The Selection Problem


 Stephen Wood
 1 years ago
 Views:
Transcription
1 The Selection Problem The selection problem: given an integer k and a list x 1,..., x n of n elements find the kth smallest element in the list Example: the 3rd smallest element of the following list is An O(n log n) solution: sort the list (O(n log n)) pick the kth element of the sorted list (O(1)) Can we find the kth smallest element faster?
2 QuickSelect Quickselect of the kth smallest element in the list S: based on pruneandsearch paradigm Prune: pick random element x (pivot) from S and split S in: L elements < x, E elements == x, G elements > x Search: L E G partitioning into L, E and G works precisely as for quicksort if k L then return quickselect(k, L) if L < k L + E then return x if k > L + E then return quickselect(k L E, G)
3 QuickSelect Visualization Quickselect can be displayed by a sequence of nodes: each node represents recursive call and stores: k, the sequence, and the pivot element k = 5, S = ( ) k = 2, S = ( ) k = 2, S = ( ) k = 1, S = (7 6 5) found 5
4 QuickSelect: Running Time The worstcase running is O(n 2 ) time: if the pivot is always the minimal or maximal element The expected running time is O(n) (compare with quicksort): with probability 0.5 the recursive call is good: (3/4)n size T (n) b a n + T ( 3 4 n) a is the time steps for partitioning per element b is the expected number of calls until a good call b = 2 (average number of coins to toss until head shows up) Let m = 2 a n, then T (n) 2 a n + T ((3/4)n) 2 a n + 2 a (3/4) n + 2 a (3/4) 2 n... = 8 a n O(n) (geometric series)
5 QuickSelect: Median of Medians We can do selection in O(n) worstcase time. Idea: recursively use select itself to find a good pivot: divide S into n/5 sets of 5 elements find a median in each set (baby median) recursively use select to find the median of the medians min. size of L min. size of G 3 The minimal size of L and G is 0.3 n.
6 QuickSelect: Median of Medians We know: The minimal size of L and G is 0.3 n. Thus the maximal size of L and G is 0.7 n. Let b N such that: partitioning of a list of size n takes at most b n time, finding the baby medians takes at most b n time, the base case n 1 takes at most b time. We derive a recurrence equation for the time complexity: b if n 1 T (0.7 n) + T (0.2 n) + 2 b n if n > 1 We will see how to solve recurrence equations...
7 Fundamental Techniques We have considered algorithms for solving special problems. Now we consider a few fundamental techniques: DevideandConquer The Greedy Method Dynamic Programming
8 DivideandConquer DivideandConquer is a general algorithm design paradigm: Divide: divide input S into k 2 disjoint subsets S 1,...,S k Recur: recursively solve the subproblems S 1,...,S k Conquer: combine solutions for S 1,...,S k to solution for S (the base case of the recursion are problems of size 0 or 1) We have already seen examples: merge sort quick sort bottomup heap construction Focus now: analysing time complexity by recurrence equations.
9 Methods for Solving Recurrence Equation We consider 3 methods for solving recurrence equation: Iterative substitution method, Recursion tree method, Guessandtest method, and Master method.
10 Iterative Substitution Iterative substitution technique works as follows: iteratively apply the recurrence equations to itself hope to find a pattern Example 1 if n 1 T (n 1) + 2 if n > 1 We start with T (n) and apply the recursive case: T (n 1) + 2 = T (n 2) + 4 = T (n k) + 2 k For k = n 1 we reach the base case T (n k) = 1 thus: (n 1) = 2 n 1
11 MergeSort Review Mergesort of a list S with n elements works as follows: Divide: divide S into two lists S 1, S 2 of n/2 elements Recur: recursively sort S 1, S 2 Conquer: merge S 1 and S 2 into a sorting of S Let b N such that: merging two lists of size n/2 takes at most b n time, and the base case n 1 takes at most b time We obtain the following recurrence equation for mergesort: b if n 1 2 T (n/2) + b n if n > 1 We search a closed solution for the equation, that is:... where T (n) does not occur in the right side
12 Example: MergeSort b if n 1 2 T (n/2) + b n if n > 1 We assume that n is a power of 2: n = 2 k (that is, k = log 2 n) allowed since we are interested in asymptotic behaviour We start with T (n) and apply the recursive case: 2 T (n/2) + b n = 2 (2 T (n/2 2 ) + b (n/2) ) + b n = 2 2 T (n/2 2 ) + 2 b n = 2 3 T (n/2 3 ) + 3 b n = 2 k T (n/2 k ) + k b n = n b + (log 2 n) b n Thus b (n + n log 2 n) O(n log 2 n).
13 The Recursion Tree Method The recursion tree method is a visual approach: draw the recursion tree and hope to find a pattern b if n 2 3 T (n/3) + b n if n > 2 For a node with input size k: work at this node is b k. depth nodes size 0 1 n 1 3 n/3 i 3 i n/3 i Thus the work at depth i is 3 i b n/3 i = b n. The height of the tree is log 3 n. Thus T (n) is O(n log 3 n).
14 GuessandTest Method The guessandtest method works as follows: we guess a solution (or an upper bound) we prove that the solution is true by induction Example 1 if n 1 T (n/2) + 2 n if n > 1 Guess: T (n) 2 n for n = 1 it holds for n > 1 we have: T (n/2) + 2 n 2 n/2 + 2 n = 3 n Wrong guess: we cannot make 3 n smaller or equal to 2 n.
15 Example, continued 1 if n 1 T (n/2) + 2 n if n > 1 New guess: T (n) 4 n for n = 1 it holds for n > 1 we have: T (n/2) + 2 n 4 n/2 + 2 n = 4 n (by induction hypothesis) This time the guess was good: 4 n 4 n. Thus T (n) 4 n.
16 Example: QuickSelect with Median of Median b if n 1 T (0.7 n) + T (0.2 n) + 2 b n if n > 1 Guess: T (n) 20 b n for n = 1 it holds b 20 b 1 for n > 1 we have: T (0.7 n) + T (0.2 n) + 2 b n b n b n + 2 b n = b n + 2 b n = 18 b n + 2 b n = 20 b n (by IH) Thus the guess was good. This shows that quickselect with median of median is O(n).
17 Master method Many divideandconquer recurrence equations have the form: c if n < d at (n/b) + f (n) if n d Theorem (The Master Theorem) 1. if f (n) is O(n log b a ε ), then T (n) is Θ(n log b a ) 2. if f (n) is Θ(n log b a log k n), then T (n) is Θ(n log b a log k+1 n) 3. if f (n) is Ω(n log b a+ε ), then T (n) is Θ(f (n)), provided af (n/b) δf (n) for some δ < 1.
18 Master Method: Example 1 c at (n/b) + f (n) if n < d if n d Theorem (The Master Theorem) 1. if f (n) is O(n log b a ε ), then T (n) is Θ(n log b a ) 2. if f (n) is Θ(n log b a log k n), then T (n) is Θ(n log b a log k+1 n) 3. if f (n) is Ω(n log b a+ε ), then T (n) is Θ(f (n)), provided af (n/b) δf (n) for some δ < 1. Example 4 T (n/2) + n Solution: log b a = 2, thus case 1 says Θn 2.
19 Master Method: Example 2 c at (n/b) + f (n) if n < d if n d Theorem (The Master Theorem) 1. if f (n) is O(n log b a ε ), then T (n) is Θ(n log b a ) 2. if f (n) is Θ(n log b a log k n), then T (n) is Θ(n log b a log k+1 n) 3. if f (n) is Ω(n log b a+ε ), then T (n) is Θ(f (n)), provided af (n/b) δf (n) for some δ < 1. Example 2 T (n/2) + n log n Solution: log b a = 1, thus case 2 says Θn log 2 n.
20 Master Method: Example 3 c at (n/b) + f (n) if n < d if n d Theorem (The Master Theorem) 1. if f (n) is O(n log b a ε ), then T (n) is Θ(n log b a ) 2. if f (n) is Θ(n log b a log k n), then T (n) is Θ(n log b a log k+1 n) 3. if f (n) is Ω(n log b a+ε ), then T (n) is Θ(f (n)), provided af (n/b) δf (n) for some δ < 1. Example T (n/3) + n log n Solution: log b a = 0, thus case 3 says Θn log n.
21 Master Method: Example 4 c at (n/b) + f (n) if n < d if n d Theorem (The Master Theorem) 1. if f (n) is O(n log b a ε ), then T (n) is Θ(n log b a ) 2. if f (n) is Θ(n log b a log k n), then T (n) is Θ(n log b a log k+1 n) 3. if f (n) is Ω(n log b a+ε ), then T (n) is Θ(f (n)), provided af (n/b) δf (n) for some δ < 1. Example 8 T (n/2) + n 2 Solution: log b a = 3, thus case 1 says Θn 3.
22 Master Method: Example 5 c at (n/b) + f (n) if n < d if n d Theorem (The Master Theorem) 1. if f (n) is O(n log b a ε ), then T (n) is Θ(n log b a ) 2. if f (n) is Θ(n log b a log k n), then T (n) is Θ(n log b a log k+1 n) 3. if f (n) is Ω(n log b a+ε ), then T (n) is Θ(f (n)), provided af (n/b) δf (n) for some δ < 1. Example 9 T (n/3) + n 3 Solution: log b a = 2, thus case 3 says Θn 3.
23 Master Method: Example 6 c at (n/b) + f (n) if n < d if n d Theorem (The Master Theorem) 1. if f (n) is O(n log b a ε ), then T (n) is Θ(n log b a ) 2. if f (n) is Θ(n log b a log k n), then T (n) is Θ(n log b a log k+1 n) 3. if f (n) is Ω(n log b a+ε ), then T (n) is Θ(f (n)), provided af (n/b) δf (n) for some δ < 1. Example T (n/2) + 1 binary search Solution: log b a = 0, thus case 2 says Θlog n.
24 Master Method: Example 7 c at (n/b) + f (n) if n < d if n d Theorem (The Master Theorem) 1. if f (n) is O(n log b a ε ), then T (n) is Θ(n log b a ) 2. if f (n) is Θ(n log b a log k n), then T (n) is Θ(n log b a log k+1 n) 3. if f (n) is Ω(n log b a+ε ), then T (n) is Θ(f (n)), provided af (n/b) δf (n) for some δ < 1. Example 2 T (n/2) + log n heap construction Solution: log b a = 0, thus case 1 says Θn.
Merge 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 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 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 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 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 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 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 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 informationCS473  Algorithms I
CS473  Algorithms I Lecture 4 The DivideandConquer Design Paradigm View in slideshow mode 1 Reminder: Merge Sort Input array A sort this half sort this half Divide Conquer merge two sorted halves Combine
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 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 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 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 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 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 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 informationDivide 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 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 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 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 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 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 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 informationSimplified Analyses of Randomized Algorithms for Searching, Sorting, and Selection
Simplified Analyses of Randomized Algorithms for Searching, Sorting, and Selection i Michael T. Goodrich Roberto Tamassia Dept. of Info. & Computer Science Dept. of Computer Science University of California
More informationthe recursiontree method
the recursion method recurrence into a 1 recurrence into a 2 MCS 360 Lecture 39 Introduction to Data Structures Jan Verschelde, 22 November 2010 recurrence into a The for consists of two steps: 1 Guess
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 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 informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NPComplete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More informationCSE 326, Data Structures. Sample Final Exam. Problem Max Points Score 1 14 (2x7) 2 18 (3x6) 3 4 4 7 5 9 6 16 7 8 8 4 9 8 10 4 Total 92.
Name: Email ID: CSE 326, Data Structures Section: Sample Final Exam Instructions: The exam is closed book, closed notes. Unless otherwise stated, N denotes the number of elements in the data structure
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 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 informationA Note on Maximum Independent Sets in Rectangle Intersection Graphs
A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,
More informationTo My Parents Laxmi and Modaiah. To My Family Members. To My Friends. To IIT Bombay. To All Hard Workers
To My Parents Laxmi and Modaiah To My Family Members To My Friends To IIT Bombay To All Hard Workers Copyright 2010 by CareerMonk.com All rights reserved. Designed by Narasimha Karumanchi Printed in
More informationSORTING. Review of Sorting. Merge Sort. Sets. sorting
Review of Sorting Merge Sort Sets SORTING 1 Sorting Algorithms Selection Sort uses a priority queue P implemented with an unsorted sequence:  Phase 1: the insertion of an item into P takes O(1) time;
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 informationAlgorithms. Margaret M. Fleck. 18 October 2010
Algorithms Margaret M. Fleck 18 October 2010 These notes cover how to analyze the running time of algorithms (sections 3.1, 3.3, 4.4, and 7.1 of Rosen). 1 Introduction The main reason for studying bigo
More informationCMSC 451 Design and Analysis of Computer Algorithms 1
CMSC 4 Design and Analysis of Computer Algorithms David M. Mount Department of Computer Science University of Maryland Fall 003 Copyright, David M. Mount, 004, Dept. of Computer Science, University of
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 informationAPP INVENTOR. Test Review
APP INVENTOR Test Review Main Concepts App Inventor Lists Creating Random Numbers Variables Searching and Sorting Data Linear Search Binary Search Selection Sort Quick Sort Abstraction Modulus Division
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 informationSorting algorithm. Sorting algorithms used in computer science are often classified by:
Sorting algorithm In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most used orders are numerical order and lexicographical
More information1) The postfix expression for the infix expression A+B*(C+D)/F+D*E is ABCD+*F/DE*++
Answer the following 1) The postfix expression for the infix expression A+B*(C+D)/F+D*E is ABCD+*F/DE*++ 2) Which data structure is needed to convert infix notations to postfix notations? Stack 3) The
More informationDESIGN AND ANALYSIS OF ALGORITHMS
CPS 3 DESIGN AND ANALYSIS OF ALGORITHMS Fall 8 Instructor: Herbert Edelsbrunner Teaching Assistant: Zhiqiang Gu CPS 3 Fall Semester of 8 Table of Contents Introduction 3 I DESIGN TECHNIQUES DivideandConquer
More informationKCover of Binary sequence
KCover of Binary sequence Prof Amit Kumar Prof Smruti Sarangi Sahil Aggarwal Swapnil Jain Given a binary sequence, represent all the 1 s in it using at most k covers, minimizing the total length of all
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
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
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 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 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 informationSorting Algorithms. Nelson PaduaPerez Bill Pugh. Department of Computer Science University of Maryland, College Park
Sorting Algorithms Nelson PaduaPerez Bill Pugh Department of Computer Science University of Maryland, College Park Overview Comparison sort Bubble sort Selection sort Tree sort Heap sort Quick sort Merge
More informationSection IV.1: Recursive Algorithms and Recursion Trees
Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller
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 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 informationDivide and Conquer Paradigm
Divide and Conquer Paradigm Melissa Manley California State University Northridge Author s address: 18111 Nordhoff Street, Northridge, CA, 91330 Permission to make digital/hard copy of all or part of this
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 informationCost Model: Work, Span and Parallelism. 1 The RAM model for sequential computation:
CSE341T 08/31/2015 Lecture 3 Cost Model: Work, Span and Parallelism In this lecture, we will look at how one analyze a parallel program written using Cilk Plus. When we analyze the cost of an algorithm
More informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More informationLecture 4 Online and streaming algorithms for clustering
CSE 291: Geometric algorithms Spring 2013 Lecture 4 Online and streaming algorithms for clustering 4.1 Online kclustering To the extent that clustering takes place in the brain, it happens in an online
More informationAlgorithm Design and Analysis Homework #1 Due: 5pm, Friday, October 4, 2013 TA email: ada@csie.ntu.edu.tw. === Homework submission instructions ===
Algorithm Design and Analysis Homework #1 Due: 5pm, Friday, October 4, 2013 TA email: ada@csie.ntu.edu.tw === Homework submission instructions === For Problem 1, commit your source code and a brief documentation
More informationAlgorithms and Data Structures
Algorithm Analysis Page 1 BFHTI: Softwareschule Schweiz Algorithm Analysis Dr. CAS SD01 Algorithm Analysis Page 2 Outline Course and Textbook Overview Analysis of Algorithm PseudoCode and Primitive Operations
More informationPARALLEL PROGRAMMING
PARALLEL PROGRAMMING TECHNIQUES AND APPLICATIONS USING NETWORKED WORKSTATIONS AND PARALLEL COMPUTERS 2nd Edition BARRY WILKINSON University of North Carolina at Charlotte Western Carolina University MICHAEL
More informationInduction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition
Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing
More informationRecurrences and the Complexity of Divide and Conquer Algorithms
2006 Antonio Crznig Recurrences nd the Complexity of Divide nd Conquer Algorithms Antonio Crznig Fculty of Informtics Università dell Svizzer itlin October 12, 2010 2006 Antonio Crznig Outline Anlysis
More informationExam study sheet for CS2711. List of topics
Exam study sheet for CS2711 Here is the list of topics you need to know for the final exam. For each data structure listed below, make sure you can do the following: 1. Give an example of this data structure
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 informationQuestions 1 through 25 are worth 2 points each. Choose one best answer for each.
Questions 1 through 25 are worth 2 points each. Choose one best answer for each. 1. For the singly linked list implementation of the queue, where are the enqueues and dequeues performed? c a. Enqueue in
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 informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora
princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of
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 informationExternal Sorting. Chapter 13. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1
External Sorting Chapter 13 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Why Sort? A classic problem in computer science! Data requested in sorted order e.g., find students in increasing
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 information28 ClosestPoint Problems 
28 ClosestPoint Problems  Geometric problems involving points on the plane usually involve implicit or explicit treatment of distances
More information1 23 Trees: The Basics
CS10: Data Structures and ObjectOriented Design (Fall 2013) November 1, 2013: 23 Trees: Inserting and Deleting Scribes: CS 10 Teaching Team Lecture Summary In this class, we investigated 23 Trees in
More informationBiostatistics 615/815
Merge Sort Biostatistics 615/815 Lecture 8 Notes on Problem Set 2 Union Find algorithms Dynamic Programming Results were very ypositive! You should be gradually becoming g y g comfortable compiling, debugging
More informationDivide and Conquer Examples. Topdown recursive mergesort. Gravitational Nbody problem.
Divide and Conquer Divide the problem into several subproblems of equal size. Recursively solve each subproblem in parallel. Merge the solutions to the various subproblems into a solution for the original
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 information6 March 2007 1. Array Implementation of Binary Trees
Heaps CSE 0 Winter 00 March 00 1 Array Implementation of Binary Trees Each node v is stored at index i defined as follows: If v is the root, i = 1 The left child of v is in position i The right child of
More informationMODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.
MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on
More informationCS5310 Algorithms 3 credit hours 2 hours lecture and 2 hours recitation every week
CS5310 Algorithms 3 credit hours 2 hours lecture and 2 hours recitation every week This course is a continuation of the study of data structures and algorithms, emphasizing methods useful in practice.
More informationSmall Maximal Independent Sets and Faster Exact Graph Coloring
Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected
More informationLearning Example. Machine learning and our focus. Another Example. An example: data (loan application) The data and the goal
Learning Example Chapter 18: Learning from Examples 22c:145 An emergency room in a hospital measures 17 variables (e.g., blood pressure, age, etc) of newly admitted patients. A decision is needed: whether
More information6. Standard Algorithms
6. Standard Algorithms The algorithms we will examine perform Searching and Sorting. 6.1 Searching Algorithms Two algorithms will be studied. These are: 6.1.1. inear Search The inear Search The Binary
More informationBalanced Binary Search Tree
AVL Trees / Slide 1 Balanced Binary Search Tree Worst case height of binary search tree: N1 Insertion, deletion can be O(N) in the worst case We want a tree with small height Height of a binary tree with
More informationData Structures Fibonacci Heaps, Amortized Analysis
Chapter 4 Data Structures Fibonacci Heaps, Amortized Analysis Algorithm Theory WS 2012/13 Fabian Kuhn Fibonacci Heaps Lacy merge variant of binomial heaps: Do not merge trees as long as possible Structure:
More informationExternal Memory Geometric Data Structures
External Memory Geometric Data Structures Lars Arge Department of Computer Science University of Aarhus and Duke University Augues 24, 2005 1 Introduction Many modern applications store and process datasets
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 information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationCpt S 223. School of EECS, WSU
Priority Queues (Heaps) 1 Motivation Queues are a standard mechanism for ordering tasks on a firstcome, firstserved basis However, some tasks may be more important or timely than others (higher priority)
More information5.4 Closest Pair of Points
5.4 Closest Pair of Points Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Fundamental geometric primitive. Graphics, computer
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 informationInternational Journal of Advanced Research in Computer Science and Software Engineering
Volume 3, Issue 7, July 23 ISSN: 2277 28X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Greedy Algorithm:
More informationLecture 6 Online and streaming algorithms for clustering
CSE 291: Unsupervised learning Spring 2008 Lecture 6 Online and streaming algorithms for clustering 6.1 Online kclustering To the extent that clustering takes place in the brain, it happens in an online
More informationClustering UE 141 Spring 2013
Clustering UE 141 Spring 013 Jing Gao SUNY Buffalo 1 Definition of Clustering Finding groups of obects such that the obects in a group will be similar (or related) to one another and different from (or
More informationDecisionTree Learning
DecisionTree Learning Introduction ID3 Attribute selection Entropy, Information, Information Gain Gain Ratio C4.5 Decision Trees TDIDT: TopDown Induction of Decision Trees Numeric Values Missing Values
More informationTreemaps with bounded aspect ratio
technische universiteit eindhoven Department of Mathematics and Computer Science Master s Thesis Treemaps with bounded aspect ratio by Vincent van der Weele Supervisor dr. B. Speckmann Eindhoven, July
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture  17 ShannonFanoElias Coding and Introduction to Arithmetic Coding
More informationHeaps & Priority Queues in the C++ STL 23 Trees
Heaps & Priority Queues in the C++ STL 23 Trees CS 3 Data Structures and Algorithms Lecture Slides Friday, April 7, 2009 Glenn G. Chappell Department of Computer Science University of Alaska Fairbanks
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 information1/18/2013. 5.5year Ph.D. student internships in. Done job hunting recently. Will join Yahoo! Labs soon. Interviewed with
Liangjie Hong Ph.D. Candidate Dept. of Computer Science and Engineering 5.5year Ph.D. student internships in a local software company (2008, 2 months) Yahoo! Labs (2010, 3 months) Yahoo! Labs (2011, 3
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More information