Welcome to the course Algorithm Design
|
|
|
- Kevin Blankenship
- 8 years ago
- Views:
Transcription
1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Welcome to the course Algorithm Design Summer Term 2011 Friedhelm Meyer auf der Heide Lecture 6, Friedhelm Meyer auf der Heide 1
2 Topics HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity - Divide & conquer - Dynamic programming - Greedy Algorithms - Approximation Algorithms - Randomized Algorithms - Online Algorithms Friedhelm Meyer auf der Heide 2
3 For which problems are Greedy-Algorithms optimal? HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Consider a finite set E and a system U of subsets of E. (E,U) is called a subset-system, if the following holds: (i)? 2 U (ii) For each B 2 U, also each subset of B is in U. B2 U is called maximal, if no proper superset of B is in U. The Optimization problem corresponding to (E,U) is : Given a weight function w:e Q +,compute a maximal set B 2 U with maximizes w(b) = e2 B w(e). (Minimization problems are described analogously.) Friedhelm Meyer auf der Heide 3
4 Subset-Systems and the Canonical Greedy-Algorithm HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Canonical Greedy ((E,U)) (1) Sort E such that w(e 1 ) w(e n ). (2) B?. (3) For k=1 to n if B [ {e k } 2 U then B B [ {e k } (4) Return B Friedhelm Meyer auf der Heide 4
5 Greedy-Algorithms und Matroids HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity A system of subsets (E,U) is a matroid, if in addition, the following exchange property holds: (iii) For all A,B 2 U with A < B, there is x2 B-A such that A[{x} 2 U. Remark: All maximal sets of a matroid have the same size. (homework) Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal for (E,U) for every weight function w, if and only if (E,U) is a matroid. Friedhelm Meyer auf der Heide 5
6 Matroids and the canonical Greedy Algorithm HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal for (E,U) for every weight function w, if and only if (E,U) is a matroid. Proof: ( : Consider a matroid (E,U) and a weight function w. Let w(e 1 ) w(e n ). - Consider an optimal solution T ={e i1,, e ik }. - Assume the solution T={e j1,, e jk } found by Canonical Greedy were not optimal, i.e. w(t )> w(t). - Then there is an index p with w(e ip ) > w(e jp ). Let p be minimal with this property. Note that i p < j p, because the items are sorted w.r.t weight. - Apply the exchange property to A = {e j1,,e jp-1 } and B ={e i1,, e ip } : As A < B, there is e iq 2 B-A with A [ {e iq } 2 U. - As w(e iq ) w(e ip ) > w(e jp ), Canonical Greedy would have chosen e iq before e jp, a contradiction. Friedhelm Meyer auf der Heide 6
7 Matroids and the canonical Greedy Algorithm HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal for (E,U) for every weight function w, if and only if (E,U) is a matroid. Proof: ) : Assume that, for some A,B 2 U with A < B, it holds that A[{e} is not in U, for every e2b-a. Let B =r and consider the weight function w with - w(e) = r+1 for e2a - w(e) = r for e2b-a - w(e) = 0 else Then, Canonical Greedy would select a solution T with T µ A and T Å (B-A) =?. As w(t)= A (r+1) (r-1)(r+1) r 2-1, T is not optimal, because B is a solution with larger weight B r = r 2. Friedhelm Meyer auf der Heide 7
8 Topics HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity - Divide & conquer - Dynamic programming - Greedy Algorithms - Approximation Algorithms - Randomized Algorithms - Online Algorithms Friedhelm Meyer auf der Heide 8
9
10 Approximation Algorithms Greedy Techniques Load-Balancing Problem Center Selection Problem Pricing Method Vertex Cover Problem Linear Programming and Rounding Vertex Cover Problem Generalized Load-Balancing Problem Polynomial Time Approximation Scheme Knapsack Problem
11
12
13 Load Balancing: List Scheduling A B C D E F G H I J Machine 1 Machine 2 Machine 3 0 Time
14 Load Balancing: List Scheduling B C D E F G H I J A Machine 1 Machine 2 Machine 3 0 Time
15 Load Balancing: List Scheduling C D E F G H I J A Machine 1 B Machine 2 Machine 3 0 Time
16 Load Balancing: List Scheduling D E F G H I J A Machine 1 B Machine 2 C Machine 3 0 Time
17 Load Balancing: List Scheduling E F G H I J A Machine 1 B D Machine 2 C Machine 3 0 Time
18 Load Balancing: List Scheduling F G H I J A E Machine 1 B D Machine 2 C Machine 3 0 Time
19 Load Balancing: List Scheduling G H I J A E Machine 1 B D Machine 2 C F Machine 3 0 Time
20 Load Balancing: List Scheduling H I J A E Machine 1 B D Machine 2 G C F Machine 3 0 Time
21 Load Balancing: List Scheduling I J A E H Machine 1 B D Machine 2 G C F Machine 3 0 Time
22 Load Balancing: List Scheduling J A E H Machine 1 I B D Machine 2 G G C F Machine 3 0 Time
23 Load Balancing: List Scheduling A E H Machine 1 I B D Machine 2 G C F Machine 3 J 0 Time
24 Load Balancing: List Scheduling D E H Machine 1 C B A Machine G 2 I F Machine 3 J 0 Optimal Schedule A E H Machine 1 I B D Machine 2 G C F Machine 3 J 0 List schedule
25
26
27
28
29
30
31
32
33 Approximation Algorithms Greedy Techniques Load-Balancing Problem Center Selection Problem Pricing Method Vertex Cover Problem Linear Programming and Rounding Vertex Cover Problem Generalized Load-Balancing Problem Polynomial Time Approximation Scheme Knapsack Problem
34
35
36
37
38
39
40 Approximation Algorithms Greedy Techniques Load-Balancing Problem Center Selection Problem Pricing Method Vertex Cover Problem Linear Programming and Rounding Vertex Cover Problem Generalized Load-Balancing Problem Polynomial Time Approximation Scheme Knapsack Problem
41
42
43
44
45
46 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Thank you for your attention! Friedhelm Meyer auf der Heide Heinz Nixdorf Institute & Computer Science Department University of Paderborn Fürstenallee Paderborn, Germany Tel.: +49 (0) 52 51/ Fax: +49 (0) 52 51/ Friedhelm Meyer auf der Heide 46
Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one
! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of
Applied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three
Approximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
Algorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li
Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order
Data Management in Hierarchical Bus Networks
Data Management in Hierarchical Bus Networks F. Meyer auf der Heide Department of Mathematics and Computer Science and Heinz Nixdorf Institute, Paderborn University, Germany fmadh@upb.de H. Räcke Department
MapReduce and Distributed Data Analysis. Sergei Vassilvitskii Google Research
MapReduce and Distributed Data Analysis Google Research 1 Dealing With Massive Data 2 2 Dealing With Massive Data Polynomial Memory Sublinear RAM Sketches External Memory Property Testing 3 3 Dealing With
Introduction to Markov Chain Monte Carlo
Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem
The van Hoeij Algorithm for Factoring Polynomials
The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial
Module Handbook 2009 Version 2
Module Handbook 2009 Version 2 Bachelor and Master Program Computer Science Faculty of Computer Science, Electrical Engineering and Mathematics of the University of Paderborn Bachelor Degree Course Computer
11. APPROXIMATION ALGORITHMS
11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005
Scheduling Shop Scheduling. Tim Nieberg
Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations
Corollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality
![Corollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality](/thumbs/26/7077664.jpg "Corollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality")
Corollary For equidistant knots, i.e., u i = a + i (b-a)/n, we obtain with (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality 120202: ESM4A - Numerical Methods
Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
How Asymmetry Helps Load Balancing
How Asymmetry Helps oad Balancing Berthold Vöcking nternational Computer Science nstitute Berkeley, CA 947041198 voecking@icsiberkeleyedu Abstract This paper deals with balls and bins processes related
Why Study NP- hardness. NP Hardness/Completeness Overview. P and NP. Scaling 9/3/13. Ron Parr CPS 570. NP hardness is not an AI topic
Why Study NP- hardness NP Hardness/Completeness Overview Ron Parr CPS 570 NP hardness is not an AI topic It s important for all computer scienhsts Understanding it will deepen your understanding of AI
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
Large induced subgraphs with all degrees odd
Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order
Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06
CS880: Approximations Algorithms Scribe: Matt Elder Lecturer: Shuchi Chawla Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06 3.1 Set Cover The Set Cover problem is: Given a set of
Complexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar
Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples
Lecture 7: NP-Complete Problems
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit
FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM
International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 34-48 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT
Discrete Optimization
Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using
LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method
LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right
CSC2420 Fall 2012: Algorithm Design, Analysis and Theory
CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms
Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique
On the representability of the bi-uniform matroid
On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large
Notes on Complexity Theory Last updated: August, 2011. Lecture 1
Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look
NOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
Near Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
COMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:
COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative
Sample or Random Security A Security Model for Segment-Based Visual Cryptography
Sample or Random Security A Security Model for Segment-Based Visual Cryptography Sebastian Pape Dortmund Technical University March 5th, 2014 Financial Cryptography and Data Security 2014 Sebastian Pape
A linear algebraic method for pricing temporary life annuities
A linear algebraic method for pricing temporary life annuities P. Date (joint work with R. Mamon, L. Jalen and I.C. Wang) Department of Mathematical Sciences, Brunel University, London Outline Introduction
International Graduate School
International International Graduate School Graduate School echanical Engineering Dynamic Electrical Intelligent Engineering Systems Mathematics Mechanical Engineering Dynamic Electrical Intelligent Engineering
ABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).
![ABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).](/thumbs/26/7227512.jpg "ABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).")
Degrees of Freedom Versus Dimension for Containment Orders Noga Alon 1 Department of Mathematics Tel Aviv University Ramat Aviv 69978, Israel Edward R. Scheinerman 2 Department of Mathematical Sciences
On the Relationship between Classes P and NP
Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,
Big Data & Scripting Part II Streaming Algorithms
Big Data & Scripting Part II Streaming Algorithms 1, 2, a note on sampling and filtering sampling: (randomly) choose a representative subset filtering: given some criterion (e.g. membership in a set),
CS473 - 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
Exponential time algorithms for graph coloring
Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].
. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the
Scheduling to Minimize Power Consumption using Submodular Functions
Scheduling to Minimize Power Consumption using Submodular Functions Erik D. Demaine MIT edemaine@mit.edu Morteza Zadimoghaddam MIT morteza@mit.edu ABSTRACT We develop logarithmic approximation algorithms
x 2 f 2 e 1 e 4 e 3 e 2 q f 4 x 4 f 3 x 3
Karl-Franzens-Universitat Graz & Technische Universitat Graz SPEZIALFORSCHUNGSBEREICH F 003 OPTIMIERUNG und KONTROLLE Projektbereich DISKRETE OPTIMIERUNG O. Aichholzer F. Aurenhammer R. Hainz New Results
Welcome to... Problem Analysis and Complexity Theory 716.054, 3 VU
Welcome to... Problem Analysis and Complexity Theory 716.054, 3 VU Birgit Vogtenhuber Institute for Software Technology email: bvogt@ist.tugraz.at office hour: Tuesday 10:30 11:30 slides: http://www.ist.tugraz.at/pact.html
5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson
Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams
Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams André Ciré University of Toronto John Hooker Carnegie Mellon University INFORMS 2014 Home Health Care Home health care delivery
NP-Completeness and Cook s Theorem
NP-Completeness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:
ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015
ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving
Euclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li. Advised by: Dave Mount. May 22, 2014
Euclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li Advised by: Dave Mount May 22, 2014 1 INTRODUCTION In this report we consider the implementation of an efficient
Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.
Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.
A Practical Scheme for Wireless Network Operation
A Practical Scheme for Wireless Network Operation Radhika Gowaikar, Amir F. Dana, Babak Hassibi, Michelle Effros June 21, 2004 Abstract In many problems in wireline networks, it is known that achieving
A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number
Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and
CRYPTOG NETWORK SECURITY
CRYPTOG NETWORK SECURITY PRINCIPLES AND PRACTICES FOURTH EDITION William Stallings Prentice Hall Upper Saddle River, NJ 07458 'jkfetmhki^^rij^jibwfcmf «MMr""'-^.;
CMPSCI611: Approximating MAX-CUT Lecture 20
CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to
Small 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
Embedded Systems 20 BF - ES
Embedded Systems 20-1 - Multiprocessor Scheduling REVIEW Given n equivalent processors, a finite set M of aperiodic/periodic tasks find a schedule such that each task always meets its deadline. Assumptions:
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
Load-Balanced Virtual Backbone Construction for Wireless Sensor Networks
Load-Balanced Virtual Backbone Construction for Wireless Sensor Networks Jing (Selena) He, Shouling Ji, Yi Pan, Zhipeng Cai Department of Computer Science, Georgia State University, Atlanta, GA, USA, {jhe9,
CSC2420 Spring 2015: Lecture 3
CSC2420 Spring 2015: Lecture 3 Allan Borodin January 22, 2015 1 / 1 Announcements and todays agenda Assignment 1 due next Thursday. I may add one or two additional questions today or tomorrow. Todays agenda
(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7
(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition
How To Find An Optimal Search Protocol For An Oblivious Cell
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
THE SCHEDULING OF MAINTENANCE SERVICE
THE SCHEDULING OF MAINTENANCE SERVICE Shoshana Anily Celia A. Glass Refael Hassin Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single
Lecture 13: The Knapsack Problem
Lecture 13: The Knapsack Problem Outline of this Lecture Introduction of the 0-1 Knapsack Problem. A dynamic programming solution to this problem. 1 0-1 Knapsack Problem Informal Description: We have computed
Scheduling Single Machine Scheduling. Tim Nieberg
Scheduling Single Machine Scheduling Tim Nieberg Single machine models Observation: for non-preemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe
Embedded Systems 20 REVIEW. Multiprocessor Scheduling
Embedded Systems 0 - - Multiprocessor Scheduling REVIEW Given n equivalent processors, a finite set M of aperiodic/periodic tasks find a schedule such that each task always meets its deadline. Assumptions:
Can linear programs solve NP-hard problems?
Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf Linear programs Can linear programs solve NP-hard problems? p. 2/9 Can linear programs solve
9th Max-Planck Advanced Course on the Foundations of Computer Science (ADFOCS) Primal-Dual Algorithms for Online Optimization: Lecture 1
9th Max-Planck Advanced Course on the Foundations of Computer Science (ADFOCS) Primal-Dual Algorithms for Online Optimization: Lecture 1 Seffi Naor Computer Science Dept. Technion Haifa, Israel Introduction
1 Formulating The Low Degree Testing Problem
6.895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Linearity Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz In the last lecture, we proved a weak PCP Theorem,
Semantics and Verification of Software
Semantics and Verification of Software Lecture 21: Nondeterminism and Parallelism IV (Equivalence of CCS Processes & Wrap-Up) Thomas Noll Lehrstuhl für Informatik 2 (Software Modeling and Verification)
Systems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
Max-Min Representation of Piecewise Linear Functions
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 297-302. Max-Min Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department,
Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information
Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information Lawrence Ong School of Electrical Engineering and Computer Science, The University of Newcastle, Australia Email: lawrence.ong@cantab.net
7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
Distributed Computing over Communication Networks: Maximal Independent Set
Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.
(a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9
![(a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9](/thumbs/40/21231804.jpg "(a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9")
Homework #01, due 1/20/10 = 9.1.2, 9.1.4, 9.1.6, 9.1.8, 9.2.3 Additional problems for study: 9.1.1, 9.1.3, 9.1.5, 9.1.13, 9.2.1, 9.2.2, 9.2.4, 9.2.5, 9.2.6, 9.3.2, 9.3.3 9.1.1 (This problem was not assigned
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
Online and Offline Selling in Limit Order Markets
Online and Offline Selling in Limit Order Markets Kevin L. Chang 1 and Aaron Johnson 2 1 Yahoo Inc. klchang@yahoo-inc.com 2 Yale University ajohnson@cs.yale.edu Abstract. Completely automated electronic
An Introduction to Linear Programming
An Introduction to Linear Programming Steven J. Miller March 31, 2007 Mathematics Department Brown University 151 Thayer Street Providence, RI 02912 Abstract We describe Linear Programming, an important
Load balancing of temporary tasks in the l p norm
Load balancing of temporary tasks in the l p norm Yossi Azar a,1, Amir Epstein a,2, Leah Epstein b,3 a School of Computer Science, Tel Aviv University, Tel Aviv, Israel. b School of Computer Science, The
Lecture 11: The Goldreich-Levin Theorem
COM S 687 Introduction to Cryptography September 28, 2006 Lecture 11: The Goldreich-Levin Theorem Instructor: Rafael Pass Scribe: Krishnaprasad Vikram Hard-Core Bits Definition: A predicate b : {0, 1}
The Classes P and NP. mohamed@elwakil.net
Intractable Problems The Classes P and NP Mohamed M. El Wakil mohamed@elwakil.net 1 Agenda 1. What is a problem? 2. Decidable or not? 3. The P class 4. The NP Class 5. TheNP Complete class 2 What is a
Truthful Mechanisms for Value-Based Scheduling in Cloud Computing
Truthful Mechanisms for Value-Based Scheduling in Cloud Computing Jonathan Yaniv Truthful Mechanisms for Value-Based Scheduling in Cloud Computing Research Thesis Submitted in partial fulfillment of the
2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]
![2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]](/thumbs/40/21097758.jpg "2. (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)
Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
Class constrained bin covering
Class constrained bin covering Leah Epstein Csanád Imreh Asaf Levin Abstract We study the following variant of the bin covering problem. We are given a set of unit sized items, where each item has a color
Definition 11.1. Given a graph G on n vertices, we define the following quantities:
Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define
Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
Page 1. CSCE 310J Data Structures & Algorithms. CSCE 310J Data Structures & Algorithms. P, NP, and NP-Complete. Polynomial-Time Algorithms
CSCE 310J Data Structures & Algorithms P, NP, and NP-Complete Dr. Steve Goddard goddard@cse.unl.edu CSCE 310J Data Structures & Algorithms Giving credit where credit is due:» Most of the lecture notes
ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER
Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group
The Data Management Strategy
Approximation Algorithms for Data Management in Networks Christof Krick Harald Räcke Matthias Westermann Abstract This paper deals with static data management in computer systems connected by networks.
Primality - Factorization
Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010. Class 4 Nancy Lynch
6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 4 Nancy Lynch Today Two more models of computation: Nondeterministic Finite Automata (NFAs)
Part II: Bidding, Dynamics and Competition. Jon Feldman S. Muthukrishnan
Part II: Bidding, Dynamics and Competition Jon Feldman S. Muthukrishnan Campaign Optimization Budget Optimization (BO): Simple Input: Set of keywords and a budget. For each keyword, (clicks, cost) pair.
The Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
Stationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
Minimizing the Number of Machines in a Unit-Time Scheduling Problem
Minimizing the Number of Machines in a Unit-Time Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.bas-net.by Frank