Intractable Problems Intractable Problems The Classes P and NP Mohamed M. El Wakil Mohamed M. El Wakil mohamed@elwakil.net 1 Agenda 1. 2. 3. 4. 5. 5 What is a problem? Decidable or not? Decidable or not? The P class The NP Class The NP Complete class The NP‐Complete class 2 What is a problem? What is a problem? • A problem is a question to be answered. – What is the value of X/Y? • A problem usually has parameters. p y p – X‚ and Y • A decision problem is a version of the
Premium Computational complexity theory Algorithm
A Guide for Text Complexity Analysis 1. Fill in the title and author information in the upper left side of the placemat. 2. Complete the Text Description. (Green Box) 3. Identify the Quantitative Measure. (Red Box) Use Lexile.com (or the quantitative measure your district uses) to find the quantitative measure of the text. Use the chart below to determine the grade band alignment for the quantitative measure of the text. Enter the Lexile Measure and Complexity Band Level on the placemat
Premium Quantitative research Computational complexity theory Scientific method
Path Complexity of the Class Binary Search Tree Contents Page No. Abstract List of Symbols and Abbreviations List of Figures List of Tables V VI VII VII 1. Introduction 1.1 1.2 General Organization of the Thesis 1 1 3 4 4 4 5 5 7 9 9 11 15 21 22 22 24 30 31 2. Preliminaries 2.1. 2.2. 2.3. Introduction Terminology and Notations Path complexity of a class 2.3.1. Introduction 2.3.2. The class Stack 3. Path complexity of the class BST 3.1. 3.2. 3.3. 3.4. State representation of
Premium Graph theory Trees Computational complexity theory
INTRODUCTION TO THE THEORY OF COMPUTATION‚ SECOND EDITION MICHAEL SIPSER MassachusettsInstitute of Technology THOMSON COURSE TECHNOLOGY Australia * Canada * Mexico * Singapore * Spain * United Kingdom * United States THOIVISON COURSE TECHNOLOGY Introduction to the Theory of Computation‚ Second Edition by Michael Sipser Senior Product Manager: Alyssa Pratt Executive Editor: Mac Mendelsohn Associate Production Manager: Aimee Poirier Senior Marketing Manager: Karen Seitz COPYRIGHT
Premium Computational complexity theory Algorithm Computer science
Operations research An introduction to solution methods Ecole des Mines de Nantes Master MOST 2012-2013 Olivier Péton - 1- Problem Min f ( x ) xS An optimization problem S is the solution set that represents all feasible solutions of a problem. f is the objective function that maps S to R. It evaluates each feasible solution. Also called evaluation function or cost function Minimization = maximization ! max f ( x) min ( f ( x)) xS xS - 2- Mathematical
Premium Computational complexity theory Graph theory Optimization
But there is difference between computer science and information technology ‚ and both are not the same things as we most of us think . “Computer Science” is the mixture and application of “Applied Mathematics”‚ “Electrical Engineering“‚ and “Complexity Theory/Algorithms” to understand and/or model information. In other words‚ the “field of computation”. “Information Technology” is the mixture and application of “Programming”‚ “Hardware Administration”‚ “Software Administration”‚ “Networking“‚ “Network
Premium Computer science Computer Algorithm
derivative or a look-up table which may need a large amount of memory due to channel variations. To reduce the computational and/or hardware complexity of Filho’s algorithm‚ in this paper‚ an improved method for the decision-directed algorithm is proposed. It is shown that the proposed scheme‚ when it is combined with decisiondirected algorithm‚ reduces the computational complexity drastically while it retains the convergence and steadystate performance of the Filho’s algorithm. Figure 1
Premium Computational complexity theory Derivative Signal processing
In today’s lecture 1 Recall... Computational Complexity Time Complexity 2 Example: Binary Search Best case Worst case Average case 3 Exercise: Linear Search 4 Example: Bubble sort Most of the content is based on Section 2.3 of Rosen’s Discrete Mathematics‚ and 2.2 of Cormen et al. Introduction to Algorithms. CS204/209 — Lecture 4: Best‚ Worst‚ and Average Case Complexity 1/11 Recall... Computational Complexity Our goal is to be able to compare algorithms and determine which
Premium Computational complexity theory Algorithm
THE SEVEN SORTING SCHEME 1. BUBBLE SORT The bubble sort is generally considered to be the simplest sorting algorithm. Because of its simplicity and ease of visualization‚ it is often taught in introductory computer science courses. Because of its abysmal O(n2) performance‚ it is not used often for large (or even medium-sized) datasets. ALGORITHM: for i = 1:n‚ swapped = false for j = n:i+1‚ if a[j] < a[j-1]‚ swap a[j‚j-1] swapped = true
Premium Computational complexity theory
branch of our work aims to construct code based on abstract assumptions about the embedding layer.1 Another is embedding-focused and does not explore code structures.2 Our study shows that the code-based strategy has the advantage of low computational complexity‚ but the embedding-based scheme holds the benefit of high collusion resistance‚ which is measured by the number of colluders that can be caught within a certain probability of detection. We describe a design that considers both coding and
Premium Copyright infringement Fair use Computational complexity theory