Introduction to Algorithms 6.046J/18.401J/SMA5503
|
|
|
- Adam Allison
- 5 years ago
- Views:
Transcription
1 Introduction to Algorithms 6.046J/8.40J/SMA5503 Lecture 3 Prof. Piotr Indyk
2 The divide-and-conquer design paradigm. Divide the problem (instance) into subproblems. 2. Conquer the subproblems by solving them recursively. 3. Combine subproblem solutions. February, 2003 (c) Charles Leiserson and Piotr Indyk L3.2
3 Example: merge sort. Divide: Trivial. 2. Conquer: Recursively sort 2 subarrays. 3. Combine: Linear-time merge. T(n) = 2 T(n/2) + O(n) # subproblems subproblem size work dividing and combining February, 2003 (c) Charles Leiserson and Piotr Indyk L3.3
4 Binary search Find an element in a sorted array:. Divide: Check middle element. 2. Conquer: Recursively search subarray. 3. Combine: Trivial. Example: Find February, 2003 (c) Charles Leiserson and Piotr Indyk L3.4
5 Binary search Find an element in a sorted array:. Divide: Check middle element. 2. Conquer: Recursively search subarray. 3. Combine: Trivial. Example: Find February, 2003 (c) Charles Leiserson and Piotr Indyk L3.5
6 Binary search Find an element in a sorted array:. Divide: Check middle element. 2. Conquer: Recursively search subarray. 3. Combine: Trivial. Example: Find February, 2003 (c) Charles Leiserson and Piotr Indyk L3.6
7 Binary search Find an element in a sorted array:. Divide: Check middle element. 2. Conquer: Recursively search subarray. 3. Combine: Trivial. Example: Find February, 2003 (c) Charles Leiserson and Piotr Indyk L3.7
8 Binary search Find an element in a sorted array:. Divide: Check middle element. 2. Conquer: Recursively search subarray. 3. Combine: Trivial. Example: Find February, 2003 (c) Charles Leiserson and Piotr Indyk L3.8
9 Binary search Find an element in a sorted array:. Divide: Check middle element. 2. Conquer: Recursively search subarray. 3. Combine: Trivial. Example: Find February, 2003 (c) Charles Leiserson and Piotr Indyk L3.9
10 Recurrence for binary search T(n) = T(n/2) + Θ() # subproblems subproblem size work dividing and combining n log ba = n log 2 = n 0 = CASE 2 (k = 0) T(n) = Θ(lg n). February, 2003 (c) Charles Leiserson and Piotr Indyk L3.0
11 Powering a number Problem: Compute a n, where n N. Naive algorithm: Θ(n). Divide-and-conquer algorithm: a n = a n/2 a n/2 a (n )/2 a (n )/2 a if n is even; if n is odd. T(n) = T(n/2) + Θ() T(n) = Θ(lg n). February, 2003 (c) Charles Leiserson and Piotr Indyk L3.
12 Polynomial multiplication Input: a(x)=a 0 +a x+ +a n x n, b(x)=b 0 +b x+ +b n x n, Output: c(x)= a(x)*b(x) =c 0 +c x+ +c 2n x 2n c i =a 0 b i +a b i- + + a i- b +a i b 0 Example: (a 0 +a x) * (b 0 +b x) = a 0 b 0 + (a 0 b +a b 0 )x + a b x 2 = c 0 + c x + c 2 x 2 February, 2003 (c) Charles Leiserson and Piotr Indyk L3.2
13 Motivation (more in recitations) Essentially equivalent to multiplying large integers: 6046*600 = (6* *0 + 0* *0 3 ) * (* * 0 + 0* *0 3 ) = a(0) * b(0) = c(0), where c(x)=a(x)*b(x) c(0) = c c c The coefficients of c form the digits of the product c(0) February, 2003 (c) Charles Leiserson and Piotr Indyk L3.3
14 How to multiply two polynomials From the definition: Θ(n 2 ) time Faster? Use divide and conquer Divide: a(x) = a 0 +a x+ +a n x n = (a a n/2 x n/2 ) + x n/2 (a n/2 x a n x n/2 ) = p(x) + x n/2 q(x) = p + x n/2 q In the same way: b(x) = s+x n/2 t February, 2003 (c) Charles Leiserson and Piotr Indyk L3.4
15 Conquer Observe that: a*b = (p+x n/2 q) * (s+x n/2 t) = p*s + x n/2 (p*t+q*s) + x n q*t But p,q,s,t have degree n/2 can compute the products recursively! (and then perform Θ(n) additions) February, 2003 (c) Charles Leiserson and Piotr Indyk L3.5
16 The great moment T(n) = 4 T(n/2) + Θ(n) # subproblems subproblem size work dividing and combining n log ba = n log 2 4 = n 2 CASE T(n) = Θ(n 2 ). No better than the ordinary algorithm??? February, 2003 (c) Charles Leiserson and Piotr Indyk L3.6
17 Need to be more clever Compute: p*s q*t (p+q) * (s+t) = p*s + (p*t + q*s) + q*t (all polynomials have degree n/2 ) Can extract (p*t + q*s) without any additional multiplications! February, 2003 (c) Charles Leiserson and Piotr Indyk L3.7
18 The truly great moment T(n) = 3T(n/2) + Θ(n) # subproblems subproblem size work adding and subtracting n log ba = n log 2 3 = n Much better than Θ(n 2 )! February, 2003 (c) Charles Leiserson and Piotr Indyk L3.8
19 Matrix multiplication Input: A = [a ij ], B = [b ij ]. Output: C = [c ij ] = A B. i, j =, 2,, n. c c 2 c c 2 22 c c n 2n = a a 2 a a 2 22 a a n 2n b b 2 b b 2 22 b b n 2n c n c n2 c nn a n a n2 a nn b n b n2 b nn c ij = n k= a ik b kj February, 2003 (c) Charles Leiserson and Piotr Indyk L3.9
20 Standard algorithm for i to n do for j to n do c ij 0 for k to n do c ij c ij + a ik b kj Running time = Θ(n 3 ) February, 2003 (c) Charles Leiserson and Piotr Indyk L3.20
21 Divide-and-conquer algorithm IDEA: n n matrix = 2 2 matrix of (n/2) (n/2) submatrices: r t s u = a c b d e g f h r = ae + bg s = af + bh t = ce + dg u = cf + dh C = A B 8 mults of (n/2) (n/2) submatrices 4 adds of (n/2) (n/2) submatrices February, 2003 (c) Charles Leiserson and Piotr Indyk L3.2
22 Analysis of D&C algorithm T(n) = 8 T(n/2) + Θ(n 2 ) # submatrices submatrix size work adding submatrices n log ba = n log 2 8 = n 3 CASE T(n) = Θ(n 3 ). No better than the ordinary algorithm. February, 2003 (c) Charles Leiserson and Piotr Indyk L3.22
23 Strassen s idea (969) Multiply 2 2 matrices with only 7 recursive mults. P = a ( f h) P 2 = (a + b) h P 3 = (c + d) e P 4 = d (g e) P 5 = (a + d) (e + h) P 6 = (b d) (g + h) P 7 = (a c) (e + f ) r = P 5 + P 4 P 2 + P 6 s = P + P 2 t = P 3 + P 4 u = P 5 + P P 3 P 7 7 mults, 8 adds/subs. Note: No reliance on commutativity of mult! February, 2003 (c) Charles Leiserson and Piotr Indyk L3.23
24 Strassen s idea Multiply 2 2 matrices with only 7 recursive mults. P = a ( f h) P 2 = (a + b) h P 3 = (c + d) e P 4 = d (g e) P 5 = (a + d) (e + h) P 6 = (b d) (g + h) P 7 = (a c) (e + f ) r = P 5 + P 4 P 2 + P 6 = (a + d) (e + h) + d (g e) (a + b) h + (b d) (g + h) = ae + ah + de + dh + dg de ah bh + bg + bh dg dh = ae + bg February, 2003 (c) Charles Leiserson and Piotr Indyk L3.24
25 Strassen s algorithm. Divide: Partition A and B into (n/2) (n/2) submatrices. Form terms to be multiplied using + and. 2. Conquer: Perform 7 multiplications of (n/2) (n/2) submatrices recursively. 3. Combine: Form C using + and on (n/2) (n/2) submatrices. T(n) = 7 T(n/2) + Θ(n 2 ) February, 2003 (c) Charles Leiserson and Piotr Indyk L3.25
26 Analysis of Strassen T(n) = 7 T(n/2) + Θ(n 2 ) n log ba = n log 2 7 n 2.8 CASE T(n) = Θ(n lg 7 ). The number 2.8 may not seem much smaller than 3, but because the difference is in the exponent, the impact on running time is significant. In fact, Strassen s algorithm beats the ordinary algorithm on today s machines for n 30 or so. Best to date (of theoretical interest only): Θ(n ). February, 2003 (c) Charles Leiserson and Piotr Indyk L3.26
27 Conclusion Divide and conquer is just one of several powerful techniques for algorithm design. Divide-and-conquer algorithms can be analyzed using recurrences and the master method (so practice this math). Can lead to more efficient algorithms February, 2003 (c) Charles Leiserson and Piotr Indyk L3.27
28 VLSI layout Problem: Embed a complete binary tree with n leaves in a grid using minimal area. W(n) H(n) H(n) = H(n/2) + Θ() W(n) = 2W(n/2) + Θ() = Θ(lg n) = Θ(n) Area = Θ(n lg n) February, 2003 (c) Charles Leiserson and Piotr Indyk L3.28
29 H-tree embedding L(n) L(n) L(n) = 2L(n/4) + Θ() = Θ( n) Area = Θ(n) L(n/4) Θ() L(n/4) February, 2003 (c) Charles Leiserson and Piotr Indyk L3.29
30 Master theorem (reprise) T(n) = a T(n/b) + f (n) CASE : f(n) = O(n log ba ε ) T(n) = Θ(n log ba ). CASE 2: f (n) = Θ(n log ba lg k n) T(n) = Θ(n log ba lg k+ n). CASE 3: f (n) = Ω(n log ba + ε ) and a f (n/b) c f (n) T(n) = Θ( f (n)). Merge sort: a = 2, b = 2 n log ba = n CASE 2 (k = 0) T(n) = Θ(n lg n). February, 2003 (c) Charles Leiserson and Piotr Indyk L3.30
31 Fibonacci numbers Recursive definition: 0 if n = 0; F n = if n = ; F n + F n 2 if n Naive recursive algorithm: Ω(φ n ) (exponential time), where φ = ( + is the golden ratio. 5)/2 February, 2003 (c) Charles Leiserson and Piotr Indyk L3.3
32 Computing Fibonacci numbers Naive recursive squaring: F n = φ n / 5 rounded to the nearest integer. Recursive squaring: Θ(lg n) time. This method is unreliable, since floating-point arithmetic is prone to round-off errors. Bottom-up: Compute F 0, F, F 2,, F n in order, forming each number by summing the two previous. Running time: Θ(n). February, 2003 (c) Charles Leiserson and Piotr Indyk L3.32
33 February, 2003 (c) Charles Leiserson and Piotr Indyk L3.33 Recursive squaring n F F F F n n n n = + 0 Theorem:. Proof of theorem. (Induction on n.) Base (n = ): = F F F F Algorithm: Recursive squaring. Time = Θ(lg n).
34 February, 2003 (c) Charles Leiserson and Piotr Indyk L3.34 Recursive squaring.. Inductive step (n 2): n n F F F F F F F F n n n n n n n n = = =
Divide-and-conquer algorithm
Divide-and-conquer algorithm IDEA: n n matrix = 2 2 matrix of (n/2) (n/2) submatrices: r=ae+bg s=af+bh t =ce+dh u=cf+dg r t s u = a c e g September 15, 2004 Introduction to Algorithms L3.31 b d C = A B
The Divide-and-Conquer Design Paradigm
CS473- Algorithms I Lecture 4 The Divide-and-Conquer Design Paradigm CS473 Lecture 4 1 The Divide-and-Conquer Design Paradigm 1. Divide the problem (instance) into subproblems. 2. Conquer the subproblems
CS60020: Foundations of Algorithm Design and Machine Learning. Sourangshu Bhattacharya
CS60020: Foundations of Algorithm Design and Machine Learning Sourangshu Bhattacharya Matrix multiplication September 14, 2005 L2.27 Standard algorithm for i 1 to n do for j 1 ton do c ij 0 for k 1 to
CMPS 2200 Fall Divide-and-Conquer. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk
CMPS 2200 Fall 2017 Divide-and-Conquer Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk 1 The divide-and-conquer design paradigm 1. Divide the problem (instance)
Algorithms Design & Analysis. Divide & Conquer
Algorithms Desig & Aalysis Divide & Coquer Recap Direct-accessible table Hash tables Hash fuctios Uiversal hashig Perfect Hashig Ope addressig 2 Today s topics The divide-ad-coquer desig paradigm Revised
Introduction to Algorithms 6.046J/18.401J LECTURE 3 Divide and conquer Binary search Powering a number Fibonacci numbers Matrix multiplication
Itroductio to Algorithms 6.046J/8.40J LECTURE 3 Divide ad coquer Biary search Powerig a umber Fiboacci umbers Matrix multiplicatio Strasse s algorithm VLSI tree layout Prof. Charles E. Leiserso The divide-ad-coquer
Chapter 2. Recurrence Relations. Divide and Conquer. Divide and Conquer Strategy. Another Example: Merge Sort. Merge Sort Example. Merge Sort Example
Recurrence Relations Chapter 2 Divide and Conquer Equation or an inequality that describes a function by its values on smaller inputs. Recurrence relations arise when we analyze the running time of iterative
CS 577 Introduction to Algorithms: Strassen s Algorithm and the Master Theorem
CS 577 Introduction to Algorithms: Jin-Yi Cai University of Wisconsin Madison In the last class, we described InsertionSort and showed that its worst-case running time is Θ(n 2 ). Check Figure 2.2 for
Divide and Conquer. Maximum/minimum. Median finding. CS125 Lecture 4 Fall 2016
CS125 Lecture 4 Fall 2016 Divide and Conquer We have seen one general paradigm for finding algorithms: the greedy approach. We now consider another general paradigm, known as divide and conquer. We have
Chapter 4 Divide-and-Conquer
Chapter 4 Divide-and-Conquer 1 About this lecture (1) Recall the divide-and-conquer paradigm, which we used for merge sort: Divide the problem into a number of subproblems that are smaller instances of
Divide and Conquer: Polynomial Multiplication Version of October 1 / 7, 24201
Divide and Conquer: Polynomial Multiplication Version of October 7, 2014 Divide and Conquer: Polynomial Multiplication Version of October 1 / 7, 24201 Outline Outline: Introduction The polynomial multiplication
Algorithm Design and Analysis
Algorithm Design and Analysis LECTURE 9 Divide and Conquer Merge sort Counting Inversions Binary Search Exponentiation Solving Recurrences Recursion Tree Method Master Theorem Sofya Raskhodnikova S. Raskhodnikova;
CS 470/570 Divide-and-Conquer. Format of Divide-and-Conquer algorithms: Master Recurrence Theorem (simpler version)
CS 470/570 Divide-and-Conquer Format of Divide-and-Conquer algorithms: Divide: Split the array or list into smaller pieces Conquer: Solve the same problem recursively on smaller pieces Combine: Build the
Fast Convolution; Strassen s Method
Fast Convolution; Strassen s Method 1 Fast Convolution reduction to subquadratic time polynomial evaluation at complex roots of unity interpolation via evaluation at complex roots of unity 2 The Master
Divide and Conquer. Andreas Klappenecker. [based on slides by Prof. Welch]
![Divide and Conquer. Andreas Klappenecker. [based on slides by Prof. Welch]](/thumbs/84/89083836.jpg "Divide and Conquer. Andreas Klappenecker. [based on slides by Prof. Welch]")
Divide and Conquer Andreas Klappenecker [based on slides by Prof. Welch] Divide and Conquer Paradigm An important general technique for designing algorithms: divide problem into subproblems recursively
A design paradigm. Divide and conquer: (When) does decomposing a problem into smaller parts help? 09/09/ EECS 3101
A design paradigm Divide and conquer: (When) does decomposing a problem into smaller parts help? 09/09/17 112 Multiplying complex numbers (from Jeff Edmonds slides) INPUT: Two pairs of integers, (a,b),
Class Note #14. In this class, we studied an algorithm for integer multiplication, which. 2 ) to θ(n
Class Note #14 Date: 03/01/2006 [Overall Information] In this class, we studied an algorithm for integer multiplication, which improved the running time from θ(n 2 ) to θ(n 1.59 ). We then used some of
Divide and Conquer Algorithms
Divide and Conquer Algorithms Introduction There exist many problems that can be solved using a divide-and-conquer algorithm. A divide-andconquer algorithm A follows these general guidelines. Divide Algorithm
Matrix Multiplication
Matrix Multiplication Matrix Multiplication Matrix multiplication. Given two n-by-n matrices A and B, compute C = AB. n c ij = a ik b kj k=1 c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn = a 11 a 12 a 1n
Data Structures and Algorithms CMPSC 465
Data Structures and Algorithms CMPSC 465 LECTURE 9 Solving recurrences Substitution method Adam Smith S. Raskhodnikova and A. Smith; based on slides by E. Demaine and C. Leiserson Review question Draw
Answer the following questions: Q1: ( 15 points) : A) Choose the correct answer of the following questions: نموذج اإلجابة
Benha University Final Exam Class: 3 rd Year Students Subject: Design and analysis of Algorithms Faculty of Computers & Informatics Date: 10/1/2017 Time: 3 hours Examiner: Dr. El-Sayed Badr Answer the
Lecture 8 - Algebraic Methods for Matching 1
CME 305: Discrete Mathematics and Algorithms Instructor: Professor Aaron Sidford (sidford@stanford.edu) February 1, 2018 Lecture 8 - Algebraic Methods for Matching 1 In the last lecture we showed that
The divide-and-conquer strategy solves a problem by: 1. Breaking it into subproblems that are themselves smaller instances of the same type of problem
Chapter 2. Divide-and-conquer algorithms The divide-and-conquer strategy solves a problem by: 1. Breaking it into subproblems that are themselves smaller instances of the same type of problem. 2. Recursively
The maximum-subarray problem. Given an array of integers, find a contiguous subarray with the maximum sum. Very naïve algorithm:
The maximum-subarray problem Given an array of integers, find a contiguous subarray with the maximum sum. Very naïve algorithm: Brute force algorithm: At best, θ(n 2 ) time complexity 129 Can we do divide
V. Adamchik 1. Recurrences. Victor Adamchik Fall of 2005
V. Adamchi Recurrences Victor Adamchi Fall of 00 Plan Multiple roots. More on multiple roots. Inhomogeneous equations 3. Divide-and-conquer recurrences In the previous lecture we have showed that if the
Design and Analysis of Algorithms
CSE 101, Winter 2018 Design and Analysis of Algorithms Lecture 4: Divide and Conquer (I) Class URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ Divide and Conquer ( DQ ) First paradigm or framework DQ(S)
MA008/MIIZ01 Design and Analysis of Algorithms Lecture Notes 3
MA008 p.1/37 MA008/MIIZ01 Design and Analysis of Algorithms Lecture Notes 3 Dr. Markus Hagenbuchner markus@uow.edu.au. MA008 p.2/37 Exercise 1 (from LN 2) Asymptotic Notation When constants appear in exponents
Analysis of Multithreaded Algorithms
Analysis of Multithreaded Algorithms Marc Moreno Maza University of Western Ontario, London, Ontario (Canada) CS 4435 - CS 9624 (Moreno Maza) Analysis of Multithreaded Algorithms CS 4435 - CS 9624 1 /
Problem Set 1 Solutions
Introduction to Algorithms September 24, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Piotr Indyk and Charles E. Leiserson Handout 7 Problem Set 1 Solutions Exercise 1-1. Do Exercise
Divide-and-conquer: Order Statistics. Curs: Fall 2017
Divide-and-conquer: Order Statistics Curs: Fall 2017 The divide-and-conquer strategy. 1. Break the problem into smaller subproblems, 2. recursively solve each problem, 3. appropriately combine their answers.
Divide and Conquer algorithms
Divide and Conquer algorithms Another general method for constructing algorithms is given by the Divide and Conquer strategy. We assume that we have a problem with input that can be split into parts in
CPS 616 DIVIDE-AND-CONQUER 6-1
CPS 616 DIVIDE-AND-CONQUER 6-1 DIVIDE-AND-CONQUER Approach 1. Divide instance of problem into two or more smaller instances 2. Solve smaller instances recursively 3. Obtain solution to original (larger)
CS 4407 Algorithms Lecture 3: Iterative and Divide and Conquer Algorithms
CS 4407 Algorithms Lecture 3: Iterative and Divide and Conquer Algorithms Prof. Gregory Provan Department of Computer Science University College Cork 1 Lecture Outline CS 4407, Algorithms Growth Functions
Analysis of Algorithms I: Asymptotic Notation, Induction, and MergeSort
Analysis of Algorithms I: Asymptotic Notation, Induction, and MergeSort Xi Chen Columbia University We continue with two more asymptotic notation: o( ) and ω( ). Let f (n) and g(n) are functions that map
CS 4407 Algorithms Lecture 2: Iterative and Divide and Conquer Algorithms
CS 4407 Algorithms Lecture 2: Iterative and Divide and Conquer Algorithms Prof. Gregory Provan Department of Computer Science University College Cork 1 Lecture Outline CS 4407, Algorithms Growth Functions
COMP 382: Reasoning about algorithms
Fall 2014 Unit 4: Basics of complexity analysis Correctness and efficiency So far, we have talked about correctness and termination of algorithms What about efficiency? Running time of an algorithm For
CS 5321: Advanced Algorithms - Recurrence. Acknowledgement. Outline. Ali Ebnenasir Department of Computer Science Michigan Technological University
CS 5321: Advanced Algorithms - Recurrence Ali Ebnenasir Department of Computer Science Michigan Technological University Acknowledgement Eric Torng Moon Jung Chung Charles Ofria Outline Motivating example:
Chapter 1 Divide and Conquer Algorithm Theory WS 2016/17 Fabian Kuhn
Chapter 1 Divide and Conquer Algorithm Theory WS 2016/17 Fabian Kuhn Formulation of the D&C principle Divide-and-conquer method for solving a problem instance of size n: 1. Divide n c: Solve the problem
Data Structures and Algorithms Chapter 3
Data Structures and Algorithms Chapter 3 1. Divide and conquer 2. Merge sort, repeated substitutions 3. Tiling 4. Recurrences Recurrences Running times of algorithms with recursive calls can be described
Introduction to Algorithms 6.046J/18.401J
Itrodutio to Algorithms.04J/8.40J The divide-d-oquer desig prdigm. Divide the problem (iste) ito subproblems.. Coquer the subproblems by solvig them reursively. 3. Combie subproblem solutios. Leture 3
CS 5321: Advanced Algorithms Analysis Using Recurrence. Acknowledgement. Outline
CS 5321: Advanced Algorithms Analysis Using Recurrence Ali Ebnenasir Department of Computer Science Michigan Technological University Acknowledgement Eric Torng Moon Jung Chung Charles Ofria Outline Motivating
Algorithms and Data Structures Strassen s Algorithm. ADS (2017/18) Lecture 4 slide 1
Algorithms and Data Structures Strassen s Algorithm ADS (2017/18) Lecture 4 slide 1 Tutorials Start in week (week 3) Tutorial allocations are linked from the course webpage http://www.inf.ed.ac.uk/teaching/courses/ads/
Introduction to Algorithms 6.046J/18.401J
Introduction to Algorithms 6.046J/8.40J Lecture Prof. Piotr Indyk Welcome to Introduction to Algorithms, Spring 08 Handouts. Course Information. Calendar 3. Signup sheet (PLEASE return at the end of this
CMPSCI611: Three Divide-and-Conquer Examples Lecture 2
CMPSCI611: Three Divide-and-Conquer Examples Lecture 2 Last lecture we presented and analyzed Mergesort, a simple divide-and-conquer algorithm. We then stated and proved the Master Theorem, which gives
Data Structures and Algorithms CSE 465
Data Structures and Algorithms CSE 465 LECTURE 3 Asymptotic Notation O-, Ω-, Θ-, o-, ω-notation Divide and Conquer Merge Sort Binary Search Sofya Raskhodnikova and Adam Smith /5/0 Review Questions If input
Tutorials. Algorithms and Data Structures Strassen s Algorithm. The Master Theorem for solving recurrences. The Master Theorem (cont d)
DS 2018/19 Lecture 4 slide 3 DS 2018/19 Lecture 4 slide 4 Tutorials lgorithms and Data Structures Strassen s lgorithm Start in week week 3 Tutorial allocations are linked from the course webpage http://www.inf.ed.ac.uk/teaching/courses/ads/
Algorithm efficiency analysis
Algorithm efficiency analysis Mădălina Răschip, Cristian Gaţu Faculty of Computer Science Alexandru Ioan Cuza University of Iaşi, Romania DS 2017/2018 Content Algorithm efficiency analysis Recursive function
CS/COE 1501 cs.pitt.edu/~bill/1501/ Integer Multiplication
CS/COE 1501 cs.pitt.edu/~bill/1501/ Integer Multiplication Integer multiplication Say we have 5 baskets with 8 apples in each How do we determine how many apples we have? Count them all? That would take
Design and Analysis of Algorithms
Design and Analysis of Algorithms CSE 5311 Lecture 5 Divide and Conquer: Fast Fourier Transform Junzhou Huang, Ph.D. Department of Computer Science and Engineering CSE5311 Design and Analysis of Algorithms
The Master Theorem for solving recurrences. Algorithms and Data Structures Strassen s Algorithm. Tutorials. The Master Theorem (cont d)
The Master Theorem for solving recurrences lgorithms and Data Structures Strassen s lgorithm 23rd September, 2014 Theorem Let n 0 N, k N 0 and a, b R with a > 0 and b > 1, and let T : N R satisfy the following
Review Of Topics. Review: Induction
Review Of Topics Asymptotic notation Solving recurrences Sorting algorithms Insertion sort Merge sort Heap sort Quick sort Counting sort Radix sort Medians/order statistics Randomized algorithm Worst-case
Chapter 2. Divide-and-conquer. 2.1 Strassen s algorithm
Chapter 2 Divide-and-conquer This chapter revisits the divide-and-conquer paradigms and explains how to solve recurrences, in particular, with the use of the master theorem. We first illustrate the concept
A SUMMARY OF RECURSION SOLVING TECHNIQUES
A SUMMARY OF RECURSION SOLVING TECHNIQUES KIMMO ERIKSSON, KTH These notes are meant to be a complement to the material on recursion solving techniques in the textbook Discrete Mathematics by Biggs. In
Computational Complexity - Pseudocode and Recursions
Computational Complexity - Pseudocode and Recursions Nicholas Mainardi 1 Dipartimento di Elettronica e Informazione Politecnico di Milano nicholas.mainardi@polimi.it June 6, 2018 1 Partly Based on Alessandro
8/13/16. Data analysis and modeling: the tools of the trade. Ø Set of numbers. Ø Binary representation of numbers. Ø Floating points.
Data analysis and modeling: the tools of the trade Patrice Koehl Department of Biological Sciences National University of Singapore http://www.cs.ucdavis.edu/~koehl/teaching/bl5229 koehl@cs.ucdavis.edu
Lecture 7: Dynamic Programming I: Optimal BSTs
5-750: Graduate Algorithms February, 06 Lecture 7: Dynamic Programming I: Optimal BSTs Lecturer: David Witmer Scribes: Ellango Jothimurugesan, Ziqiang Feng Overview The basic idea of dynamic programming
CSE 421 Algorithms. T(n) = at(n/b) + n c. Closest Pair Problem. Divide and Conquer Algorithms. What you really need to know about recurrences
CSE 421 Algorithms Richard Anderson Lecture 13 Divide and Conquer What you really need to know about recurrences Work per level changes geometrically with the level Geometrically increasing (x > 1) The
Problem Set 2 Solutions
Introduction to Algorithms October 1, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Piotr Indyk and Charles E. Leiserson Handout 9 Problem Set 2 Solutions Reading: Chapters 5-9,
shelat divide&conquer closest points matrixmult median
L6feb 9 2016 shelat 4102 divide&conquer closest points matrixmult median 7.5 7.8 9.5 3.7 26.1 closest pair of points simple brute force approach takes 14 1 2 7 8 9 4 13 3 10 11 5 12 6 solve the large problem
Design and Analysis of Algorithms Recurrence. Prof. Chuhua Xian School of Computer Science and Engineering
Design and Analysis of Algorithms Recurrence Prof. Chuhua Xian Email: chhxian@scut.edu.cn School of Computer Science and Engineering Course Information Instructor: Chuhua Xian ( 冼楚华 ) Email: chhxian@scut.edu.cn
Solving Recurrences. Lecture 23 CS2110 Fall 2011
Solving Recurrences Lecture 23 CS2110 Fall 2011 1 Announcements Makeup Prelim 2 Monday 11/21 7:30-9pm Upson 5130 Please do not discuss the prelim with your classmates! Quiz 4 next Tuesday in class Topics:
Chapter 1 Divide and Conquer Polynomial Multiplication Algorithm Theory WS 2015/16 Fabian Kuhn
Chapter 1 Divide and Conquer Polynomial Multiplication Algorithm Theory WS 2015/16 Fabian Kuhn Formulation of the D&C principle Divide-and-conquer method for solving a problem instance of size n: 1. Divide
Divide-and-Conquer Algorithms and Recurrence Relations. Niloufar Shafiei
Divide-and-Conquer Algorithms and Recurrence Relations Niloufar Shafiei Divide-and-conquer algorithms Divide-and-conquer algorithms: 1. Dividing the problem into smaller sub-problems 2. Solving those sub-problems
Asymptotic Analysis and Recurrences
Appendix A Asymptotic Analysis and Recurrences A.1 Overview We discuss the notion of asymptotic analysis and introduce O, Ω, Θ, and o notation. We then turn to the topic of recurrences, discussing several
CSCI Honor seminar in algorithms Homework 2 Solution
CSCI 493.55 Honor seminar in algorithms Homework 2 Solution Saad Mneimneh Visiting Professor Hunter College of CUNY Problem 1: Rabin-Karp string matching Consider a binary string s of length n and another
The divide-and-conquer strategy solves a problem by: Chapter 2. Divide-and-conquer algorithms. Multiplication
The divide-and-conquer strategy solves a problem by: Chapter 2 Divide-and-conquer algorithms 1 Breaking it into subproblems that are themselves smaller instances of the same type of problem 2 Recursively
Divide and Conquer. Andreas Klappenecker
Divide and Conquer Andreas Klappenecker The Divide and Conquer Paradigm The divide and conquer paradigm is important general technique for designing algorithms. In general, it follows the steps: - divide
CS483 Design and Analysis of Algorithms
CS483 Design and Analysis of Algorithms Lecture 6-8 Divide and Conquer Algorithms Instructor: Fei Li lifei@cs.gmu.edu with subject: CS483 Office hours: STII, Room 443, Friday 4:00pm - 6:00pm or by appointments
Recurrence Relations
Recurrence Relations Winter 2017 Recurrence Relations Recurrence Relations A recurrence relation for the sequence {a n } is an equation that expresses a n in terms of one or more of the previous terms
COMP 9024, Class notes, 11s2, Class 1
COMP 90, Class notes, 11s, Class 1 John Plaice Sun Jul 31 1::5 EST 011 In this course, you will need to know a bit of mathematics. We cover in today s lecture the basics. Some of this material is covered
shelat 16f-4800 sep Matrix Mult, Median, FFT
L5 shelat 16f-4800 sep 23 2016 Matrix Mult, Median, FFT merge-sort (A, p, r) if p
Taking Stock. IE170: Algorithms in Systems Engineering: Lecture 3. Θ Notation. Comparing Algorithms
Taking Stock IE170: Algorithms in Systems Engineering: Lecture 3 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University January 19, 2007 Last Time Lots of funky math Playing
Data Structures and Algorithms Chapter 3
1 Data Structures and Algorithms Chapter 3 Werner Nutt 2 Acknowledgments The course follows the book Introduction to Algorithms, by Cormen, Leiserson, Rivest and Stein, MIT Press [CLRST]. Many examples
Algorithm Analysis Recurrence Relation. Chung-Ang University, Jaesung Lee
Algorithm Analysis Recurrence Relation Chung-Ang University, Jaesung Lee Recursion 2 Recursion 3 Recursion in Real-world Fibonacci sequence = + Initial conditions: = 0 and = 1. = + = + = + 0, 1, 1, 2,
1 Substitution method
Recurrence Relations we have discussed asymptotic analysis of algorithms and various properties associated with asymptotic notation. As many algorithms are recursive in nature, it is natural to analyze
A matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
Asymptotic Algorithm Analysis & Sorting
Asymptotic Algorithm Analysis & Sorting (Version of 5th March 2010) (Based on original slides by John Hamer and Yves Deville) We can analyse an algorithm without needing to run it, and in so doing we can
feb abhi shelat Matrix, FFT
L7 feb 11 2016 abhi shelat Matrix, FFT userid: = Using the standard method, how many multiplications does it take to multiply two NxN matrices? cos( /4) = cos( /2) = sin( /4) = sin( /2) = Mergesort Karatsuba
Chapter 5 Divide and Conquer
CMPT 705: Design and Analysis of Algorithms Spring 008 Chapter 5 Divide and Conquer Lecturer: Binay Bhattacharya Scribe: Chris Nell 5.1 Introduction Given a problem P with input size n, P (n), we define
Divide and Conquer. Arash Rafiey. 27 October, 2016
27 October, 2016 Divide the problem into a number of subproblems Divide the problem into a number of subproblems Conquer the subproblems by solving them recursively or if they are small, there must be
Chapter. Divide-and-Conquer. Contents
Chapter 11 Divide-and-Conquer Grand Canyon from South Rim, 1941. Ansel Adams. U.S. government image. U.S. National Archives and Records Administration. Contents 11.1RecurrencesandtheMasterTheorem...305
Chapter 5. Divide and Conquer CLRS 4.3. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 5 Divide and Conquer CLRS 4.3 Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 Divide-and-Conquer Divide-and-conquer. Break up problem into several parts. Solve
Data Structures and Algorithms CSE 465
Data Structures and Algorithms CSE 465 LECTURE 8 Analyzing Quick Sort Sofya Raskhodnikova and Adam Smith Reminder: QuickSort Quicksort an n-element array: 1. Divide: Partition the array around a pivot
Chapter 5. Divide and Conquer CLRS 4.3. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 5 Divide and Conquer CLRS 4.3 Slides by Kevin Wayne. Copyright 25 Pearson-Addison Wesley. All rights reserved. Divide-and-Conquer Divide-and-conquer. Break up problem into several parts. Solve
Algorithms Design & Analysis. Analysis of Algorithm
Algorithms Design & Analysis Analysis of Algorithm Review Internship Stable Matching Algorithm 2 Outline Time complexity Computation model Asymptotic notions Recurrence Master theorem 3 The problem of
Recurrence Relations
Recurrence Relations Analysis Tools S.V. N. (vishy) Vishwanathan University of California, Santa Cruz vishy@ucsc.edu January 15, 2016 S.V. N. Vishwanathan (UCSC) CMPS101 1 / 29 Recurrences Outline 1 Recurrences
Copyright 2000, Kevin Wayne 1
Divide-and-Conquer Chapter 5 Divide and Conquer Divide-and-conquer. Break up problem into several parts. Solve each part recursively. Combine solutions to sub-problems into overall solution. Most common
What we have learned What is algorithm Why study algorithm The time and space efficiency of algorithm The analysis framework of time efficiency Asympt
Lecture 3 The Analysis of Recursive Algorithm Efficiency What we have learned What is algorithm Why study algorithm The time and space efficiency of algorithm The analysis framework of time efficiency
CSE 613: Parallel Programming. Lecture 8 ( Analyzing Divide-and-Conquer Algorithms )
CSE 613: Parallel Programming Lecture 8 ( Analyzing Divide-and-Conquer Algorithms ) Rezaul A. Chowdhury Department of Computer Science SUNY Stony Brook Spring 2012 A Useful Recurrence Consider the following
CSE 613: Parallel Programming. Lectures ( Analyzing Divide-and-Conquer Algorithms )
CSE 613: Parallel Programming Lectures 13 14 ( Analyzing Divide-and-Conquer Algorithms ) Rezaul A. Chowdhury Department of Computer Science SUNY Stony Brook Spring 2015 A Useful Recurrence Consider the
Divide and Conquer Strategy
Divide and Conquer Strategy Algorithm design is more an art, less so a science. There are a few useful strategies, but no guarantee to succeed. We will discuss: Divide and Conquer, Greedy, Dynamic Programming.
Divide and Conquer. Chapter Overview. 1.2 Divide and Conquer
Chapter 1 Divide and Conquer 1.1 Overview This chapter introduces Divide and Conquer, which is a technique for designing recursive algorithms that are (sometimes) asymptotically efficient. It also presents
Outline. 1 Introduction. Merging and MergeSort. 3 Analysis. 4 Reference
Outline Computer Science 331 Sort Mike Jacobson Department of Computer Science University of Calgary Lecture #25 1 Introduction 2 Merging and 3 4 Reference Mike Jacobson (University of Calgary) Computer
Discrete Mathematics U. Waterloo ECE 103, Spring 2010 Ashwin Nayak May 17, 2010 Recursion
Discrete Mathematics U. Waterloo ECE 103, Spring 2010 Ashwin Nayak May 17, 2010 Recursion During the past week, we learnt about inductive reasoning, in which we broke down a problem of size n, into one
Divide and Conquer. CSE21 Winter 2017, Day 9 (B00), Day 6 (A00) January 30,
Divide and Conquer CSE21 Winter 2017, Day 9 (B00), Day 6 (A00) January 30, 2017 http://vlsicad.ucsd.edu/courses/cse21-w17 Merging sorted lists: WHAT Given two sorted lists a 1 a 2 a 3 a k b 1 b 2 b 3 b
CS483 Design and Analysis of Algorithms
CS483 Design and Analysis of Algorithms Chapter 2 Divide and Conquer Algorithms Instructor: Fei Li lifei@cs.gmu.edu with subject: CS483 Office hours: Room 5326, Engineering Building, Thursday 4:30pm -
Chapter 4. Recurrences
Chapter 4. Recurrences Outline Offers three methods for solving recurrences, that is for obtaining asymptotic bounds on the solution In the substitution method, we guess a bound and then use mathematical
Grade 11/12 Math Circles Fall Nov. 5 Recurrences, Part 2
1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 11/12 Math Circles Fall 2014 - Nov. 5 Recurrences, Part 2 Running time of algorithms In computer science,
Introduction to Algorithms
Lecture 1 Introduction to Algorithms 1.1 Overview The purpose of this lecture is to give a brief overview of the topic of Algorithms and the kind of thinking it involves: why we focus on the subjects that
COMP Analysis of Algorithms & Data Structures
COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Lecture 4 - Jan. 10, 2018 CLRS 1.1, 1.2, 2.2, 3.1, 4.3, 4.5 University of Manitoba Picture is from the cover of the textbook CLRS. 1 /