1.225J
1.225J (ESD 225) Transportation Flow Systems
Lecture 4
Introduction to Network Models
and
Shortest Paths
Profs. Ismail Chabini and Amedeo Odoni
Lecture 4 Outline
‰ Conceptual Networks: Definitions
‰ Representation of an Urban Road Network (Supply)
‰ Shortest Paths (Reading: pp. 359-367, 6.2.3 and 6.2.4 of R6)
• Introduction
• Dijkstra’s algorithm: example
• Dijkstra’s algorithm: statement
• Observations
‰ Extensions to Classical Shortest Path Problems
‰ All-or-nothing traffic assignment
‰ Zoning and Analysis Periods (Demand)
‰ Motivation for more advanced traffic assignment models
‰ Summary
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Lecture 4, Page 2
1
Conceptual Networks: Definitions
‰ A network is:
• a set of nodes N and a set of links A
• nodes are also called vertices or points
• links are also called arcs or edges
‰ Examples:
Net 2
Net 1
1
2
1
2
3
4
3
4
5
‰ Directed networks: all links are directed
‰ Path: a sequence of links from one node to another node
(i.e., (5,4)-(4,3)-(3,2))
‰ A network is connected if there is at least one path from one node to
another node (Net1 is connected whereas Net2 is not)
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Lecture 4, Page 3
Representation of an Urban Road Network
‰ Physical
Intersections
Streets
Zones
‰ Conceptual
Nodes
Links
Centroids
‰ Simple node representation
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2
Intersection 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 and
depends on:
• type of analysis
• data availability to build, validate, and apply model
• accuracy vs. computation time trade-off
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Lecture 4, Page 5
Shortest Path Problems
‰ Basic problem: find a shortest path and the shortest distance between
two nodes
‰ Basic problem is called the one-to-one shortest path problem
‰ Types 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 cost
‰ There are hundreds of shortest path algorithms, but they are similar
‰ Some algorithms work for non-negative costs only
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3
Dijkstra’
Dijkstra’s Shortest Paths Algorithm: Example
7
b
Mixed network
f
6
5
5
3
h
5
2
1
a
c
8
e
8
j
1
5
i
d
Directed network
7
6
f
5
5
a
c
h
5
j
1
6
6
2
2
1
e
8
8
5
4
g
4
b
3
6
6
2
i
d
g
4
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4
Lecture 4, Page 7
First Shortest Path Algorithm (Dijkstra
(Dijkstra’’s Algorithm)
‰ Notation:
• s: source node
• d(j): length of shortest path from s to j discovered so far
• p(j): immediate predecessor to node j on shortest path from s to j
discovered so far
• k: last node selected by algorithm
‰ Step 1: Initialization
• d(s) = 0, p(s) = *
• d(j) = ∝, p(j) = -, for all other nodes j ≠ s
• k=s
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Lecture 4, Page 8
4
Dijkstra’
Dijkstra’s Shortest Paths Algorithm: Example
(3,a)+
1st iteration
3
(∝,-)
f
7
b
5
5
6
(∝,-)
h
5
(∝,-)
(0,*)+
a
c
d
(5,-)
1
g
(∝,-)
4
6
5
5
i
(∝,-)
4
(10,-)
f
7
(3,a)+ b
3
j (∝,-)
6
2
5
2nd iteration
e
8
(8,-)
8
6
(∝,-)
h
5
(∝,-)
(0,*)+
a
c
(8,-)
8
d
(5,a)+
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j (∝,-)
1
6
6
4
g
(∝,-)
2
1
e
8
2
5
2
1
4
i
(∝,-)
Lecture 4, Page 9
First Shortest Path Algorithm (Dijkstra
(Dijkstra’’s Algorithm)
‰ Step 2: Update labels of neighbors in open state
• For all (k, j), if j is open do:
If d(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 node j* 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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Lecture 4, Page 10
5
Shortest Paths Algorithm: Example
3
(10,-)
f
7
(3,a)+ b
3rd iteration
6
5
5
(∝,-)
h
5
(∝,-)
(0,*)+
a
c
8
d
(5,a)+
1
g
(9,-)
4
6
5
5
i
(∝,-)
4
(10,-)
f
7
(3,a)+ b
3
j (∝,-)
6
2
5
4th iteration
e
8
(7,d)+
6
(∝,-)
h
5
(15,-)
(0,*)+
a
c
2
1
e
8
(7,d)+
8
j (∝,-)
1
6
6
2
5
2
1
d
g
(9,d)+
4
(5,a)+
i
(∝,-)
4
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Lecture 4, Page 11
Shortest Paths Algorithm: Example
3
(10,b)+
f
7
(3,a)+ b
5th iteration
6
5
5
(∝,-)
h
5
(15,-)
(0,*)+
a
c
8
d
(5,a)+
1
g
(9,d)+
4
6
5
5
i
(13,-)
4
(10,b)+
f
7
(3,a)+ b
3
j (∝,-)
6
2
5
6th iteration
e
8
(7,d)+
6
(16,-)
h
5
(15,-)
(0,*)+
a
c
(7,d)+
8
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j (∝,-)
1
6
6
d
(5,a)+
4
g
(9,d)+
2
1
e
8
2
5
2
1
4
i
(13,g)+
Lecture 4, Page 12
6
Shortest Paths Algorithm: Example
3
(0,*)+
5
a
c
(14,i)+
e
8
(7,d)+
g
(9,d)+
6
5
5
c
6
5
(15,e)+
h
(14,i)+
2
1
e
8
(7,d)+
8
i
(13,g)+
4
(10,b)+
f
7
(3,a)+ b
j (∝,-)
1
6
6
2
5
j (19,-)
1
4
a
2
1
6
d
(5,a)+
3
(16,-)
h
5
2
5
8th iteration
6
5
8
(0,*)+
(10,b)+
f
7
(3,a)+ b
7th iteration
d
g
(9,d)+
4
(5,a)+
i
(13,g)+
4
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Lecture 4, Page 13
Shortest Paths Algorithm: Example
3
(0,*)+
5
5
a
c
8
(7,d)+
6
5
(15,e)+
h
(14,i)+
j (17,h)+
1
6
6
d
(5,a)+
4
4
(10,b)+
f
7
(3,a)+ b
g
(9,d)+
i
(13,g)+
(15,e)+
h
3
(14,i)+
(0,*)+
a
c
1
e
(7,d)+
d
(5,a)+
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4
2
j (17,h)+
1
2
5
2
1
e
8
2
5
Shortest-path tree
(10,b)+
f
7
(3,a)+ b
9th iteration
g
(9,d)+
4
i
(13,g)+
Lecture 4, Page 14
7
Observations about Dijkstra’
Dijkstra’s Algorithm
‰ Dijkstra’s algorithm is in general not valid if some l(i, j) < 0
‰ Shortest paths form a tree
‰ The algorithm can also solve the all-to-one problem
‰ If 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 and
intuitive to look at it as a network problem as we did in class
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Lecture 4, Page 15
Extensions of Shortest Path Problem
‰ There is a huge number of potential extensions of the classical
shortest 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 the
state 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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Lecture 4, Page 16
8
A Traffic Assignment Problem
2
2
1
1
4
8
1
4
5
4
2
3
6
3
1
2
3
4
5
1
-
30
35
40
15
2
10
-
15
12
10
3
50
40
-
35
20
4
25
30
35
-
40
5
45
30
35
40
-
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Lecture 4, Page 17
“AllAll-oror-nothing”
nothing” Traffic Assignment
130
47
85
2
115
130
52
120
235
127
130
4
1
130
150
85
50
180
170
35
3
120
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5
145
Does this make
sense?
Lecture 4, Page 18
9
Traffic Assignment Models
‰ Conceptual definition:
Supply
(input)
(input)
Supply/Demand
Interaction
• Network representation of
transportation network
• Link performance functions
Demand
• Origin-destination flows
• Zoning
output
Flows and Travel Times
‰ 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
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Lecture 4, Page 19
Zoning
‰ Physical zones
10
‰ Zone-to-zone Flows
11
Zone 1
13
12
Zone 2
15
14
Zone 3
16
Zone 4
17
18
Zone 1
Zone 2
Zone 3
Zone 4
‰ Centroid nodes and Connectors
10
11
13
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15
4
3
16
12
14
17
Zone 2
90
0
180
70
Zone 3
120
60
0
150
Zone 4
80
130
50
0
‰ O/D Flows
2
1
Zone 1
0
100
120
40
18
1
2
3
4
1
0
100
120
40
2
90
0
180
70
3
120
60
0
150
4
80
130
50
0
Lecture 4, Page 20
10
Analysis Periods
flows
Morning-peak
period
Midday period
time of day
Evening-peak
period
‰ Over an analysis period, flows are assumed constant in order for steadystate analysis to apply
‰ The duration of a period is longer than a trip
‰ Typical analysis periods: morning-peak, midday, evening-peak
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Lecture 4, Page 21
Lecture 4 Summary
‰ Conceptual Networks: Definitions
‰ Representation of an Urban Road Network (Supply)
‰ Shortest Paths (Reading: pp. 359-367, 6.2.3 and 6.2.4 of R6)
• Introduction
• Dijkstra’s algorithm: example
• Dijkstra’s algorithm: statement
• Observations
‰ Extensions to Classical Shortest Path Problems
‰ All-or-nothing traffic assignment
‰ Zoning and Analysis Periods (Demand)
‰ Motivation for more advanced traffic assignment models
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Lecture 4, Page 22
11