Data Structures and Algorithms CMPSC 465
|
|
- Marjory Owen
- 6 years ago
- Views:
Transcription
1 Data Structures and Algorithms CMPSC 465 LECTURES 7-8 More Divide and Conquer Multiplication Adam Smith S. Raskhodnikova and A. Smith; based on slides by E. Demaine and C. Leiserson
2 John McCarthy ( ) Pioneer in artificial intelligence In fact, he coined the term Lots of work on how reasoning can be automated Meta-lesson: you learn a lot about something when you teach a computer to do it.
3 McCarthy s 91 function Why be so careful about correctness with recursive procedures? M(n) = { n 10, if n>100 M(M(n + 11)), if n 100 This function is tricky to reason about because it doesn t always make recursive calls on smaller inputs Programmer s goal: code that is easy to understand Do not write circular programs! S. Raskhodnikova and A. Smith; based on slides by E. Demaine and C. Leiserson 3
4 Reminder: geometric series x x x = for x < x x x x x n n = for x 1
5 Review For each tree: How many leaves? How many nodes in total? (big-theta) H
6 Multiplying large integers Given n-bit integers a, b (in binary), compute c=ab Naive (grade-school) algorithm: a n-1 a n-2 a 0 Write a,b in binary Compute n intermediate products Do n additions Running time? Total work: (n 2 ) n bits b n-1 b n-2 b 0 n bits n bits 2n bit output
7 Divide-and-conquer design 1. Divide the problem (instance) into subproblems. 2. Conquer the subproblems by solving them recursively. 3. Combine subproblem solutions.
8 Multiplying large integers Divide and Conquer (Attempt #1): Write a = A 1 2 n /2 + A 0 b = B 1 2 n /2 + B 0 We want ab = A 1 B 1 2 n + (A 1 B 0 + B 1 A 0 ) 2 n /2 + A 0 B 0 Multiply n/2 bit integers recursively T(n) = 4T(n/2) + (n) Alas! this is still (n 2 ). (Exercise: write out the recursion tree.)
9 Multiplying large integers Divide and Conquer (Attempt #1): Write a = A 1 2 n /2 + A 0 b = B 1 2 n /2 + B 0 We want ab = A 1 B 1 2 n + (A 1 B 0 + B 1 A 0 ) 2 n /2 + A 0 B 0 Multiply n/2 bit integers recursively T(n) Consider = 4T(n/2) the expression + (n) (A Alas! this is still (n 2 0 +A 1 ) (B 0 + B 1 ) = A 0 ) B. 0 + A 1 B 1 + (A 0 B 1 + B 1 A 0 ) (Exercise: We can get write away out with the 3 recursion multiplications! tree.) (in yellow) x = A 1 B 1 y = A 0 B 0 z = (A 0 +A 1 )(B 0 +B 1 ) Now we use ab = A 1 B 1 2 n + (A 1 B 0 + B 1 A 0 ) 2 n /2 + A 0 B 0 = x 2n + (z x y) 2 n /2 + y
10 Multiplying large integers MULTIPLY (n, a, b) a and b are n-bit integers Assume n is a power of 2 for simplicity 1. If n 2 then use grade-school algorithm else 2. A 1 a div 2 n /2 ; B 1 b div 2 n /2 ; 3. A 0 a mod 2 n /2 ; B 0 b mod 2 n /2. 4. x MULTIPLY(n/2, A 1, B 1 ) 5. y MULTIPLY(n/2, A 0, B 0 ) 6. z MULTIPLY(n/2, A 1 +A 0, B 1 +B 0 ) 7. Output x 2 n + (z x y)2 n /2 + y
11 Multiplying large integers The resulting recurrence T(n) = 3T(n/2) + (n) We will see that T(n) = (n log 23 ) = (n 1.59 ) Note: There is a (n log n) algorithm for multiplication (assumes constant-time word ops) Actually, (n log(n) 2 log*(n) ) bit-level operations (due to Martin Fürer, 2007)
12 Reminder: recursion trees Technique for guessing solution to recurrences Write out tree of recursive calls Each node gets assigned the work done during that call to the procedure (dividing and combining) Total work is sum of work at all nodes
13 Recursion tree for multiplication Solve T(n) = 3T(n/2) + cn, where c > 0 is constant. cn cn lg 2 n cn/2 cn/4 cn/4 (1) cn/4 cn/2 cn/2 At bottom level: (3/2)cn (9/4)cn At level k: (3/2) k cn (3/2) log n cn = c n log 2 3
14 Recursion tree for multiplication lg 2 n Solve T(n) = 3T(n/2) + cn, where c > 0 is constant. (1) cn/2 cn/4 cn/4 cn/4 Total work log 2 n = k= 0 cn cn/2 3 2 k cn cn/2 = (3 log 2 n ) = ( n log cn (3/2)cn (9/4)cn Geometric series: When base of exponent is constant (e.g. 3/2), the largest term is a constant fraction of total 2 3 )
15 Divide-and-conquer design 1. Divide the problem (instance) into subproblems. 2. Conquer the subproblems by solving them recursively. 3. Combine subproblem solutions.
16 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
17 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
18 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
19 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
20 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
21 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
22 Binary Search BINARYSEARCH(b, A[1.. n]) find b in sorted (increasing order) array A 1. If n=0 then return not found 2. If A[ n/2 ] = b then return n/2 3. If A[ n/2 ] > b then 4. return BINARYSEARCH(A[1.. n/2 ]) 5. Else 6. return n/2 + BINARYSEARCH(A[ n/ n])
23 Proof of correctness Omitted here, similar to Merge-Sort
24 Recurrence for binary search T(n) = 1 T(n/2) + (1) # subproblems subproblem size work dividing and combining
25 Recurrence for binary search T(n) = 1 T(n/2) + (1) # subproblems subproblem size work dividing and combining T(n) = T(n/2) + c = T(n/4) + 2c = c log n = (lg n).
26 Exponentiation Problem: Compute a b, where b N is n bits long Question: How many multiplications? Naive algorithm: (b) = (2 n ) (exponential in the input length!) Divide-and-conquer algorithm: a b = a b/2 a b/2 if b is even; a (b 1)/2 a (b 1)/2 a if b is odd. T(b) = T(b/2) + (1) T(b) = (log b) = (n).
27 So far: 3 recurrences Merge Sort T(n) = 2 T(n/2) + (n) = (n log n) Binary Search / Exponentiation T(n) = 1 T(n/2) + (1) = (log n) Multiplication T(n) = 3 T(n/2) + (n) = (n log 2 3 ) Coming soon: systematic ways to solve these.
Data Structures and Algorithms CSE 465
Data Structures and Algorithms CSE 465 LECTURE 4 More Divide and Conquer Binary Search Exponentiation Multiplication Sofya Raskhodnikova and Adam Smith Review questions How long does Merge Sort take on
More informationCS473 - Algorithms I
CS473 - Algorithms I Lecture 4 The Divide-and-Conquer Design Paradigm View in slide-show mode 1 Reminder: Merge Sort Input array A sort this half sort this half Divide Conquer merge two sorted halves Combine
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 16 Dynamic Programming (plus FFT Recap) Adam Smith 9/24/2008 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne Discrete Fourier Transform
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 13 Divide and Conquer Closest Pair of Points Convex Hull Strassen Matrix Mult. Adam Smith 9/24/2008 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova,
More informationWe can use a max-heap to sort data.
Sorting 7B N log N Sorts 1 Heap Sort We can use a max-heap to sort data. Convert an array to a max-heap. Remove the root from the heap and store it in its proper position in the same array. Repeat until
More informationCSC Design and Analysis of Algorithms
CSC 8301- Design and Analysis of Algorithms Lecture 6 Divide and Conquer Algorithm Design Technique Divide-and-Conquer The most-well known algorithm design strategy: 1. Divide a problem instance into two
More informationCSC Design and Analysis of Algorithms. Lecture 6. Divide and Conquer Algorithm Design Technique. Divide-and-Conquer
CSC 8301- Design and Analysis of Algorithms Lecture 6 Divide and Conquer Algorithm Design Technique Divide-and-Conquer The most-well known algorithm design strategy: 1. Divide a problem instance into two
More informationAlgorithms with numbers (1) CISC4080, Computer Algorithms CIS, Fordham Univ.! Instructor: X. Zhang Spring 2017
Algorithms with numbers (1) CISC4080, Computer Algorithms CIS, Fordham Univ. Instructor: X. Zhang Spring 2017 Acknowledgement The set of slides have used materials from the following resources Slides for
More informationLECTURE NOTES OF ALGORITHMS: DESIGN TECHNIQUES AND ANALYSIS
Department of Computer Science University of Babylon LECTURE NOTES OF ALGORITHMS: DESIGN TECHNIQUES AND ANALYSIS By Faculty of Science for Women( SCIW), University of Babylon, Iraq Samaher@uobabylon.edu.iq
More informationPlotting run-time graphically. Plotting run-time graphically. CS241 Algorithmics - week 1 review. Prefix Averages - Algorithm #1
CS241 - week 1 review Special classes of algorithms: logarithmic: O(log n) linear: O(n) quadratic: O(n 2 ) polynomial: O(n k ), k 1 exponential: O(a n ), a > 1 Classifying algorithms is generally done
More informationCS 171: Introduction to Computer Science II. Quicksort
CS 171: Introduction to Computer Science II Quicksort Roadmap MergeSort Recursive Algorithm (top-down) Practical Improvements Non-recursive algorithm (bottom-up) Analysis QuickSort Algorithm Analysis Practical
More informationCS583 Lecture 01. Jana Kosecka. some materials here are based on Profs. E. Demaine, D. Luebke A.Shehu, J-M. Lien and Prof. Wang s past lecture notes
CS583 Lecture 01 Jana Kosecka some materials here are based on Profs. E. Demaine, D. Luebke A.Shehu, J-M. Lien and Prof. Wang s past lecture notes Course Info course webpage: - from the syllabus on http://cs.gmu.edu/
More informationDivide-and-Conquer. Dr. Yingwu Zhu
Divide-and-Conquer Dr. Yingwu Zhu Divide-and-Conquer The most-well known algorithm design technique: 1. Divide instance of problem into two or more smaller instances 2. Solve smaller instances independently
More informationDivide and Conquer. Algorithm Fall Semester
Divide and Conquer Algorithm 2014 Fall Semester Divide-and-Conquer The most-well known algorithm design strategy: 1. Divide instance of problem into two or more smaller instances 2. Solve smaller instances
More informationParallel Algorithms for (PRAM) Computers & Some Parallel Algorithms. Reference : Horowitz, Sahni and Rajasekaran, Computer Algorithms
Parallel Algorithms for (PRAM) Computers & Some Parallel Algorithms Reference : Horowitz, Sahni and Rajasekaran, Computer Algorithms Part 2 1 3 Maximum Selection Problem : Given n numbers, x 1, x 2,, x
More informationAdvanced Algorithms and Data Structures
Advanced Algorithms and Data Structures Prof. Tapio Elomaa Course Basics A new 7 credit unit course Replaces OHJ-2156 Analysis of Algorithms We take things a bit further than OHJ-2156 We will assume familiarity
More informationCSC Design and Analysis of Algorithms. Lecture 6. Divide and Conquer Algorithm Design Technique. Divide-and-Conquer
CSC 8301- Design and Analysis of Algorithms Lecture 6 Divide and Conuer Algorithm Design Techniue Divide-and-Conuer The most-well known algorithm design strategy: 1. Divide a problem instance into two
More informationLecture 2: Getting Started
Lecture 2: Getting Started Insertion Sort Our first algorithm is Insertion Sort Solves the sorting problem Input: A sequence of n numbers a 1, a 2,..., a n. Output: A permutation (reordering) a 1, a 2,...,
More informationChapter 4. Divide-and-Conquer. Copyright 2007 Pearson Addison-Wesley. All rights reserved.
Chapter 4 Divide-and-Conquer Copyright 2007 Pearson Addison-Wesley. All rights reserved. Divide-and-Conquer The most-well known algorithm design strategy: 2. Divide instance of problem into two or more
More informationCOE428 Lecture Notes Week 1 (Week of January 9, 2017)
COE428 Lecture Notes: Week 1 1 of 10 COE428 Lecture Notes Week 1 (Week of January 9, 2017) Table of Contents COE428 Lecture Notes Week 1 (Week of January 9, 2017)...1 Announcements...1 Topics...1 Informal
More informationRecurrences and Divide-and-Conquer
Recurrences and Divide-and-Conquer Frits Vaandrager Institute for Computing and Information Sciences 16th November 2017 Frits Vaandrager 16th November 2017 Lecture 9 1 / 35 Algorithm Design Strategies
More informationDivide-and-Conquer. The most-well known algorithm design strategy: smaller instances. combining these solutions
Divide-and-Conquer The most-well known algorithm design strategy: 1. Divide instance of problem into two or more smaller instances 2. Solve smaller instances recursively 3. Obtain solution to original
More informationAdvanced Algorithms and Data Structures
Advanced Algorithms and Data Structures Prof. Tapio Elomaa tapio.elomaa@tut.fi Course Prerequisites A seven credit unit course Replaced OHJ-2156 Analysis of Algorithms We take things a bit further than
More informationChapter 1 Programming: A General Overview
Chapter 1 Programming: A General Overview 2 Introduction This class is an introduction to the design, implementation, and analysis of algorithms. Examples: sorting large amounts of data organizing information
More informationJana Kosecka. Linear Time Sorting, Median, Order Statistics. Many slides here are based on E. Demaine, D. Luebke slides
Jana Kosecka Linear Time Sorting, Median, Order Statistics Many slides here are based on E. Demaine, D. Luebke slides Insertion sort: Easy to code Fast on small inputs (less than ~50 elements) Fast on
More informationCSE 421 Algorithms: Divide and Conquer
CSE 42 Algorithms: Divide and Conquer Larry Ruzzo Thanks to Paul Beame, Kevin Wayne for some slides Outline: General Idea algorithm design paradigms: divide and conquer Review of Merge Sort Why does it
More information6/12/2013. Introduction to Algorithms (2 nd edition) Overview. The Sorting Problem. Chapter 2: Getting Started. by Cormen, Leiserson, Rivest & Stein
Introduction to Algorithms (2 nd edition) by Cormen, Leiserson, Rivest & Stein Chapter 2: Getting Started (slides enhanced by N. Adlai A. DePano) Overview Aims to familiarize us with framework used throughout
More informationPRAM Divide and Conquer Algorithms
PRAM Divide and Conquer Algorithms (Chapter Five) Introduction: Really three fundamental operations: Divide is the partitioning process Conquer the the process of (eventually) solving the eventual base
More informationLecture 13: Divide and Conquer (1997) Steven Skiena. skiena
Lecture 13: Divide and Conquer (1997) Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Problem Solving Techniques Most
More informationDivide and Conquer Algorithms
Divide and Conquer Algorithms T. M. Murali February 19, 2009 Divide and Conquer Break up a problem into several parts. Solve each part recursively. Solve base cases by brute force. Efficiently combine
More informationTo begin this textbook, we need to start with a refresher of the topics of numbers and numbering systems.
1.1 Integers To begin this textbook, we need to start with a refresher of the topics of numbers and numbering systems. We will start, here, with a recap of the simplest of numbering systems, the integers.
More informationAlgorithms (IX) Guoqiang Li. School of Software, Shanghai Jiao Tong University
Algorithms (IX) Guoqiang Li School of Software, Shanghai Jiao Tong University Q: What we have learned in Algorithm? Algorithm Design Algorithm Design Basic methodologies: Algorithm Design Algorithm Design
More informationLecture 2 Algorithms with numbers
Advanced Algorithms Floriano Zini Free University of Bozen-Bolzano Faculty of Computer Science Academic Year 2013-2014 Lecture 2 Algorithms with numbers 1 RSA Algorithm Why does RSA work? RSA is based
More informationThe divide-and-conquer paradigm involves three steps at each level of the recursion: Divide the problem into a number of subproblems.
2.3 Designing algorithms There are many ways to design algorithms. Insertion sort uses an incremental approach: having sorted the subarray A[1 j - 1], we insert the single element A[j] into its proper
More informationDIVIDE & CONQUER. Problem of size n. Solution to sub problem 1
DIVIDE & CONQUER Definition: Divide & conquer is a general algorithm design strategy with a general plan as follows: 1. DIVIDE: A problem s instance is divided into several smaller instances of the same
More informationDivide and Conquer 4-0
Divide and Conquer 4-0 Divide-and-Conquer The most-well known algorithm design strategy: 1. Divide instance of problem into two or more smaller instances 2. Solve smaller instances recursively 3. Obtain
More informationCS/COE 1501 cs.pitt.edu/~bill/1501/ More Math
CS/COE 1501 cs.pitt.edu/~bill/1501/ More Math Exponentiation x y Can easily compute with a simple algorithm: Runtime? ans = 1 i = y while i > 0: ans = ans * x i-- 2 Just like with multiplication, let s
More informationCSE 332: Data Structures & Parallelism Lecture 12: Comparison Sorting. Ruth Anderson Winter 2019
CSE 332: Data Structures & Parallelism Lecture 12: Comparison Sorting Ruth Anderson Winter 2019 Today Sorting Comparison sorting 2/08/2019 2 Introduction to sorting Stacks, queues, priority queues, and
More informationTheory and Algorithms Introduction: insertion sort, merge sort
Theory and Algorithms Introduction: insertion sort, merge sort Rafael Ramirez rafael@iua.upf.es Analysis of algorithms The theoretical study of computer-program performance and resource usage. What s also
More informationLecture 3: Sorting 1
Lecture 3: Sorting 1 Sorting Arranging an unordered collection of elements into monotonically increasing (or decreasing) order. S = a sequence of n elements in arbitrary order After sorting:
More informationSelection (deterministic & randomized): finding the median in linear time
Lecture 4 Selection (deterministic & randomized): finding the median in linear time 4.1 Overview Given an unsorted array, how quickly can one find the median element? Can one do it more quickly than bysorting?
More informationDivide-and-Conquer Algorithms
Divide-and-Conquer Algorithms Divide and Conquer Three main steps Break input into several parts, Solve the problem in each part recursively, and Combine the solutions for the parts Contribution Applicable
More informationII (Sorting and) Order Statistics
II (Sorting and) Order Statistics Heapsort Quicksort Sorting in Linear Time Medians and Order Statistics 8 Sorting in Linear Time The sorting algorithms introduced thus far are comparison sorts Any comparison
More informationChapter 1 Divide and Conquer Algorithm Theory WS 2013/14 Fabian Kuhn
Chapter 1 Divide and Conquer Algorithm Theory WS 2013/14 Fabian Kuhn Divide And Conquer Principle Important algorithm design method Examples from Informatik 2: Sorting: Mergesort, Quicksort Binary search
More informationAlgorithms and Data Structures (INF1) Lecture 7/15 Hua Lu
Algorithms and Data Structures (INF1) Lecture 7/15 Hua Lu Department of Computer Science Aalborg University Fall 2007 This Lecture Merge sort Quick sort Radix sort Summary We will see more complex techniques
More informationDivide and Conquer. Design and Analysis of Algorithms Andrei Bulatov
Divide and Conquer Design and Analysis of Algorithms Andrei Bulatov Algorithms Divide and Conquer 4-2 Divide and Conquer, MergeSort Recursive algorithms: Call themselves on subproblem Divide and Conquer
More information08 A: Sorting III. CS1102S: Data Structures and Algorithms. Martin Henz. March 10, Generated on Tuesday 9 th March, 2010, 09:58
08 A: Sorting III CS1102S: Data Structures and Algorithms Martin Henz March 10, 2010 Generated on Tuesday 9 th March, 2010, 09:58 CS1102S: Data Structures and Algorithms 08 A: Sorting III 1 1 Recap: Sorting
More informationAnalyzing Complexity of Lists
Analyzing Complexity of Lists Operation Sorted Array Sorted Linked List Unsorted Array Unsorted Linked List Search( L, x ) O(logn) O( n ) O( n ) O( n ) Insert( L, x ) O(logn) O( n ) + O( 1 ) O( 1 ) + O(
More informationDivide and Conquer Algorithms. Sathish Vadhiyar
Divide and Conquer Algorithms Sathish Vadhiyar Introduction One of the important parallel algorithm models The idea is to decompose the problem into parts solve the problem on smaller parts find the global
More informationn = 1 What problems are interesting when n is just 1?
What if n=1??? n = 1 What problems are interesting when n is just 1? Sorting? No Median finding? No Addition? How long does it take to add one pair of numbers? Multiplication? How long does it take to
More informationAlgorithm Design Techniques part I
Algorithm Design Techniques part I Divide-and-Conquer. Dynamic Programming DSA - lecture 8 - T.U.Cluj-Napoca - M. Joldos 1 Some Algorithm Design Techniques Top-Down Algorithms: Divide-and-Conquer Bottom-Up
More informationCS302 Topic: Algorithm Analysis #2. Thursday, Sept. 21, 2006
CS302 Topic: Algorithm Analysis #2 Thursday, Sept. 21, 2006 Analysis of Algorithms The theoretical study of computer program performance and resource usage What s also important (besides performance/resource
More informationDynamic Programming Group Exercises
Name: Name: Name: Dynamic Programming Group Exercises Adapted from material by Cole Frederick Please work the following problems in groups of 2 or 3. Use additional paper as needed, and staple the sheets
More informationPresentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, Merge Sort & Quick Sort
Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 Merge Sort & Quick Sort 1 Divide-and-Conquer Divide-and conquer is a general algorithm
More informationLecture 1. Introduction / Insertion Sort / Merge Sort
Lecture 1. Introduction / Insertion Sort / Merge Sort T. H. Cormen, C. E. Leiserson and R. L. Rivest Introduction to Algorithms, 3nd Edition, MIT Press, 2009 Sungkyunkwan University Hyunseung Choo choo@skku.edu
More informationHow invariants help writing loops Author: Sander Kooijmans Document version: 1.0
How invariants help writing loops Author: Sander Kooijmans Document version: 1.0 Why this document? Did you ever feel frustrated because of a nasty bug in your code? Did you spend hours looking at the
More informationLecture 3. Recurrences / Heapsort
Lecture 3. Recurrences / Heapsort T. H. Cormen, C. E. Leiserson and R. L. Rivest Introduction to Algorithms, 3rd Edition, MIT Press, 2009 Sungkyunkwan University Hyunseung Choo choo@skku.edu Copyright
More informationMidterm CSE 21 Spring 2012
Signature Name Student ID Midterm CSE 21 Spring 2012 Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 _ (20 points) _ (15 points) _ (13 points) _ (23 points) _ (10 points) _ (8 points) Total _ (89 points) (84
More informationDivide and Conquer. Algorithm D-and-C(n: input size)
Divide and Conquer Algorithm D-and-C(n: input size) if n n 0 /* small size problem*/ Solve problem without futher sub-division; else Divide into m sub-problems; Conquer the sub-problems by solving them
More informationRandomized Algorithms, Quicksort and Randomized Selection
CMPS 2200 Fall 2017 Randomized Algorithms, Quicksort and Randomized Selection Carola Wenk Slides by Carola Wenk and Charles Leiserson CMPS 2200 Intro. to Algorithms 1 Deterministic Algorithms Runtime for
More informationCOMP 250 Fall recurrences 2 Oct. 13, 2017
COMP 250 Fall 2017 15 - recurrences 2 Oct. 13, 2017 Here we examine the recurrences for mergesort and quicksort. Mergesort Recall the mergesort algorithm: we divide the list of things to be sorted into
More informationChoice of C++ as Language
EECS 281: Data Structures and Algorithms Principles of Algorithm Analysis Choice of C++ as Language All algorithms implemented in this book are in C++, but principles are language independent That is,
More informationDynamic Programming. Introduction, Weighted Interval Scheduling, Knapsack. Tyler Moore. Lecture 15/16
Dynamic Programming Introduction, Weighted Interval Scheduling, Knapsack Tyler Moore CSE, SMU, Dallas, TX Lecture /6 Greedy. Build up a solution incrementally, myopically optimizing some local criterion.
More informationTaking Stock. IE170: Algorithms in Systems Engineering: Lecture 5. The Towers of Hanoi. Divide and Conquer
Taking Stock IE170: Algorithms in Systems Engineering: Lecture 5 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University January 24, 2007 Last Time In-Place, Out-of-Place Count
More informationCSC236 Week 5. Larry Zhang
CSC236 Week 5 Larry Zhang 1 Logistics Test 1 after lecture Location : IB110 (Last names A-S), IB 150 (Last names T-Z) Length of test: 50 minutes If you do really well... 2 Recap We learned two types of
More informationCSE 202: Design and Analysis of Algorithms Lecture 5
CSE 202: Design and Analysis of Algorithms Lecture 5 Instructor: Kamalika Chaudhuri Announcements Kamalika s Office hours today moved to tomorrow, 12:15-1:15pm Homework 1 due now Midterm on Feb 14 Algorithm
More informationMidterm 1. CS Intermediate Data Structures and Algorithms. October 23, 2013
Midterm 1 CS 141 - Intermediate Data Structures and Algorithms October 23, 2013 By taking this exam, I affirm that all work is entirely my own. I understand what constitutes cheating, and that if I cheat
More informationPseudo code of algorithms are to be read by.
Cs502 Quiz No1 Complete Solved File Pseudo code of algorithms are to be read by. People RAM Computer Compiler Approach of solving geometric problems by sweeping a line across the plane is called sweep.
More informationTest 1 Review Questions with Solutions
CS3510 Design & Analysis of Algorithms Section A Test 1 Review Questions with Solutions Instructor: Richard Peng Test 1 in class, Wednesday, Sep 13, 2017 Main Topics Asymptotic complexity: O, Ω, and Θ.
More informationDesign and Analysis of Algorithms
Design and Analysis of Algorithms Instructor: SharmaThankachan Lecture 10: Greedy Algorithm Slides modified from Dr. Hon, with permission 1 About this lecture Introduce Greedy Algorithm Look at some problems
More informationSo far... Finished looking at lower bounds and linear sorts.
So far... Finished looking at lower bounds and linear sorts. Next: Memoization -- Optimization problems - Dynamic programming A scheduling problem Matrix multiplication optimization Longest Common Subsequence
More informationIntroduction to Algorithms 6.046J/18.401J
Introduction to Algorithms 6.046J/18.401J LECTURE 1 Analysis of Algorithms Insertion sort Merge sort Prof. Charles E. Leiserson Course information 1. Staff. Prerequisites 3. Lectures 4. Recitations 5.
More informationLecture 15 : Review DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/10/2011 Lecture 15 : Review Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Today slectureservesasareviewofthematerialthatwillappearonyoursecondmidtermexam.
More informationData Structures and Algorithms. Part 2
1 Data Structures and Algorithms Part 2 Werner Nutt 2 Acknowledgments The course follows the book Introduction to Algorithms, by Cormen, Leiserson, Rivest and Stein, MIT Press [CLRST]. Many examples displayed
More informationQB LECTURE #1: Algorithms and Dynamic Programming
QB LECTURE #1: Algorithms and Dynamic Programming Adam Siepel Nov. 16, 2015 2 Plan for Today Introduction to algorithms Simple algorithms and running time Dynamic programming Soon: sequence alignment 3
More informationThe divide and conquer strategy has three basic parts. For a given problem of size n,
1 Divide & Conquer One strategy for designing efficient algorithms is the divide and conquer approach, which is also called, more simply, a recursive approach. The analysis of recursive algorithms often
More informationUnit-2 Divide and conquer 2016
2 Divide and conquer Overview, Structure of divide-and-conquer algorithms, binary search, quick sort, Strassen multiplication. 13% 05 Divide-and- conquer The Divide and Conquer Paradigm, is a method of
More informationDynamic Programming Intro
Dynamic Programming Intro Imran Rashid University of Washington February 15, 2008 Dynamic Programming Outline: General Principles Easy Examples Fibonacci, Licking Stamps Meatier examples RNA Structure
More informationReading 8 : Recursion
CS/Math 40: Introduction to Discrete Mathematics Fall 015 Instructors: Beck Hasti, Gautam Prakriya Reading 8 : Recursion 8.1 Recursion Recursion in computer science and mathematics refers to the idea of
More informationDesign and Analysis of Algorithms
Design and Analysis of Algorithms CSE 5311 Lecture 8 Sorting in Linear Time Junzhou Huang, Ph.D. Department of Computer Science and Engineering CSE5311 Design and Analysis of Algorithms 1 Sorting So Far
More informationMultiple-choice (35 pt.)
CS 161 Practice Midterm I Summer 2018 Released: 7/21/18 Multiple-choice (35 pt.) 1. (2 pt.) Which of the following asymptotic bounds describe the function f(n) = n 3? The bounds do not necessarily need
More informationMerge Sort. Algorithm Analysis. November 15, 2017 Hassan Khosravi / Geoffrey Tien 1
Merge Sort Algorithm Analysis November 15, 2017 Hassan Khosravi / Geoffrey Tien 1 The story thus far... CPSC 259 topics up to this point Priority queue Abstract data types Stack Queue Dictionary Tools
More information(Refer Slide Time: 01.26)
Data Structures and Algorithms Dr. Naveen Garg Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture # 22 Why Sorting? Today we are going to be looking at sorting.
More informationAlgorithm Design and Recursion. Search and Sort Algorithms
Algorithm Design and Recursion Search and Sort Algorithms Objectives To understand the basic techniques for analyzing the efficiency of algorithms. To know what searching is and understand the algorithms
More informationChapter 3:- Divide and Conquer. Compiled By:- Sanjay Patel Assistant Professor, SVBIT.
Chapter 3:- Divide and Conquer Compiled By:- Assistant Professor, SVBIT. Outline Introduction Multiplying large Integers Problem Problem Solving using divide and conquer algorithm - Binary Search Sorting
More informationUNIT 1 BASICS OF AN ALGORITHM
UNIT 1 BASICS OF AN ALGORITHM Basics of an Algorithm Structure Page Nos. 1.0 Introduction 5 1.1 Objectives 6 1.2. Analysis and Complexity of Algorithms 6 1.3 Basic Technique for Design of Efficient Algorithms
More informationAlgorithms and Data Structures, or
Algorithms and Data Structures, or... Classical Algorithms of the 50s, 60s and 70s Mary Cryan A&DS Lecture 1 1 Mary Cryan Our focus Emphasis is Algorithms ( Data Structures less important). Most of the
More informationChapter 1 Divide and Conquer Algorithm Theory WS 2014/15 Fabian Kuhn
Chapter 1 Divide and Conquer Algorithm Theory WS 2014/15 Fabian Kuhn Divide And Conquer Principle Important algorithm design method Examples from Informatik 2: Sorting: Mergesort, Quicksort Binary search
More informationSORTING AND SELECTION
2 < > 1 4 8 6 = 9 CHAPTER 12 SORTING AND SELECTION ACKNOWLEDGEMENT: THESE SLIDES ARE ADAPTED FROM SLIDES PROVIDED WITH DATA STRUCTURES AND ALGORITHMS IN JAVA, GOODRICH, TAMASSIA AND GOLDWASSER (WILEY 2016)
More informationTowards a Memory-Efficient Knapsack DP Algorithm
Towards a Memory-Efficient Knapsack DP Algorithm Sanjay Rajopadhye The 0/1 knapsack problem (0/1KP) is a classic problem that arises in computer science. The Wikipedia entry http://en.wikipedia.org/wiki/knapsack_problem
More informationMergesort again. 1. Split the list into two equal parts
Quicksort Mergesort again 1. Split the list into two equal parts 5 3 9 2 8 7 3 2 1 4 5 3 9 2 8 7 3 2 1 4 Mergesort again 2. Recursively mergesort the two parts 5 3 9 2 8 7 3 2 1 4 2 3 5 8 9 1 2 3 4 7 Mergesort
More informationWe will give examples for each of the following commonly used algorithm design techniques:
Review This set of notes provides a quick review about what should have been learned in the prerequisite courses. The review is helpful to those who have come from a different background; or to those who
More informationLecture #2. 1 Overview. 2 Worst-Case Analysis vs. Average Case Analysis. 3 Divide-and-Conquer Design Paradigm. 4 Quicksort. 4.
COMPSCI 330: Design and Analysis of Algorithms 8/28/2014 Lecturer: Debmalya Panigrahi Lecture #2 Scribe: Yilun Zhou 1 Overview This lecture presents two sorting algorithms, quicksort and mergesort, that
More informationCSE373: Data Structures and Algorithms Lecture 4: Asymptotic Analysis. Aaron Bauer Winter 2014
CSE373: Data Structures and Algorithms Lecture 4: Asymptotic Analysis Aaron Bauer Winter 2014 Previously, on CSE 373 We want to analyze algorithms for efficiency (in time and space) And do so generally
More informationThe Limits of Sorting Divide-and-Conquer Comparison Sorts II
The Limits of Sorting Divide-and-Conquer Comparison Sorts II CS 311 Data Structures and Algorithms Lecture Slides Monday, October 12, 2009 Glenn G. Chappell Department of Computer Science University of
More informationCSC Design and Analysis of Algorithms. Lecture 5. Decrease and Conquer Algorithm Design Technique. Decrease-and-Conquer
CSC 8301- Design and Analysis of Algorithms Lecture 5 Decrease and Conquer Algorithm Design Technique Decrease-and-Conquer This algorithm design technique is based on exploiting a relationship between
More informationESc101 : Fundamental of Computing
ESc101 : Fundamental of Computing I Semester 2008-09 Lecture 37 Analyzing the efficiency of algorithms. Algorithms compared Sequential Search and Binary search GCD fast and GCD slow Merge Sort and Selection
More informationCS Divide and Conquer
CS483-07 Divide and Conquer Instructor: Fei Li Room 443 ST II Office hours: Tue. & Thur. 1:30pm - 2:30pm or by appointments lifei@cs.gmu.edu with subject: CS483 http://www.cs.gmu.edu/ lifei/teaching/cs483_fall07/
More informationIs the statement sufficient? If both x and y are odd, is xy odd? 1) xy 2 < 0. Odds & Evens. Positives & Negatives. Answer: Yes, xy is odd
Is the statement sufficient? If both x and y are odd, is xy odd? Is x < 0? 1) xy 2 < 0 Positives & Negatives Answer: Yes, xy is odd Odd numbers can be represented as 2m + 1 or 2n + 1, where m and n are
More informationEECS 2011M: Fundamentals of Data Structures
M: Fundamentals of Data Structures Instructor: Suprakash Datta Office : LAS 3043 Course page: http://www.eecs.yorku.ca/course/2011m Also on Moodle Note: Some slides in this lecture are adopted from James
More information