4_ Networks Model and Shortest Path

8/9/2019 4_ Networks Model and Shortest Path

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Lecture 4Lecture 4

Introduction to Network ModelsIntroduction to Network Models

andand

Shortest PathsShortest Paths

Profs.Profs. Ismail ChabiniIsmail Chabini andand Amedeo OdoniAmedeo Odoni

1.2251.225J (ESD 225) Transportation Flow SystemsJ (ESD 225) Transportation Flow Systems

1.225, 11/07/02 Lecture 4, Page 2

Lecture 4 OutlineLecture 4 Outline

Conceptual Networks: DefinitionsRepresentation of an Urban Road Network (Supply)Shortest Paths (Reading: pp. 359-367, 6.2.3 and 6.2.4 of R6)

Introduction

Dijkstras algorithm: example

Dijkstras algorithm: statement

ObservationsExtensions to Classical Shortest Path ProblemsAll-or-nothing traffic assignmentZoning and Analysis Periods (Demand)Motivation for more advanced traffic assignment modelsSummary


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1.225, 11/07/02 Lecture 4, Page 3

Conceptual Networks: DefinitionsConceptual Networks: Definitions

Anetwork is: a set ofnodesNand a set oflinksA

nodes

are also called vertices orpoints links are also calledarcs or edges

Directed networks: all links are directedPath: a sequence of links from one node to another node(i.e., (5,4)-(4,3)-(3,2))A network isconnectedif there is at least one path from one node to

another node (Net1 is connected whereas Net2 is not)

Examples:2

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Net 1

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Net 2

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Representation of an Urban Road NetworkRepresentation of an Urban Road Network

Physical ConceptualIntersections Nodes

Streets Links

Zones Centroids

Simple node representation


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Intersection RepresentationsIntersection Representations

Simple node representation: no direction differenciation

no conflicting movement

Subnetwork representation: explicit direction representation

conflicting turns in an intersection

are captured by internal links and

their impedances

Conceptual representation is not unique anddepends on:

type of analysis

data availability to build, validate, and apply model

accuracy vs. computation time trade-off

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Shortest Path ProblemsShortest Path Problems

Basic problem: find a shortest path and the shortest distance betweentwo nodes

Basic problem is called the one-to-one shortest path problemTypes of shortest path problems:

One-to-one

One-to-all: find shortest paths from one node to all nodes

All-to-one: find shortest paths from all nodes to one node

Many-to-many: find shortest paths from many nodes to many

other nodes

All-to-all: find shortest paths from all node pairs

Shortest also denotes minimum general costThere are hundreds of shortest path algorithms, but they are similarSome algorithms work for non-negative costs only


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1.225, 11/07/02 Lecture 4, Page 7

DijkstraDijkstras Shortest Paths Algorithm: Examples Shortest Paths Algorithm: Example

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First Shortest Path Algorithm (First Shortest Path Algorithm (DijkstraDijkstras Algorithm)s Algorithm)

Notation: s: source node

d(j): length of shortest path from s toj discovered so far

p(j): immediate predecessor to nodej on shortest path from s toj

discovered so far

k: last node selected by algorithm

Step 1: Initialization d(s) = 0,p(s) = *

d(j) = ,p(j) = -, for all other nodesj s

k= s


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DijkstraDijkstras Shortest Paths Algorithm: Examples Shortest Paths Algorithm: Example

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(0,*)+

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1st iteration(3,a)+

(10,-)

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1.225, 11/07/02 Lecture 4, Page 10

First Shortest Path Algorithm (First Shortest Path Algorithm (DijkstraDijkstras Algorithm)s Algorithm)

Step 2: Update labels of neighbors in open state For all (k,j), ifj is open do:

Ifd(j) < d(k) + l(k,j) then

d(j) = d(k) + l(k,j)

p(j) = k

Step 3 Find a open state node i such that d(i) = min{d(j),j is an open node)}

Step 4 Find a closed state nodej* such that d(i) = d(j*) + l(j*,i)

Step 5 Node i is closed. If no node in open state , STOP.

Otherwise k= i, return to Step 2


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1.225, 11/07/02 Lecture 4, Page 11

Shortest Paths Algorithm: ExampleShortest Paths Algorithm: Example

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(9,d)+

1.225, 11/07/02 Lecture 4, Page 12


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1.225, 11/07/02 Lecture 4, Page 13


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1.225, 11/07/02 Lecture 4, Page 14


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Shortest-path tree


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1.225, 11/07/02 Lecture 4, Page 15

Observations aboutObservations about DijkstraDijkstras Algorithms Algorithm

Dijkstras algorithm is in general not valid if some l(i,j) < 0Shortest paths form a treeThe algorithm can also solve the all-to-one problemIf you solve for a one-to-many problem, stop the algorithm when all

destination nodes are closed

Shortest path problem is an LP problem, but it is more efficient andintuitive to look at it as a network problem as we did in class

1.225, 11/07/02 Lecture 4, Page 16

Extensions of Shortest Path ProblemExtensions of Shortest Path Problem

There is a huge number of potential extensions of the classicalshortest path problem

Problems on dynamic networks (link lengths change over time)Problems on probabilistic networks (link lengths are random

variables assuming discrete values or a continuous range of values)

Combinations thereof

Solutions to these problems depend on the assumptions regarding thestate of knowledge and on the relative magnitude of the parameters

involved

The meaning of shortest path is also an issue in some cases


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1.225, 11/07/02 Lecture 4, Page 17

A Traffic Assignment ProblemA Traffic Assignment Problem

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-403530455

40-3530254

2035-40503

101215-102

15403530-1

54321

1.225, 11/07/02 Lecture 4, Page 18

AllAll--oror--nothingnothing Traffic AssignmentTraffic Assignment

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170

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145120

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52235

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85

150

85

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Does this makesense?


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Traffic Assignment ModelsTraffic Assignment Models

Conceptual definition:

Principles of assignment to represent the interaction User Optimal (U.O.): O-D flows are assigned to paths with

minimum travel time

System Optimal (S.O.): O-D flows are assigned such that total

travel time on the network is minimum

Supply/DemandInteraction

Flows and Travel Times

(input)

output

Supply

Network representation of

transportation network

Link performance functions

Demand

Origin-destination flows

Zoning

(input)

1.225, 11/07/02 Lecture 4, Page 20

ZoningZoning

Physical zones10 11 12

13 14 15

16 17 18

Zone 1

Zone 3

Zone 2

Zone 4

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Centroid nodes and Connectors

Zone-to-zone Flows

O/D Flows

Zone 1 Zone 2 Zone 3 Zone 4

Zon e 1 0 90 120 80

Zon e 2 100 0 60 130

Zon e 3 120 180 0 50

Zon e 4 40 70 150 0

1 2 3 4

1 0 90 120 80

2 100 0 60 130

3 120 180 0 50

4 40 70 150 0


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Analysis PeriodsAnalysis Periods

Over an analysis period, flows are assumed constant in order for steady-state analysis to apply

The duration of a period is longer than a tripTypical analysis periods: morning-peak, midday, evening-peak

time of day

flows

Morning-peak

period

Midday period Evening-peak

period

1.225, 11/07/02 Lecture 4, Page 22

Lecture 4 SummaryLecture 4 Summary

Conceptual Networks: DefinitionsRepresentation of an Urban Road Network (Supply)Shortest Paths (Reading: pp. 359-367, 6.2.3 and 6.2.4 of R6)

Introduction

Dijkstras algorithm: example

Dijkstras algorithm: statement

ObservationsExtensions to Classical Shortest Path ProblemsAll-or-nothing traffic assignmentZoning and Analysis Periods (Demand)Motivation for more advanced traffic assignment models

Documents

4_ Networks Model and Shortest Path