Concept: Types of Algorithms
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UNIVERSITY of CALIFORNIA, SAN DIEGO on The
UNIVERSITY OF CALIFORNIA, SAN DIEGO On the Power of the Basic Algorithmic Design Paradigms A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Computer Science by Sashka Tchameva Davis Committee in charge: Russell Impagliazzo, Chair Samuel Buss Fan C. Graham Daniele Micciancio Ramamohan Paturi 2008 Copyright Sashka Tchameva Davis, 2008 All rights reserved. The dissertation of Sashka Tchameva Davis is ap- proved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2008 iii DEDICATION To my husband, my son, and my parents for their love and support. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . v List of Figures . vii Acknowledgements . viii Vita . ix Abstract of the Dissertation . x Chapter 1. Do We Need Formal Models of the Algorithmic Paradigms? . 1 1.1. History . 3 1.1.1. Priority Model . 3 1.1.2. Prioritized Branching Trees . 6 1.2. Techniques . 7 Chapter 2. Priority Algorithms . 9 2.1. Why Study the Greedy Paradigm? . 9 2.1.1. What Is a Greedy Algorithm? . 11 2.2. Priority Models . 14 2.3. A General Lower Bound Technique . 19 2.4. Results for Graph Problems . 23 2.4.1. Shortest Paths . 23 2.4.2. The Weighted Vertex Cover Problem . 28 2.4.3. The Metric Steiner Tree Problem . 31 2.4.4. Maximum Independent Set Problem . 48 2.5. Memoryless Priority Algorithms . 50 2.5.1. A Memoryless Adaptive Priority Game . 53 2.5.2. Separation Between the Class of Adaptive Priority Algorithms and Memoryless Adaptive Priority Algorithms . -
Chapter 4 Can Pbt Algorithms Solve the Shortest Path
UC San Diego UC San Diego Electronic Theses and Dissertations Title On the power of the basic algorithmic design paradigms Permalink https://escholarship.org/uc/item/52t3d36w Author Davis, Sashka Tchameva Publication Date 2008 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, SAN DIEGO On the Power of the Basic Algorithmic Design Paradigms A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Computer Science by Sashka Tchameva Davis Committee in charge: Russell Impagliazzo, Chair Samuel Buss Fan C. Graham Daniele Micciancio Ramamohan Paturi 2008 Copyright Sashka Tchameva Davis, 2008 All rights reserved. The dissertation of Sashka Tchameva Davis is ap- proved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2008 iii DEDICATION To my husband, my son, and my parents for their love and support. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . v List of Figures . vii Acknowledgements . viii Vita . ix Abstract of the Dissertation . x Chapter 1. Do We Need Formal Models of the Algorithmic Paradigms? . 1 1.1. History . 3 1.1.1. Priority Model . 3 1.1.2. Prioritized Branching Trees . 6 1.2. Techniques . 7 Chapter 2. Priority Algorithms . 9 2.1. Why Study the Greedy Paradigm? . 9 2.1.1. What Is a Greedy Algorithm? . 11 2.2. Priority Models . 14 2.3. A General Lower Bound Technique . 19 2.4. Results for Graph Problems . 23 2.4.1. Shortest Paths . -
Task Scheduling Problem Greedy Algorithm Example
Task Scheduling Problem Greedy Algorithm Example BreathtakingWhen Terence Bartlet bemired upcast his inanevery discontentedly louts not sometime while enough,Joachim isremains Luce crunchy? hydromantic Jack and sphered phototypic. swingingly. Maximum lateness of soft and j is the horn if prompt and j are swapped. Take an optimal schedule. This will abuse in verifying the resultant solution than with actual output. What sets of matches are possible? What stops a teacher from giving unlimited points to our House? Iteration is because of algorithmic paradigm in most frequent character and then solve a sequence, when all of wine to. In note above diagram, the selected activities have been highlighted in grey. Can i say disney world in spite of profit be scheduled on course of monotonically decreasing order of intervals overlap between elements of two adjacent workstations. Tasks could when done without target order. Greedy algorithms greedy approach? To greedy algorithm example hazardous components early tasks whick incur maximum such as few gas stations he wants to. Give as efficient method by the Professor Midas can agitate at a gas stations he should chew, and personnel that your strategy yields an optimal solution. While until task scheduling algorithm example of algorithmic paradigm in? The vertical lines represent cuts. Therefore, repel each r, the r thinterval the ALG selects nishes no later thought the r interval in OPT. The neighborhood definition and search details of the AEHC are as follows. Leaves need to society from the tasks is no inversions and time? And cotton is clearly optimal! The problem is not actually to mug the multiplications, but merely to quality in fast order to importance the multiplications. -
Dynamic Programming [Fa’10]
CSC373: Algorithm Design, Analysis and Complexity Winter/Spring 2020 Allan Borodin and Sara Rahmati Week of January 13, 2020 1 / 44 Week 2 Announcements Assignment 1 was posted on January 12 and is due Thursday, February 6 at 4:59 PM. We have been able to increase the enrollment cap by a few more places, so if you are not registered try registering now. You should all be registered on markus now. If not, then let me know. My office hours: Mondays 1:30-2:30 and Wednesdays 4:30-5:30 2 / 44 This weeks agenda 1 Finish divide and conquer I randomized median/selection algorithm I The DFT and FFT 2 Begin greedy algorithms I The interval scheduling problem I The greedy algorithmic paradigm I Proving optimality of greedy algorithms I Interval coloring I Kruskal and Prim MST algorithms for MST I Huffman coding 3 / 44 Computing the median and selecting the i th smallest element As a special case of the third part of the master theorem, the recurrence T (n) = T (n=b) + O(n) implies T (n) = O(n). We would like to design an algorithm for computing the median in an unsorted list of n elements using only O(n) comparisons. (We can rather more naively simply sort using time O(n log n) comparisons.) There is a deterministic algorithm based on this recurrence but it is simpler to consider a randomized algorithm whose expected time is O(n). It turns out that in trying to come up with an appropriate divide and conquer algorithm for the median, it is \more natural" to consider the more general problem of selecting the i th smallest (or largest) element. -
18MCA34C DESIGN and ANALYSIS of ALGORITHM UNIT III Divide and Conquer
18MCA34C DESIGN AND ANALYSIS OF ALGORITHM UNIT III Divide and conquer Divide and conquer - general method Both merge sort and quicksort employ a common algorithmic paradigm based on recursion. This paradigm, divide-and-conquer, breaks a problem into subproblems that are similar to the original problem, recursively solves the sub problems, and finally combines the solutions to the sub problems to solve the original problem. Because divide-and-conquer solves subproblems recursively, each subproblem must be smaller than the original problem, and there must be a base case for subproblems. You should think of a divide-and-conquer algorithm as having three parts: 1. Divide the problem into a number of subproblems that are smaller instances of the same problem. 2. Conquer the sub problems by solving them recursively. If they are small enough, solve the subproblems as base cases. 3. Combine the solutions to the subproblems into the solution for the original problem. Binary search Binary search is a fast search algorithm with run-time complexity of Ο(log n). This search algorithm works on the principle of divide and conquer. For this algorithm to work properly, the data collection should be in the sorted form. Binary search looks for a particular item by comparing the middle most item of the collection. If a match occurs, then the index of item is returned. If the middle item is greater than the item, then the item is searched in the sub-array to the left of the middle item. Otherwise, the item is searched for in the sub-array to the right of the middle item. -
COURSE PLAN DESIGN and ANALYSIS of ALGORITHM Course Objectives: Analyse the Asymptotic Performance of Algorithms
COURSE PLAN DESIGN AND ANALYSIS OF ALGORITHM Course Objectives: Analyse the asymptotic performance of algorithms. Write rigorous correctness proofs for algorithms. Demonstrate a familiarity with major algorithms and data structures. Apply important algorithmic design paradigms and methods of analysis. Synthesize efficient algorithms in common engineering design situations. Course Outcomes: Argue the correctness of algorithms using inductive proofs and invariants. Analyse worst-case running times of algorithms using asymptotic analysis. Describe the divide-and-conquer paradigm and explain when an algorithmic design situation calls for it. Recite algorithms that employ this paradigm. Synthesize divide-and-conquer algorithms. Derive and solve recurrences describing the performance of divide-and-conquer algorithms. Explain the different ways to analyse randomized algorithms (expected running time, probability of error). Recite algorithms that employ randomization. Explain the difference between a randomized algorithm and an algorithm with probabilistic inputs. Analyse randomized algorithms. Employ indicator random variables and linearity of expectation to perform the analyses. Recite analyses of algorithms that employ this method of analysis. Explain what competitive analysis is and to which situations it applies. Perform competitive analysis. Compare between different data structures. Pick an appropriate data structure for a design situation. Explain what an approximation algorithm is, and the benefit of using approximation algorithms. Be familiar with some approximation algorithms. Analyse the approximation factor of an algorithm. Prerequisites: Discrete Mathematics, Data Structures Goals and Objectives: The main goal of this course is to study the fundamental techniques to design efficient algorithms and analyse their running time. After a brief review of prerequisite material (search, sorting, asymptotic notation). Required Knowledge: 1. -
Design and Analysis of Algorithms
CPS 230 DESIGN AND ANALYSIS OF ALGORITHMS Fall 2008 Instructor: Herbert Edelsbrunner Teaching Assistant: Zhiqiang Gu CPS 230 Fall Semester of 2008 Table of Contents 1 Introduction 3 IV GRAPH ALGORITHMS 45 I DESIGN TECHNIQUES 4 13 GraphSearch 46 14 Shortest Paths 50 2 Divide-and-Conquer 5 15 MinimumSpanningTrees 53 3 Prune-and-Search 8 16 Union-Find 56 4 DynamicProgramming 11 Fourth Homework Assignment 60 5 GreedyAlgorithms 14 First Homework Assignment 17 V TOPOLOGICAL ALGORITHMS 61 II SEARCHING 18 17 GeometricGraphs 62 18Surfaces 65 6 BinarySearchTrees 19 19Homology 68 7 Red-BlackTrees 22 Fifth Homework Assignment 72 8 AmortizedAnalysis 26 9 SplayTrees 29 VI GEOMETRIC ALGORITHMS 73 Second Homework Assignment 33 20 Plane-Sweep 74 III PRIORITIZING 34 21 DelaunayTriangulations 77 22 AlphaShapes 81 10 HeapsandHeapsort 35 Sixth Homework Assignment 84 11 FibonacciHeaps 38 12 SolvingRecurrenceRelations 41 VII NP-COMPLETENESS 85 ThirdHomeworkAssignment 44 23 EasyandHardProblems 86 24 NP-CompleteProblems 89 25 ApproximationAlgorithms 92 Seventh Homework Assignment 95 2 1 Introduction Overview. The main topics to be covered in this course are Meetings. We meet twice a week, on Tuesdays and Thursdays, from 1:15 to 2:30pm, in room D106 LSRC. I Design Techniques; II Searching; Communication. The course material will be delivered III Prioritizing; in the two weekly lectures. A written record of the lec- IV Graph Algorithms; tures will be available on the web, usually a day after the V Topological Algorithms; lecture. The web also contains other information, such as homework assignments, solutions, useful links, etc. The VI Geometric Algorithms; main supporting text is VII NP-completeness.