05shortestpaths - 15.082 and 6.855J Dijkstras Algorithm for...

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1 15.082 and 6.855J February 20, 2003 Dijkstra’s Algorithm for the Shortest Path Problem
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2 Wide Range of Shortest Path Problems ± Sources and Destinations z We will consider single source problems in this lecture ± Properties of the costs. z We will consider non-negative cost coefficients in this lecture ± Network topology. z We will consider all directed graphs
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3 Assumptions for the Problem Today ± Integral, non-negative data ± There is a directed path from source node s to all other nodes. ± Objective: find the shortest path from node s to each other node. ± Applications. z Vehicle routing z Communication systems
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4 Overview of today’s lecture ± One nice application (see the book for more) ± Dijkstra’s algorithm z animation z proof of correctness (invariants) z time bound ± Dial’s algorithm (a way of implementing Dijkstra’s algorithm) z animation z time bound
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5 Approximating Piecewise Linear Functions ± INPUT: A piecewise linear function z n points a 1 = (x 1 ,y 1 ), a 2 = (x 2 ,y 2 ),. .., a n = (x n ,y n ). z x 1 x 2 ... x n . ± Objective: approximate f with fewer points z c* is the “cost” per point included z c ij = cost of approximating the function through points a i , a i+1 , . . ., a jj by a single line joining point a i to point a j . z Find the minimum cost path from node 1 to node n. z Each path from 1 to n corresponds to an approximation of the data points a 1 to a n .
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6 c i,j is the cost of deleting points a i+1 , …, a j-1 x a 1 a 3 a a 4 a 5 a 6 a 8 a 7 a 10 a 9 f(x) f (x) 1 f (x) 2 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 1 3 c 1,3 10 2 4 6 5 7 8 9 a 1 a 2 a 3 e.g., c 1,3 = -c* + dist of a 2 to the line c 3,5 c 5,7 c 7,10
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7 A Key Step in Shortest Path Algorithms ± In this lecture, and in subsequent lectures, we let d( ) denote a vector of temporary distance labels. ± d(i) is the length of some path from the origin node 1 to node i. ± Procedure Update(i) for each (i,j) A(i) do if d(j) > d(i) + c ij then d(j) : = d(i) + c ij and pred(j) : = i; ± Update(i) ± used in Dijkstra’s algorithm and in the label correcting algorithm
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8 Update(7) d(7) = 6 at some point in the algorithm,
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This note was uploaded on 03/15/2010 for the course IE 505 taught by Professor Yok during the Spring '10 term at Galatasaray Üniversitesi.

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05shortestpaths - 15.082 and 6.855J Dijkstras Algorithm for...

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