The Classes P and NP.
|
|
- Tamsin Chase
- 2 years ago
- Views:
Transcription
1 Intractable Problems The Classes P and NP Mohamed M. El Wakil 1
2 Agenda 1. What is a problem? 2. Decidable or not? 3. The P class 4. The NP Class 5. TheNP Complete class 2
3 What is a problem? A problem is a question to be answered. What is the value of X/Y? A problem usually has parameters. X, and Y A decision problem, is a version of the problem with only two possible answers: Yes or No! Given two numbers X, and Y, does Y evenly divide X? An instance: a specific problem instance Does 3 evenly divide 6? 3
4 Decidable or not? A decidable problem, is a problem that could be solved using a computer. An undecidable problem, is a problem that t can never be solved using a computer, neither now or in the future. Only decidable problems! 4
5 Classification We need to classify problems in terms of their computability. Three classes: P class NP Class NP Complete class 5
6 P class, wrt Computers Problems with at least one algorithm that solves the problem in polynomial ltime wrt to the input size. Polynomial time The number of steps needed relates polynomially to the size of the input. O(n 2 ), O(n 9 ), O(n c ), where c is a constant. but NOT O(n!), O(2 n ), when n is the size of the input. 6
7 P class, wrt Turing Machines Problems solvable in polynomial time using a Dt Deterministic iiti Turing Machine (DTM) bl belong to the class P. Polynomial time The number of moves needed relates polynomially to the size of the input. n 2, 17n 3, 9n 4, but NOT 2 n DTM A Turing machine with a tape, head, transition function, and a set of states. 7
8 P Problem (MWST) Minimum Weight Spanning Tree Given a weighted graph G, find the minimum weight spanning tree. In other words, convert the given graph into a tree, that includes allthe nodes of the original graph, and minimizes the summation of weights of the edges in the resulting tree. 8
9 MWST Example Problem Instance Source: 9
10 Kruskal's s algorithm The MWST problem belongs bl to the P class of problems, since there is an algorithm that solves it in polynomial ltime. Kruskal's algorithm O(n 2 ) Create a forest F (a set of trees), where each vertex in the graph is a separate tree Create a set S containing all the edges in the graph While S is nonempty Remove an edge with minimum weight from S If that edge connects two different trees, then add it to the forest, combining two trees into a single tree Otherwise discard that edge 10
11 MWST Example Possible Solution Source: 11
12 NP class, wrt Turing Machines Problems solvable in polynomial time using a Non Deterministic Turing Machine (NDTM) belong to the class NP. NDTM A DTM, with two stages of processing: guessing, and checking. 12
13 Non Deterministic Turing Machine Guessing: Guess a solution, and then write it down to the tape. Checking: Evaluate the guess to decide whether it solves the problem or not. The number of guessed solutions, can be either polynomial or exponential. If the number of guessed solutions is polynomial, l then, the NDTM is equivalent to a DTM. 13
14 NP class, wrt Computers Problems that can be solved within an exponential time wrt the input size. This includes problems that can be solved din polynomial time. 14
15 Important A DTM is a NDTM that has a polynomial number of guesses. According to the dfiii definition of NP, the MWST problem is an NP problem. 15
16 NP Problem Example Travelling Salesman Problem (TSP) Given a number of cities and the costs of traveling from any city to any other city, what is the cheapest round trip route that visits each city exactly once andthen returns to the starting city? Source: problem 16
17 Solving the TSP There is no one single algorithm that solves this problem in polynomial time The only way, is to enumerate all possible itineraries and checking them one by one. For n cities, there are n! routes 17
18 Polynomial Time Reduction A problem P1, is polynomially reducible to problem P2, if there is a process that takes an instance of P1 as an input, and outputs a corresponding instance of P2 in polynomial time. P1: a * b P2: ((a+b) 2 a 2 b 2 )/2 18
19 NP Complete Class A problem P is NP Complete If: P is in NP For every problem L in NP, there is a polynomial time reduction from L to P. If P1 is NP Complete, and there is polynomial l time reduction from P1 to P2, then P2 is NP Complete. 19
20 NP complete lt problems family tree 20
21 The NP World Source: 21
22 Intractable Problems The Classes P and NP Mohamed M. El Wakil 22
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
Complexity Classes P and NP
Complexity Classes P and NP MATH 3220 Supplemental Presentation by John Aleshunas The cure for boredom is curiosity. There is no cure for curiosity Dorothy Parker Computational Complexity Theory In computer
CAD Algorithms. P and NP
CAD Algorithms The Classes P and NP Mohammad Tehranipoor ECE Department 6 September 2010 1 P and NP P and NP are two families of problems. P is a class which contains all of the problems we solve using
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
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
NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University
NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with
1. Nondeterministically guess a solution (called a certificate) 2. Check whether the solution solves the problem (called verification)
Some N P problems Computer scientists have studied many N P problems, that is, problems that can be solved nondeterministically in polynomial time. Traditionally complexity question are studied as languages:
Chapter 11. Theory of Computation Pearson Addison-Wesley. All rights reserved
Chapter 11 Theory of Computation 2007 Pearson Addison-Wesley. All rights reserved Chapter 11: Theory of Computation 11.1 Functions and Their Computation 11.2 Turing Machines 11.3 Universal Programming
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,
Minimum Spanning Trees
Minimum Spanning Trees Algorithms and 18.304 Presentation Outline 1 Graph Terminology Minimum Spanning Trees 2 3 Outline Graph Terminology Minimum Spanning Trees 1 Graph Terminology Minimum Spanning Trees
Informatique Fondamentale IMA S8
Informatique Fondamentale IMA S8 Cours 4 : graphs, problems and algorithms on graphs, (notions of) NP completeness Laure Gonnord http://laure.gonnord.org/pro/teaching/ Laure.Gonnord@polytech-lille.fr Université
NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics
NP-complete? NP-hard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer
Introduction 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. divide-and-conquer algorithms
Computational complexity theory
Computational complexity theory Goal: A general theory of the resources needed to solve computational problems What types of resources? Time What types of computational problems? decision problem Decision
Quantum and Non-deterministic computers facing NP-completeness
Quantum and Non-deterministic computers facing NP-completeness Thibaut University of Vienna Dept. of Business Administration Austria Vienna January 29th, 2013 Some pictures come from Wikipedia Introduction
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
OHJ-2306 Introduction to Theoretical Computer Science, Fall 2012 8.11.2012
276 The P vs. NP problem is a major unsolved problem in computer science It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a $ 1,000,000 prize for the
Notes on NP Completeness
Notes on NP Completeness Rich Schwartz November 10, 2013 1 Overview Here are some notes which I wrote to try to understand what NP completeness means. Most of these notes are taken from Appendix B in Douglas
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
MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. III - Logic and Computer Science - Phokion G. Kolaitis
LOGIC AND COMPUTER SCIENCE Phokion G. Kolaitis Computer Science Department, University of California, Santa Cruz, CA 95064, USA Keywords: algorithm, Armstrong s axioms, complete problem, complexity class,
Theoretical Computer Science (Bridging Course) Complexity
Theoretical Computer Science (Bridging Course) Complexity Gian Diego Tipaldi A scenario You are a programmer working for a logistics company Your boss asks you to implement a program that optimizes the
Tutorial 8. NP-Complete Problems
Tutorial 8 NP-Complete Problems Decision Problem Statement of a decision problem Part 1: instance description defining the input Part 2: question stating the actual yesor-no question A decision problem
Introduction to Logic in Computer Science: Autumn 2006
Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding
GRAPH THEORY and APPLICATIONS. Trees
GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.
Chapter 7 Uncomputability
Chapter 7 Uncomputability 190 7.1 Introduction Undecidability of concrete problems. First undecidable problem obtained by diagonalisation. Other undecidable problems obtained by means of the reduction
Chapter 6: Graph Theory
Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.
Reductions & NP-completeness as part of Foundations of Computer Science undergraduate course
Reductions & NP-completeness as part of Foundations of Computer Science undergraduate course Alex Angelopoulos, NTUA January 22, 2015 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NP-completeness-
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].
Solutions to Exercises 8
Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.
Analysis of Computer Algorithms. Algorithm. Algorithm, Data Structure, Program
Analysis of Computer Algorithms Hiroaki Kobayashi Input Algorithm Output 12/13/02 Algorithm Theory 1 Algorithm, Data Structure, Program Algorithm Well-defined, a finite step-by-step computational procedure
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
One last point: we started off this book by introducing another famously hard search problem:
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 261 Factoring One last point: we started off this book by introducing another famously hard search problem: FACTORING, the task of finding all prime factors
2.3 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
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
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
Introduction to Algorithms. Part 3: P, NP Hard Problems
Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NP-Completeness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter
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;
Discuss the size of the instance for the minimum spanning tree problem.
3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can
Lecture 19: Introduction to NP-Completeness Steven Skiena. Department of Computer Science State University of New York Stony Brook, NY 11794 4400
Lecture 19: Introduction to NP-Completeness Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Reporting to the Boss Suppose
Heuristic Methods. Part #1. João Luiz Kohl Moreira. Observatório Nacional - MCT COAA. Observatório Nacional - MCT 1 / 14
Heuristic Methods Part #1 João Luiz Kohl Moreira COAA Observatório Nacional - MCT Observatório Nacional - MCT 1 / Outline 1 Introduction Aims Course's target Adviced Bibliography 2 Problem Introduction
Answers to some of the exercises.
Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D
! 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
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
Lecture Notes on Spanning Trees
Lecture Notes on Spanning Trees 15-122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 26, 2011 1 Introduction In this lecture we introduce graphs. Graphs provide a uniform model
! 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
Chapter 4. Trees. 4.1 Basics
Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.
A Fast Algorithm For Finding Hamilton Cycles
A Fast Algorithm For Finding Hamilton Cycles by Andrew Chalaturnyk A thesis presented to the University of Manitoba in partial fulfillment of the requirements for the degree of Masters of Science in 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
Theoretical Computer Science Bridging Course - Introduction / General Info. Summer Term 2016 Fabian Kuhn
Theoretical Computer Science Bridging Course - Introduction / General Info Summer Term 2016 Fabian Kuhn About the Course Topics Foundations of theoretical computer science Introduction to logic No lectures
Structural properties of oracle classes
Structural properties of oracle classes Stanislav Živný Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK Abstract Denote by C the class of oracles relative
Quantum Monte Carlo and the negative sign problem
Quantum Monte Carlo and the negative sign problem or how to earn one million dollar Matthias Troyer, ETH Zürich Uwe-Jens Wiese, Universität Bern Complexity of many particle problems Classical 1 particle:
Introduction to computer science
Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.
1 Basic Definitions and Concepts in Graph Theory
CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered
COMS4236: Introduction to Computational Complexity. Summer 2014
COMS4236: Introduction to Computational Complexity Summer 2014 Mihalis Yannakakis Lecture 17 Outline conp NP conp Factoring Total NP Search Problems Class conp Definition of NP is nonsymmetric with respect
NP-Completeness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness
Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce
Honors Class (Foundations of) Informatics. Tom Verhoeff. Department of Mathematics & Computer Science Software Engineering & Technology
Honors Class (Foundations of) Informatics Tom Verhoeff Department of Mathematics & Computer Science Software Engineering & Technology www.win.tue.nl/~wstomv/edu/hci c 2011, T. Verhoeff @ TUE.NL 1/20 Information
IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*
IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti
SIMS 255 Foundations of Software Design. Complexity and NP-completeness
SIMS 255 Foundations of Software Design Complexity and NP-completeness Matt Welsh November 29, 2001 mdw@cs.berkeley.edu 1 Outline Complexity of algorithms Space and time complexity ``Big O'' notation Complexity
The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge,
The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints
Graph Theory. Euler tours and Chinese postmen. John Quinn. Week 5
Graph Theory Euler tours and Chinese postmen John Quinn Week 5 Recap: connectivity Connectivity and edge-connectivity of a graph Blocks Kruskal s algorithm Königsberg, Prussia The Seven Bridges of Königsberg
On the k-path cover problem for cacti
On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we
Automata and Languages
Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful
The P versus NP Solution
The P versus NP Solution Frank Vega To cite this version: Frank Vega. The P versus NP Solution. 2015. HAL Id: hal-01143424 https://hal.archives-ouvertes.fr/hal-01143424 Submitted on 17 Apr
CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 313]
CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 313] File Structures A file is a collection of data stored on mass storage (e.g., disk or tape) Why on mass storage? too big to fit
What s next? Reductions other than kernelization
What s next? Reductions other than kernelization Dániel Marx Humboldt-Universität zu Berlin (with help from Fedor Fomin, Daniel Lokshtanov and Saket Saurabh) WorKer 2010: Workshop on Kernelization Nov
5. A full binary tree with n leaves contains [A] n nodes. [B] log n 2 nodes. [C] 2n 1 nodes. [D] n 2 nodes.
1. The advantage of.. is that they solve the problem if sequential storage representation. But disadvantage in that is they are sequential lists. [A] Lists [B] Linked Lists [A] Trees [A] Queues 2. The
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
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
Minesweeper as a Constraint Satisfaction Problem
Minesweeper as a Constraint Satisfaction Problem by Chris Studholme Introduction To Minesweeper Minesweeper is a simple one player computer game commonly found on machines with popular operating systems
Diagonalization. Ahto Buldas. Lecture 3 of Complexity Theory October 8, 2009. Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach.
Diagonalization Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach. Ahto Buldas Ahto.Buldas@ut.ee Background One basic goal in complexity theory is to separate interesting complexity
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
The Geometric Structure of Spanning Trees and Applications to Multiobjective Optimization
The Geometric Structure of Spanning Trees and Applications to Multiobjective Optimization By ALLISON KELLY O HAIR SENIOR THESIS Submitted in partial satisfaction of the requirements for Highest Honors
Steiner Tree NP-completeness Proof
Steiner Tree NP-completeness Proof Alessandro Santuari May 7, 2003 Abstract This document is an eercise for the Computational Compleity course taken at the University of Trento. We propose an NP-completeness
MGF 1107 CH 15 LECTURE NOTES Denson. Section 15.1
1 Section 15.1 Consider the house plan below. This graph represents the house. Consider the mail route below. This graph represents the mail route. 2 Definitions 1. Graph a structure that describes relationships.
1 Definitions. Supplementary Material for: Digraphs. Concept graphs
Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.
The Classes P and NP
The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the
Finite Automata and Formal Languages
Finite Automata and Formal Languages TMV026/DIT321 LP4 2011 Lecture 14 May 19th 2011 Overview of today s lecture: Turing Machines Push-down Automata Overview of the Course Undecidable and Intractable Problems
Boulder Dash is NP hard
Boulder Dash is NP hard Marzio De Biasi marziodebiasi [at] gmail [dot] com December 2011 Version 0.01:... now the difficult part: is it NP? Abstract Boulder Dash is a videogame created by Peter Liepa and
What are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
Guessing Game: NP-Complete?
Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple
Solving the Travelling Salesman Problem Using the Ant Colony Optimization
Ivan Brezina Jr. Zuzana Čičková Solving the Travelling Salesman Problem Using the Ant Colony Optimization Article Info:, Vol. 6 (2011), No. 4, pp. 010-014 Received 12 July 2010 Accepted 23 September 2011
8.1 Min Degree Spanning Tree
CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree
Analyzing the Facebook graph?
Logistics Big Data Algorithmic Introduction Prof. Yuval Shavitt Contact: shavitt@eng.tau.ac.il Final grade: 4 6 home assignments (will try to include programing assignments as well): 2% Exam 8% Big Data
Evaluation of Complexity of Some Programming Languages on the Travelling Salesman Problem
International Journal of Applied Science and Technology Vol. 3 No. 8; December 2013 Evaluation of Complexity of Some Programming Languages on the Travelling Salesman Problem D. R. Aremu O. A. Gbadamosi
Reminder: Complexity (1) Parallel Complexity Theory. Reminder: Complexity (2) Complexity-new
Reminder: Complexity (1) Parallel Complexity Theory Lecture 6 Number of steps or memory units required to compute some result In terms of input size Using a single processor O(1) says that regardless of
Reminder: Complexity (1) Parallel Complexity Theory. Reminder: Complexity (2) Complexity-new GAP (2) Graph Accessibility Problem (GAP) (1)
Reminder: Complexity (1) Parallel Complexity Theory Lecture 6 Number of steps or memory units required to compute some result In terms of input size Using a single processor O(1) says that regardless of
CIS 700: algorithms for Big Data
CIS 700: algorithms for Big Data Lecture 6: Graph Sketching Slides at http://grigory.us/big-data-class.html Grigory Yaroslavtsev http://grigory.us Sketching Graphs? We know how to sketch vectors: v Mv
The Traveling Beams Optical Solutions for Bounded NP-Complete Problems
The Traveling Beams Optical Solutions for Bounded NP-Complete Problems Shlomi Dolev, Hen Fitoussi Abstract Architectures for optical processors designed to solve bounded instances of NP-Complete problems
What is Classification? Data Mining Classification. Certainty. Usual Examples. Predictive / Definitive. Techniques
What is Classification? Data Mining Classification Kevin Swingler Assigning an object to a certain class based on its similarity to previous examples of other objects Can be done with reference to original
CS5310 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.
Why? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
Math 443/543 Graph Theory Notes 4: Connector Problems
Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we
Data Mining Classification
Data Mining Classification Jingpeng Li 1 of 26 What is Classification? Assigning an object to a certain class based on its similarity to previous examples of other objects Can be done with reference to
The K-Observer Problem in Computer Networks
The K-Observer Problem in Computer Networks H. B. Acharya 1, Taehwan Choi 1, Rida A. Bazzi 2, and Mohamed G. Gouda 1,3 1 The University of Texas at Austin, USA 2 Arizona State University, USA 3 The National
The Line Connectivity Problem
Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany RALF BORNDÖRFER MARIKA NEUMANN MARC E. PFETSCH The Line Connectivity Problem Supported by the DFG Research
A Working Knowledge of Computational Complexity for an Optimizer
A Working Knowledge of Computational Complexity for an Optimizer ORF 363/COS 323 Instructor: Amir Ali Ahmadi TAs: Y. Chen, G. Hall, J. Ye Fall 2014 1 Why computational complexity? What is computational
Computing Functions with Turing Machines
CS 30 - Lecture 20 Combining Turing Machines and Turing s Thesis Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata
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)
A tree can be defined in a variety of ways as is shown in the following theorem: 2. There exists a unique path between every two vertices of G.
7 Basic Properties 24 TREES 7 Basic Properties Definition 7.1: A connected graph G is called a tree if the removal of any of its edges makes G disconnected. A tree can be defined in a variety of ways as
Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi
Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the