Lecture4 - Graphs and Network Flows IE411 Lecture 4 Dr Ted...

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Graphs and Network Flows IE411 Lecture 4 Dr. Ted Ralphs
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IE411 Lecture 4 1 References for Today’s Lecture Required reading Miller and Boxer, Chapter 1 References AMO Sections 3.2 CLRS Sections 1.1–1.3
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IE411 Lecture 4 2 Algorithms al · go · rithm 1 1. any systematic method of solving a certain kind of problem 2. a predetermined set of instructions for solving a specific problem in a limited number of steps The concept of an algorithm is not new but formal study of efficiency is relatively new. 1 Wester’s New World Dictionary
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IE411 Lecture 4 3 Introduction to Computational Complexity What is the goal of computational complexity theory? To provide a method of comparing the difficulty of two different problems. To provide a method of comparing the efficiency of two different algorithms for the same problem. We would like to be able to rigorously define the meaning of efficient algorithm . Complexity theory is built on a basic set assumptions called the model of computation . We will not concern ourselves too much with the details of a particular model here. This topic is addressed in IE 407.
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IE411 Lecture 4 4 Elementary Operations In order to analyze the number of steps necessary to execute an algorithm, we have to say what we mean by a “step.” To define this precisely is tedious and beyond the scope of this course. A precise definition depends on the exact hardware being used. Our analysis will assume a very simple model of a computer in which the following operations take one step. arithmetic (addition, subtraction, multiplication, division) data movement (read from memory, store in memory, copy) comparison control (function calls, goto commands) This is a very idealized model, but it works in practice. We will sometimes need to simplify the model even further.
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IE411 Lecture 4 5 Problems, Instances, and Algorithms A problem P is a mapping of a set of inputs to specified outputs . An instance is a problem along with a particular input. An algorithm is a procedure for computing the output expected from a given input. An algorithm solves a problem P if that algorithm produces the expected output for any input. Example: Traveling Salesman Problem Given an undirected graph G = ( N, A ) and non-negative arc lengths d ij for all ( i, j ) A , find a cycle that visits all nodes exactly once and is of minimum total length. How do we specify an instance?
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IE411 Lecture 4 6 Computational Complexity: What is the Objective? Complexity analysis is aimed at answering two types of questions. How hard is a given problem? How efficient is a given algorithm for a given problem? Our measure of efficiency will be running time , defined as either The actual wall clock time required to execute the algorithm on a computer (problematic) or the number of elementary operations required (more on this later).
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  • Spring '14
  • TedRalphs
  • Analysis of algorithms, Computational complexity theory, Computational Complexity

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