search
More informationCSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis. Linda Shapiro Winter 2015
CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis Linda Shapiro Today Registration should be done. Homework 1 due 11:59 pm next Wednesday, January 14 Review math essential
More informationIntroduction to Algorithms March 10, 2004 Massachusetts Institute of Technology Professors Erik Demaine and Shafi Goldwasser Quiz 1.
Introduction to Algorithms March 10, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Quiz 1 Quiz 1 Do not open this quiz booklet until you are directed
More information2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]
Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)
More information8.1 Makespan Scheduling
600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programing: MinMakespan and Bin Packing Date: 2/19/15 Scribe: Gabriel Kaptchuk 8.1 Makespan Scheduling Consider an instance
More informationQuicksort is a divideandconquer sorting algorithm in which division is dynamically carried out (as opposed to static division in Mergesort).
Chapter 7: Quicksort Quicksort is a divideandconquer sorting algorithm in which division is dynamically carried out (as opposed to static division in Mergesort). The three steps of Quicksort are as follows:
More informationWeek 7: Divide and Conquer
Agenda: Divide and Conquer technique Multiplication of large integers Exponentiation Matrix multiplication 1 2 Divide and Conquer : To solve a problem we can break it into smaller subproblems, solve each
More informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
More informationQuiz 1 Solutions. (a) T F The height of any binary search tree with n nodes is O(log n). Explain:
Introduction to Algorithms March 9, 2011 Massachusetts Institute of Technology 6.006 Spring 2011 Professors Erik Demaine, Piotr Indyk, and Manolis Kellis Quiz 1 Solutions Problem 1. Quiz 1 Solutions True
More informationCMPS 102 Solutions to Homework 1
CMPS 0 Solutions to Homework Lindsay Brown, lbrown@soe.ucsc.edu September 9, 005 Problem.. p. 3 For inputs of size n insertion sort runs in 8n steps, while merge sort runs in 64n lg n steps. For which
More informationCS 102: SOLUTIONS TO DIVIDE AND CONQUER ALGORITHMS (ASSGN 4)
CS 10: SOLUTIONS TO DIVIDE AND CONQUER ALGORITHMS (ASSGN 4) Problem 1. a. Consider the modified binary search algorithm so that it splits the input not into two sets of almostequal sizes, but into three
More information1/1 7/4 2/2 12/7 10/30 12/25
Binary Heaps A binary heap is dened to be a binary tree with a key in each node such that: 1. All leaves are on, at most, two adjacent levels. 2. All leaves on the lowest level occur to the left, and all
More informationDivideandConquer Algorithms Part Four
DivideandConquer Algorithms Part Four Announcements Problem Set 2 due right now. Can submit by Monday at 2:15PM using one late period. Problem Set 3 out, due July 22. Play around with divideandconquer
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationOutline BST Operations Worst case Average case Balancing AVL Redblack Btrees. Binary Search Trees. Lecturer: Georgy Gimel farb
Binary Search Trees Lecturer: Georgy Gimel farb COMPSCI 220 Algorithms and Data Structures 1 / 27 1 Properties of Binary Search Trees 2 Basic BST operations The worstcase time complexity of BST operations
More informationZabin Visram Room CS115 CS126 Searching. Binary Search
Zabin Visram Room CS115 CS126 Searching Binary Search Binary Search Sequential search is not efficient for large lists as it searches half the list, on average Another search algorithm Binary search Very
More informationDivideandconquer algorithms
Chapter 2 Divideandconquer algorithms The divideandconquer strategy solves a problem by: 1 Breaking it into subproblems that are themselves smaller instances of the same type of problem 2 Recursively
More informationDivideandConquer. Three main steps : 1. divide; 2. conquer; 3. merge.
DivideandConquer Three main steps : 1. divide; 2. conquer; 3. merge. 1 Let I denote the (sub)problem instance and S be its solution. The divideandconquer strategy can be described as follows. Procedure
More informationIntroduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24
Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24 The final exam will cover seven topics. 1. greedy algorithms 2. divideandconquer algorithms
More informationRecursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1
Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 717 of Rosen cse35@cseunledu Recursive Algorithms
More informationUnit 4: Dynamic Programming
Unit 4: Dynamic Programming Course contents: Assemblyline scheduling Matrixchain multiplication Longest common subsequence Optimal binary search trees Applications: Rod cutting, optimal polygon triangulation,
More informationWhy? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationAnalysis of Algorithms I: Binary Search Trees
Analysis of Algorithms I: Binary Search Trees Xi Chen Columbia University Hash table: A data structure that maintains a subset of keys from a universe set U = {0, 1,..., p 1} and supports all three dictionary
More informationMath 313 Lecture #10 2.2: The Inverse of a Matrix
Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is
More information15750 Graduate Algorithms Spring 2010
15750 Graduate Algorithms Spring 2010 1 Cheap Cut (25 pts.) Suppose we are given a directed network G = (V, E) with a root node r and a set of terminal nodes T V. We would like to remove the fewest number
More informationScheduling a sequence of tasks with general completion costs
Scheduling a sequence of tasks with general completion costs Francis Sourd CNRSLIP6 4, place Jussieu 75252 Paris Cedex 05, France Francis.Sourd@lip6.fr Abstract Scheduling a sequence of tasks in the acceptation
More informationBinary Heap Algorithms
CS Data Structures and Algorithms Lecture Slides Wednesday, April 5, 2009 Glenn G. Chappell Department of Computer Science University of Alaska Fairbanks CHAPPELLG@member.ams.org 2005 2009 Glenn G. Chappell
More informationThe UnionFind Problem Kruskal s algorithm for finding an MST presented us with a problem in datastructure design. As we looked at each edge,
The UnionFind Problem Kruskal s algorithm for finding an MST presented us with a problem in datastructure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints
More informationA binary heap is a complete binary tree, where each node has a higher priority than its children. This is called heaporder property
CmSc 250 Intro to Algorithms Chapter 6. Transform and Conquer Binary Heaps 1. Definition A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called
More informationFrom Last Time: Remove (Delete) Operation
CSE 32 Lecture : More on Search Trees Today s Topics: Lazy Operations Run Time Analysis of Binary Search Tree Operations Balanced Search Trees AVL Trees and Rotations Covered in Chapter of the text From
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationDynamic Programming. Lecture 11. 11.1 Overview. 11.2 Introduction
Lecture 11 Dynamic Programming 11.1 Overview Dynamic Programming is a powerful technique that allows one to solve many different types of problems in time O(n 2 ) or O(n 3 ) for which a naive approach
More informationcsci 210: Data Structures Recursion
csci 210: Data Structures Recursion Summary Topics recursion overview simple examples Sierpinski gasket Hanoi towers Blob check READING: GT textbook chapter 3.5 Recursion In general, a method of defining
More information5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the FordFulkerson
More informationLecture 12: Chain Matrix Multiplication
Lecture 12: Chain Matrix Multiplication CLRS Section 15.2 Outline of this Lecture Recalling matrix multiplication. The chain matrix multiplication problem. A dynamic programming algorithm for chain matrix
More informationConverting a Number from Decimal to Binary
Converting a Number from Decimal to Binary Convert nonnegative integer in decimal format (base 10) into equivalent binary number (base 2) Rightmost bit of x Remainder of x after division by two Recursive
More informationNotes: Chapter 2 Section 2.2: Proof by Induction
Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case  S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example
More informationLecture 1: Course overview, circuits, and formulas
Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik
More informationResearch Tools & Techniques
Research Tools & Techniques for Computer Engineering Ron Sass http://www.rcs.uncc.edu/ rsass University of North Carolina at Charlotte Fall 2009 1/ 106 Overview of Research Tools & Techniques Course What
More informationPartitioning and Divide and Conquer Strategies
and Divide and Conquer Strategies Lecture 4 and Strategies Strategies Data partitioning aka domain decomposition Functional decomposition Lecture 4 and Strategies Quiz 4.1 For nuclear reactor simulation,
More informationSolutions to Homework 6
Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationBinary Search Trees. Data in each node. Larger than the data in its left child Smaller than the data in its right child
Binary Search Trees Data in each node Larger than the data in its left child Smaller than the data in its right child FIGURE 116 Arbitrary binary tree FIGURE 117 Binary search tree Data Structures Using
More informationRandomized algorithms
Randomized algorithms March 10, 2005 1 What are randomized algorithms? Algorithms which use random numbers to make decisions during the executions of the algorithm. Why would we want to do this?? Deterministic
More informationData Structures and Algorithms Written Examination
Data Structures and Algorithms Written Examination 22 February 2013 FIRST NAME STUDENT NUMBER LAST NAME SIGNATURE Instructions for students: Write First Name, Last Name, Student Number and Signature where
More informationB such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix
Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More information1 Determinants. Definition 1
Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described
More informationBinary Search Trees CMPSC 122
Binary Search Trees CMPSC 122 Note: This notes packet has significant overlap with the first set of trees notes I do in CMPSC 360, but goes into much greater depth on turning BSTs into pseudocode than
More informationBinary Multiplication
Binary Multiplication Q: How do we multiply two numbers? eg. 12, 345 6, 789 111105 987600 8641500 + 74070000 83, 810, 205 10111 10101 10111 00000 1011100 0000000 + 101110000 111100011 Pad, multiply and
More informationAlgebraic and Combinatorial Circuits
C H A P T E R Algebraic and Combinatorial Circuits Algebraic circuits combine operations drawn from an algebraic system. In this chapter we develop algebraic and combinatorial circuits for a variety of
More informationUnit 18 Determinants
Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of
More information, each of which contains a unique key value, say k i , R 2. such that k i equals K (or to determine that no such record exists in the collection).
The Search Problem 1 Suppose we have a collection of records, say R 1, R 2,, R N, each of which contains a unique key value, say k i. Given a particular key value, K, the search problem is to locate the
More informationAlgorithms Chapter 12 Binary Search Trees
Algorithms Chapter 1 Binary Search Trees Outline Assistant Professor: Ching Chi Lin 林 清 池 助 理 教 授 chingchi.lin@gmail.com Department of Computer Science and Engineering National Taiwan Ocean University
More informationWarshall s Algorithm: Transitive Closure
CS 0 Theory of Algorithms / CS 68 Algorithms in Bioinformaticsi Dynamic Programming Part II. Warshall s Algorithm: Transitive Closure Computes the transitive closure of a relation (Alternatively: all paths
More informationBinary Heaps * * * * * * * / / \ / \ / \ / \ / \ * * * * * * * * * * * / / \ / \ / / \ / \ * * * * * * * * * *
Binary Heaps A binary heap is another data structure. It implements a priority queue. Priority Queue has the following operations: isempty add (with priority) remove (highest priority) peek (at highest
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationLecture 18: Applications of Dynamic Programming Steven Skiena. Department of Computer Science State University of New York Stony Brook, NY 11794 4400
Lecture 18: Applications of Dynamic Programming Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Problem of the Day
More informationPolynomials and the Fast Fourier Transform (FFT) Battle Plan
Polynomials and the Fast Fourier Transform (FFT) Algorithm Design and Analysis (Wee 7) 1 Polynomials Battle Plan Algorithms to add, multiply and evaluate polynomials Coefficient and pointvalue representation
More informationAnalysis of Algorithms, I
Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadthfirst search (BFS) 4 Applications
More informationCS335 Program2: ConvexHulls A DivideandConquer Algorithm 15 points Due: Tues, October 6
CS335 Program2: ConvexHulls A DivideandConquer Algorithm 15 points Due: Tues, October 6 Convex Hulls There is a complex mathematical definition of a convex hull, but the informal definition is sufficient
More informationComputing exponents modulo a number: Repeated squaring
Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method
More informationCS 575 Parallel Processing
CS 575 Parallel Processing Lecture one: Introduction Wim Bohm Colorado State University Except as otherwise noted, the content of this presentation is licensed under the Creative Commons Attribution 2.5
More informationFast Fourier Transform: Theory and Algorithms
Fast Fourier Transform: Theory and Algorithms Lecture Vladimir Stojanović 6.973 Communication System Design Spring 006 Massachusetts Institute of Technology Discrete Fourier Transform A review Definition
More informationA Branch and Bound Algorithm for Solving the Binary Bilevel Linear Programming Problem
A Branch and Bound Algorithm for Solving the Binary Bilevel Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationData Structures. Algorithm Performance and Big O Analysis
Data Structures Algorithm Performance and Big O Analysis What s an Algorithm? a clearly specified set of instructions to be followed to solve a problem. In essence: A computer program. In detail: Defined
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More informationMathematical Induction
Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers
More informationUNIT 5C Merge Sort. Course Announcements
UNIT 5C Merge Sort 1 Course Announcements Exam rooms for Lecture 1, 2:30 3:20 Sections A, B, C, D at Rashid Sections E, F, G at Baker A51 (Giant Eagle Auditorium) Exam rooms for Lecture 2, 3:30 4:20 Sections
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
More informationAppendix: Solving Recurrences [Fa 10] Wil Wheaton: Embrace the dark side! Sheldon: That s not even from your franchise!
Change is certain. Peace is followed by disturbances; departure of evil men by their return. Such recurrences should not constitute occasions for sadness but realities for awareness, so that one may be
More informationLecture 13: The Knapsack Problem
Lecture 13: The Knapsack Problem Outline of this Lecture Introduction of the 01 Knapsack Problem. A dynamic programming solution to this problem. 1 01 Knapsack Problem Informal Description: We have computed
More informationThe Running Time of Programs
CHAPTER 3 The Running Time of Programs In Chapter 2, we saw two radically different algorithms for sorting: selection sort and merge sort. There are, in fact, scores of algorithms for sorting. This situation
More informationBinary search algorithm
Binary search algorithm Definition Search a sorted array by repeatedly dividing the search interval in half. Begin with an interval covering the whole array. If the value of the search key is less than
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationROUTING? Sorting? Image Processing? Sparse Matrices? Reconfigurable Meshes! P A R C
ROUTING? Sorting? Image Processing? Sparse Matrices? Reconfigurable Meshes! P A R C Heiko Schröder, 1998 Reconfigurable architectures FPGAs reconfigurable multibus reconfigurable networks (Transputers,
More information