Shortest Path Routing in Transportation Networks

with Time-dependent Road Speeds

Costas K. Constantinou

, Georgios Ellinas

1,2

, Christos Panayiotou

1,2

and Marios Polycarpou

1,2

KIOS Research Center for Intelligent Systems and Networks, University of Cyprus, 1678 Nicosia, Cyprus

Department of Electrical and Computer Engineering, University of Cyprus, 1678 Nicosia, Cyprus

Keywords:

Shortest Path Routing, Time-dependent Transportation Networks.

Abstract:

This paper deals with the subject of shortest (in terms of traveling time) path routing in transportation networks,

where the speed in the network’s roads is a function of the time interval. These networks are encountered in

practice when the roads’ speed has been measured for several time instants during a large period of time (e.g.,

an entire year). In this way, time-dependent speed patterns can be derived for the network’s roads, that consti-

tute an estimation of the network’s future behavior. For shortest path routing in these networks, the traveling

time on the network’s roads must be calculated according to the time instant of departure. Conventional ap-

proaches perform this calculation under the assumption that the road’s speed has a constant (possibly distinct)

value inside each time interval. In the work presented here, the assumption that the road’s speed is linear (pos-

sibly distinct) function of time inside each time interval, is considered. Under this assumption, a procedure is

proposed that derives the traveling time on the network’s roads according to the time instant of departure. It

is combined with Dijkstra’s algorithm, resulting in a practically applicable algorithm for optimal shortest path

routing for the type of networks investigated in this work.

1 INTRODUCTION

The current paper deals with the subject of short-

est path routing in time-dependent transportation net-

works, where the term “shortest path” refers to the

path with the minimum traveling time. The time-

dependent characteristic of the network is the road’s

speed, which is a function of the time interval. This is

a category of networks that is encountered in practice

when the speed on the network’s roads has been mea-

sured for certain time instants (either directly mea-

sured, or derived from vehicle density measurements

(May, 1990)), during a large time interval (e.g., an en-

tire year). In this way, time-dependent speed patterns

can be derived for the network’s roads, that constitute

an estimation of the network’s future behavior.

For shortest path routing in these networks, the

traveling time on a network’s road depends on the

time instant of departure from the origin of the road,

and must be calculated during the derivation of the

shortest path. Conventional approaches perform this

calculation under the assumption that the road’s speed

has a constant (possibly distinct) value inside each

time interval. However, this assumption leads to a dis-

continuous function of speed over time, and it is not

in accordance with practical scenarios, as described

in more detail in Section 2. In the work presented

here, the assumption that the road’s speed is a lin-

ear (possibly distinct) function of time inside each

time interval, is considered. Such a function over-

comes the aforementioned drawbacks of the conven-

tional approaches. The main contribution of the cur-

rent paper is a procedure that derives the traveling

time on a network’s road according to the time instant

of departure from the origin of the road, under the

aforementioned linearity assumption. The proposed

procedure is combined with Dijkstra’s algorithm, re-

sulting to a practically applicable algorithm for op-

timal shortest path routing for the type of networks

under investigation.

The proposed work can be easily utilized in navi-

gation systems for the derivation of the fastest routes,

considering that the network will demonstrate the

same behavior in the future, as the one estimated

by the speed measurements. Note that the proposed

work refers to optimal routing, for the networks un-

der investigation; therefore, numerical examples are

not presented.

The outline of the remaining paper is as follows.

In Section 2 the assumptions for this work are pre-

Constantinou, C., Ellinas, G., Panayiotou, C. and Polycarpou, M.

Shortest Path Routing in Transportation Networks with Time-Dependent Road Speeds.

In Proceedings of the International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2016), pages 91-98

ISBN: 978-989-758-185-4

sented and their validity examined under realistic sce-

narios. The modeling of the networks under inves-

tigation is also presented, as well as the relevant ex-

isting approach for optimal shortest path routing in

these types of networks. Further, the drawbacks of

this approach are described; these constitute the mo-

tivation for the work proposed in the paper. Section

3 describes alternative existing approaches to model

a time-dependent network, and it is described why

these approaches are not suitable for the networks in-

vestigated in this paper. The contribution of the paper

can be found in Sections 4 and 5. Speciﬁcally, in Sec-

tion 4, the case of the road’s speed being a linear func-

tion of time inside each time interval is investigated.

A procedure is proposed that derives the time needed

to traverse a network’s road according to the time in-

stant of departure, under the aforementioned assump-

tion. This procedure is then combined with Dijkstra’s

algorithm for optimal shortest path routing in the in-

vestigated networks. In addition, in Section 5, it is

proven that the model utilized in this paper satisﬁes

the First-In-First-Out (FIFO) property for the general

case of the road’s speed being an arbitrary function of

time inside each time interval. Finally, in Section 6

the conclusions for this work are presented, as well as

ongoing research.

2 UTILIZED MODEL

Throughout the paper it is considered that the speed

in the network’s roads has been measured for cer-

tain time instants (either directly measured or derived

from vehicle density measurements (May, 1990)),

during a large period of time (e.g., an entire year).

In this way, time-dependent speed patterns can be de-

rived for the network’s roads, that constitute an esti-

mation of the network’s future behavior. More pre-

cisely, it is assumed that the network in the future

demonstrates the same behavior (except for unpre-

dictable events) as the one observed during the period

of time where the roads’ speed has been measured.

The network is modeled as a directed graph con-

sisting of a set of N (n = |N|) nodes (i.e., road junc-

tions) and a set of A (m = |A|) arcs (i.e., roads). The

arc originating from node x and ending at node y is

denoted by < xy > and its length (i.e., actual length

of the corresponding road) by d

. The speed in arc

< xy > is denoted by v

. A node y is considered to

be adjacent to node x if arc < xy > exists in the graph.

The cost c

(or “traversal time” or “traveling time”)

of arc < xy > is deﬁned as the time needed to traverse

it, i.e., to move from node x to node y. Consequently,

the term “shortest path” from node a to node b refers

to the minimum traveling time path from a to b.

For the case of time-independent networks, the

cost of an arc < xy > is constant over time, equal to

. For time-dependent networks, it is a func-

tion of the time instant of departure, τ, from node

x, and it is denoted by c

(τ). Therefore, for short-

est path routing in time-dependent networks an ad-

ditional calculation must be performed, compared to

time-independent networks; that is, the calculation of

(τ) according to the time instant of departure from

node x. At this point, it must be stated that the time in-

stant of departure τ is permitted to have any real value,

rather than being a discrete-value variable. This is in

accordance with practical scenarios.

Consider that the speed in a certain arc (i.e., road)

is measured at several time instants, τ

, with k ∈ N

and 0 ≤ k ≤ K; the notation v

(τ

) = v

stands for

the measurement of speed at time instant τ

. Then, for

arc < xy > the time horizon is partitioned into (in gen-

eral not equal) non-overlapped time intervals, where

the partitioning points are the time instants where the

speed has been measured. The (k + 1)

time interval

is denoted by [τ

,τ

k+1

). According to the aforemen-

tioned notation, the number of time intervals is equal

to K + 1.

Let the set of speeds v

and time intervals

[τ

,τ

k+1

), 0 ≤ k ≤ K, for arc < xy > be denoted by

and T

respectively. Consider also that

V =

∀<xy>∈A

(1)

T =

∀<xy>∈A

(2)

Then, the network graph is denoted by G =

(N,A,T,V ).

This modeling of the time-dependent network was

proposed in (Sung et al., 2000), and it is called Flow

Speed Model (FSM). One aspect of this modeling of

a time-dependent network that is not described yet, is

the value of an arc’s speed for time instants where the

speed has not been measured. In (Sung et al., 2000),

it is assumed that the speed is constant inside each

time interval, equal to the value measured at the left

bound of the time interval. More precisely, as de-

scribed previously, the arbitrary time interval of arc

< xy > (denoted by [τ

,τ

k+1

)) was determined based

on the fact that the speed has been measured at time

instants τ

and τ

k+1

. Therefore, in (Sung et al., 2000)

it is considered that the speed in arc < xy > is equal

to v

for all time instants τ such that τ

≤ τ < τ

k+1

In other words, it is considered that for time instant

τ = τ

k+1

−ε, where ε is a positive, arbitrarily small

constant, the speed in < xy > is equal to v

(i.e., equal

to the one measured at time instant τ

), whereas for

VEHITS 2016 - International Conference on Vehicle Technology and Intelligent Transport Systems

time instant τ = τ

k+1

the speed in < xy > is equal to

k+1

(i.e., equal to the one measured at time instant

k+1

). This assumption has the following potential

drawbacks:

i) It leads to a discontinuous function of speed

over time, since

lim

ε→0

(τ

k+1

−ε)} 6= v

(τ

k+1

) (3)

(Note that in equation 3 the expression v

(τ

k+1

−

ε), as stated previously, refers to the speed at time

instant τ

k+1

−ε, i.e., it does not denote the multipli-

cation of v

with the latter.)

ii) It is not in accordance with practical scenarios,

since for time instants τ very close to τ

k+1

, this as-

sumption still gives a speed value equal to v

, rather

than a value close to v

k+1

The work presented in the current paper is based

on the assumption that the speed inside a time inter-

val must be considered as a linear function of time,

where this function must lead to speed values equal to

, v

k+1

for time instants τ

, τ

k+1

respectively. Such

a function would overcome the aforementioned draw-

backs of the conventional approach. This is exactly

the contribution of the current paper, presented in de-

tail in Section 4.

Another characteristic of the FSM that must be

stated, is that it always satisﬁes the First-In-First-Out

(FIFO) property, as proven in (Sung et al., 2000). In

simple words, the FIFO property has the following

meaning: Consider a vehicle A that can depart from

node x and traverse arc < xy >, having two possible

time instants of departure, τ

and τ

, with τ

< τ

and let τ

and τ

be the corresponding time instants

of arrival at node y. Then, if the FIFO property is

valid, τ

< τ

(the traveling time τ

−τ

, though, may

be smaller than traveling time τ

−τ

). The reader

should note that this property is necessary for a model

in order for it to objectively represent real networks,

since the scenario of departing later and arriving ear-

lier deviates from reality. Section 5 addresses pre-

cisely this point, proving that the model utilized in

this paper satisﬁes the First-In-First-Out (FIFO) prop-

erty for the general case of the road’s speed being an

arbitrary function of time inside each time interval.

As stated previously, for shortest path routing

in time-dependent networks an additional calculation

must be performed compared to time-independent

networks, that is the calculation of c

(τ), i.e., the

traversal time on arc < xy > according to the time

instant of departure, τ, from node x. In the FSM, this

is performed as described below.

2.1 AT T Procedure

In the FSM, the traversal time c

(τ) on arc < xy >,

if τ is the time instant of departure from x, is derived

using the following procedure, called Arc Traversal

Time Procedure (AT T

(τ), or simply AT T ). This

procedure has been proposed in (Sung et al., 2000) (in

this paper it is presented in an equivalent, slightly dif-

ferent way). The reader should note that in the steps

of the AT T , as well as in the succeeding illustrative

example, for simplicity index ∗

is omitted from the

relevant variables.

Steps of ATT Procedure

1. Locate index k such that τ

≤ τ < τ

k+1

;

2. If (v

·(τ

k+1

−τ) ≥ d)

c(τ) =

;

Else

{

3.(a)i. a = d −v

·(τ

k+1

−τ);

ii. k

∗

= k + 1;

(b) While (v

∗

·(τ

∗

−τ

∗

) < a)

{

i. a ← a −v

∗

·(τ

∗

−τ

∗

);

ii. k

∗

← k

∗

+ 1;

}

4. c(τ) = (τ

∗

−τ) +

∗

;

}

The necessity of the aforementioned procedure

lies on the fact that a single time interval may not be

enough for the traversal of the whole arc. This occurs

when the distance that can be traversed from the time

instant of departure till the end of the corresponding

time interval is less than the length of the arc.

Example of ATT

The example of Figure 1 illustrates the operation

of the AT T procedure. Here, the length d of arc

< xy > is equal to 170m and the departure time τ

from x is equal to 6s. The ﬁrst ﬁve time intervals

are [τ

,τ

) = [0,10), [τ

,τ

) = [10,15), [τ

,τ

) =

[15,30), [τ

,τ

) = [30,40), [τ

,τ

) = [40,50). These

time intervals, along with their corresponding speeds,

are shown in Figure 1.

For this example, the AT T procedure performs as

follows:

1. The departure time τ = 6s lies in time interval

[τ

,τ

) = [0, 10) ⇒ k = 0

Shortest Path Routing in Transportation Networks with Time-Dependent Road Speeds

5 10 15 20 25 30 35 40 45

m/s

t50 s

Figure 1: Example for the derivation of arc traversal time

(d = 170m, τ = 6s).

2. The distance that can be traversed from τ = 6s

until the end of this time interval is equal to

·(τ

−τ) = 10 ·(10 −6) = 40m. Since it is

smaller than the length of the arc, the procedure

continues.

3.(a)i. a = d −v

·(τ

−τ) = 170 −40 = 130m

ii. k

∗

= 1

(b)• v

· (τ

− τ

) = 6 · (15 −10) = 30m < a =

130m ⇒

i. a = 130 −30 = 100m

ii. k

∗

= 2

• v

·(τ

−τ

) = 8 · (30 − 15) = 120m > a =

100m ⇒ exit from the while loop

4. c(6) = (τ

−τ) + a/v

= (15 −6) + 100/8 = 9 +

12.5 = 21.5s

The traversal time is equal to 21.5s and the time

instant of arrival at node y is equal to 6+21.5 = 27.5s.

Complexity of ATT

Step 1 of the AT T procedure needs O(K) time, if the

time intervals are checked sequentially. The order of

the number of the time intervals that are checked dur-

ing the while loop (step 3b) is O(K). Therefore, the

order of the computational complexity of the ATT

procedure is O(K).

In (Sung et al., 2000), the ATT procedure is com-

bined with Dijkstra’s algorithm (Dijkstra, 1959), and

the resulting algorithm can provide optimal short-

est path routing for the networks under investiga-

tion. For completeness, this algorithm is presented

in the succeeding section. Hereafter, it is called Time-

Dependent-Dijkstra (TD-Dijkstra).

2.2 TD-Dijkstra

The input of TD-Dijkstra is the network graph G =

(N,A,T,V ), the source node s, and the time instant of

departure τ, from the source. The output is the short-

est path from the source to every other network node.

For the execution of TD-Dijkstra, the following are

used.

• G = (N,A,T,V ): Network graph

• s: Source

• τ: Time instant of departure from s

• W (x): Label of node x

• p(x): Predecessor of node x

• G

: Set of nodes adjacent to node x

• g

: Number of nodes adjacent to node x

(i.e., g

= |G

The exact steps of TD-Dijkstra are:

1.(a) W (s) = τ;

(b) p(s) = 0;

∗

= N −{s};

(d) ∀x ∈N

∗

i. If (x ∈G

)

{

Run AT T

(τ);

W (x) = τ + c

(τ);

p(x) = s ;

}

ii. Else

{

W (x) = ∞;

p(x) = 0;

}

2. While (N

∗

{

(a) Find x ∈ N

∗

such that ∀x

∈N

∗

: W (x) ≤W (x

);

(b) N

∗

← N

∗

−{x};

∈ (N

∗

∩G

i. Run AT T

(W (x));

ii. If (W (x) + c

(W (x)) < W (x

))

{

W (x

) = W (x) + c

(W (x));

p(x

) = x;

}

The TD-Dijkstra algorithm functions as the clas-

sical Dijkstra’s algorithm, with the difference that the

AT T procedure is used in steps 1(d)i and 2(c)i for the

derivation of the cost of the arbitrary arc < xy >, ac-

cording to the time instant of departure from node x.

VEHITS 2016 - International Conference on Vehicle Technology and Intelligent Transport Systems

On termination of the algorithm, the label W(x) of

the arbitrary node x gives the time instant of arrival

at x. Therefore, the cost of the shortest path from the

source s to node x is equal to W(x) −τ. The variable

p(x) gives the predecessor of x in this path.

Since the computational complexity of the ATT

procedure is of order O(K), step 1 requires O(g

K +

n) time and each iteration of step 2 requires O(n +

K) time. Since the number of repetitions of step

2 is of order O(n), and

∑

x=1

= m, the complexity

of TD-Dijkstra is of order O(g

K + n + n

+ mK) =

O(n

+ mK). The reader should note that although

the aforementioned, simple implementation is pre-

sented in (Sung et al., 2000) where this algorithm was

proposed, a faster implementation can be performed

if Fibonacci heaps are utilized (Fredman and Tarjan,

1987), (Ahuja et al., 1993). Under this data structure,

the computational complexity of TD-Dijkstra would

be of order O(nlogn + mK).

In the section that follows, existing alternative ap-

proaches to model a time-dependent transportation

network are presented, and it is explained why they

are unsuitable for optimal shortest path routing in the

networks under investigation.

3 ALTERNATIVE APPROACHES

Alternative existing approaches to model a time-

dependent network can be found, among others, in

(Cooke and Halsey, 1966), (Delling, 2011), (Nan-

nicini et al., 2012), (Delling and Nannicini, 2012),

(Delling et al., 2009), (Delling and Wagner, 2009),

(Ding et al., 2008), (Batz et al., 2013), (Chabini and

Lan, 2002), and (Chabini, 1998). All the aforemen-

tioned papers assume that for each network’s arc <

xy >, the traversal time is available for certain time

instants of departure from node x. Let this arc traver-

sal time, which is a function of the time instant of

departure, be called as traversal time function and be

denoted by f

(τ), for the arbitrary arc < xy > and for

τ as the time instant of departure. In the aforemen-

tioned papers, the traversal time function is assumed

to be known for some values of τ and these values are

utilized for the derivation of the arc traversal time for

an arbitrary time instant of departure.

However, as stated previously, in the networks in-

vestigated in the current paper, it is (realistically) as-

sumed that the time instant of departure may have

any arbitrary value. Therefore, the work presented in

(Chabini and Lan, 2002), (Chabini, 1998), which is

concentrated on discrete-time networks, cannot lead

to optimal shortest path routing in the investigated

networks. Furthermore, in the networks under investi-

gation, the traversal time function is not directly avail-

able, since, as described previously, the speed rather

than the arc traversal time, has been measured. Con-

sequently, the work found in the rest of the aforemen-

tioned papers (which are concentrated on continuous-

time networks) cannot be directly applied for optimal

shortest path routing in the networks under investiga-

tion.

Consider the case where the traversal time func-

tion is derived from the available data (i.e., the speed

measurements) for certain time instants (e.g., for the

time instants that the speed has been measured). This

would be a preprocessing step, performed using the

AT T procedure. Even under this scenario, the work

found in these papers would not lead to optimal short-

est path routing in the investigated networks. The rea-

son is that in all these works, it is assumed that the

traversal time function is a piecewise linear function

of time, having the time instants where it is known,

as breakpoints. Under this assumption, for a time

instant τ for which f

(τ) is unknown, it can be de-

rived by linear interpolation between the consecutive

breakpoints τ

, τ

k+1

such that τ

< τ < τ

k+1

, using

the known values f

(τ

) and f

(τ

k+1

), as shown in

equation 5 below.

(τ)−f

(τ

)

τ−τ

(τ

k+1

)−f

(τ

)

k+1

−τ

(4)

⇒ f

(τ) =

(τ

k+1

)−f

(τ

)

k+1

−τ

·(τ −τ

) + f

(τ

) (5)

In the networks investigated in the current paper,

though, the assumption that the traversal time func-

tion is a piecewise linear function of time, is, in gen-

eral, not valid. This can be proven theoretically; it

is not presented here, since it deviates from the pur-

pose of the current paper. Nevertheless, this can be

easily veriﬁed from the example of Figure 1. Here, if

the arc traversal time is derived for the time instants

that the speed has been measured (this must be per-

formed using the AT T procedure), then the derived

traversal time function f

(τ) for time instants equal

to 0s, 10s is equal to 20s, 22s, respectively. Using

equation 5, f

(6) = 21.2s, i.e., the assumption that

(τ) is piecewise linear function of time, leads to

arc traversal time equal to 21,2s, for τ = 6s as the

time instant of departure. However, the correct value

is equal to 21.5s, as derived in Section 2. Therefore,

using this example, it is shown that this assumption,

in general, is not valid.

Thus, according to the aforementioned analy-

sis, the alternative approaches for modeling a time-

dependent network cannot be applied for optimal

shortest path routing in the networks under investi-

gation, since for these networks they give suboptimal

solutions. Further analysis of the advantages of the

Shortest Path Routing in Transportation Networks with Time-Dependent Road Speeds

FSM model, compared to other approaches, can also

be found in (Sung et al., 2000), where this model was

proposed.

In the section that follows, the proposed model

is presented, along with the corresponding procedure

that derives the traveling time on the network’s roads

according to the time instant of departure.

4 SPEED AS LINEAR FUNCTION

OF TIME, INSIDE THE TIME

INTERVAL

As stated previously, the conventional approach

(Sung et al., 2000) assumes that the speed is consid-

ered to have constant value inside each time interval.

In this section, the case of speed being a linear func-

tion of time inside each time interval, is proposed. The

reader should note that the linearity here refers to the

function of speed over time, inside each time interval;

it must not be confused with the linearity of the traver-

sal time function that was described in Section 3.

The proposed approach overcomes the drawbacks

of the conventional one, since it leads to a continuous

function of speed over time, i.e.,

lim

ε→0

(τ

k+1

−ε)} = v

(τ

k+1

) (6)

It is in accordance with practical scenarios as well,

since for time instants τ very close to τ

k+1

, this ap-

proach leads to a speed value close to v

k+1

Let this linear function of speed be denoted by

(t) for arc < xy > and for time instant of depar-

ture t belonging to time interval [τ

,τ

k+1

), and let

) =

(t)dt (with τ

≤ t

≤ τ

k+1

The latter gives the value of the distance traversed on

arc < xy > from time instant t

to t

(as long as t

and

belong to the same time interval).

Consider that the speed has been measured at time

instants τ

and τ

k+1

with measured values equal to v

and v

k+1

respectively. Then, function g

(t) is consid-

ered to be a linear function of time t inside the time

interval [τ

,τ

k+1

), taking the values g

(τ

) = v

and

(τ

k+1

) = v

k+1

at the bounds of this time interval.

As stated previously, in the conventional approach it

is assumed that this function is constant, equal to v

for the entire time interval [τ

,τ

k+1

Since function g

(t) is considered to be linear in-

side each time interval, the following hold (τ

≤ t <

k+1

, and index ∗

is omitted for simplicity):

(t)−v

t−τ

k+1

−v

k+1

−τ

(7)

(t) −v

k+1

−v

k+1

−τ

·(t −τ

) (8)

(t) =

k+1

−v

k+1

−τ

t + (v

−

k+1

−v

k+1

−τ

) (9)

(t) =

k+1

−v

k+1

−τ

t +

k+1

−v

k+1

−τ

(10)

) = (

k+1

−v

k+1

−τ

k+1

−v

k+1

−τ

)−

−(

k+1

−v

k+1

−τ

k+1

−v

k+1

−τ

) (11)

) =

k+1

−v

k+1

−τ

−t

k+1

−v

k+1

−τ

−t

) (12)

) = R

−t

+ S

−t

) (13)

where equation 13 is derived if we set:

k+1

−v

k+1

−τ

= R

(14)

k+1

−v

k+1

−τ

= S

(15)

Let the ATT procedure under the linearity as-

sumption be denoted as Linear AT T (ATT

). Then,

the proposed AT T

is derived from the existing AT T

as follows.

• Step 1 remains the same as in AT T .

• In step 2 of the AT T , v

·(τ

k+1

−τ) is equal to the

distance traversed from τ to τ

k+1

. This distance in

AT T

is given by

(τ,τ

k+1

) = R

k+1

−τ

+ S

(τ

k+1

−τ) (16)

– If this distance is equal to or larger than d (i.e.,

if G

(τ,τ

k+1

) ≥ d), then the traversal time c(τ)

is given by the solution of equation 17, as fol-

lows.

(τ,τ + c(τ)) = d (17)

(τ+c(τ))

−τ

+ S

(τ + c(τ) −τ) = d (18)

((τ + c(τ))

−τ

) + 2S

(τ + c(τ) −τ) =

= 2d (19)

(c(τ)

+ 2τc(τ)) + 2S

c(τ) = 2d (20)

c(τ)

+ 2(R

τ + S

)c(τ) −2d = 0 (21)

c(τ) =

−2(R

τ+S

)±

√

4(R

τ+S

)

+8R

(22)

c(τ) =

−(R

τ+S

)±

√

τ+S

)

+2R

(23)

– If this distance is less than d (i.e., if

(τ,τ

k+1

) < d), the procedure continues to

step 3.

VEHITS 2016 - International Conference on Vehicle Technology and Intelligent Transport Systems

• In step 3(a)i, a is set to d −G

(τ,τ

k+1

). Step3(a)ii

remains the same.

• In step 3(b), the statement in the while loop would

be G

∗

(τ

∗

,τ

∗

) < a; every time this is valid, a

takes the value a−G

∗

(τ

∗

,τ

∗

), and k

∗

←k

∗

• When the procedure exits from the while loop, the

value of the arrival time, τ

, will lie in time in-

terval [τ

∗

,τ

∗

). let c

(τ) = τ

−τ

∗

; this is the

time needed to traverse the last part of the arc, i.e.,

while time interval [τ

∗

,τ

∗

) is under consider-

ation. Therefore, the arc traversal time will be

equal to (τ

∗

−τ) + c

(τ). The value of c

(τ) is

derived as follows.

∗

(τ

∗

,τ

) = a (24)

∗

(τ

∗

,τ

∗

+ c

(τ)) = a (25)

∗

(τ

∗

(τ))

−τ

∗

(τ

∗

+ c

(τ) −τ

∗

) = a (26)

∗

((τ

∗

+ c

(τ

∗

))

−τ

∗

+2S

∗

(τ

∗

+ c

(τ) −τ

∗

) = 2a (27)

∗

(τ)

+ 2τ

∗

(τ)) + 2S

∗

(τ) = 2a (28)

∗

(τ)

+ 2(R

∗

+ S

∗

(τ) −2a = 0(29)

(τ) =

−2(R

∗

)±

√

4(R

∗

)

+8R

∗

(30)

(τ) =

−(R

∗

)±

√

∗

)

+2R

∗

(31)

Equation 31 is used in step 4 of the ATT

, for the

derivation of the arc traversal time after the proce-

dure exits from the while loop of step 3, i.e.,

c(τ) = τ

−τ (32)

c(τ) = (τ

∗

+ c

(τ)) −τ (33)

c(τ) = (τ

∗

−τ)+

−(R

∗

)±

√

∗

)

+2R

∗

(34)

Considering the analysis above, the exact steps of

the proposed AT T

procedure are as follows.

Steps of AT T

Procedure

1. Locate index k such that τ

≤ τ < τ

k+1

;

2. If (G

(τ,τ

k+1

) ≥ d)

c(τ) =

−(R

τ+S

)±

√

τ+S

)

+2R

;

Else

{

3.(a)i. a = d −G

(τ,τ

k+1

);

ii. k

∗

= k + 1;

(b) While (G

∗

(τ

∗

,τ

∗

) < a)

{

i. a ← a −G

∗

(τ

∗

,τ

∗

);

ii. k

∗

← k

∗

+ 1;

}

c(τ) = (τ

∗

−τ)+

−(R

∗

)±

√

∗

)

+2R

∗

;

}

The computational complexity of the proposed

AT T

procedure is O(K) (i.e, the same as the one of

AT T ). The optimal shortest path routing for this case

can be performed by TD-Dijkstra, (Section 2.2) just

by substituting the ATT procedure with the AT T

pro-

cedure in the description of the algorithm.

In Section 3 it was shown that the alternative ap-

proaches give suboptimal solutions for the networks

investigated in the current paper, under the assump-

tion that the speed has constant value inside each time

interval. These approaches also give suboptimal so-

lutions for the investigated networks, under the as-

sumption that the speed is linear function of time in-

side each time interval. This can be veriﬁed using the

example of Figure 1, modiﬁed under the linearity as-

sumption.

5 FSM AND FIFO PROPERTY

FOR THE GENERALISED CASE

In (Sung et al., 2000), it was proven that the FSM

satisﬁes the FIFO property for the case of constant

speed inside each time interval. In this section, it is

proven that the FIFO property also holds for the gen-

eral case of the speed being an arbitrary function of

time (g

(τ)) inside the time interval. A direct conse-

quence of it is that the FSM satisﬁes the FIFO prop-

erty under the linearity assumption utilized in Sec-

tion 4.

Consider that for an arbitrary arc < xy >, two ve-

hicles 1, 2 depart from node x at time instants τ

, τ

respectively, with τ

> τ

, and arrive at node y at time

instants τ

, τ

respectively. Obviously, τ

> τ

and

> τ

To prove that the FIFO property is valid, it must be

proven that τ

> τ

. The following cases are possible:

Shortest Path Routing in Transportation Networks with Time-Dependent Road Speeds

• τ

≥ τ

>τ

===⇒ τ

> τ

• τ

< τ

. The distance d is traversed from

time instant τ

to τ

, and it can be split into dis-

tances a and b (a+b = d), where a is the distance

traversed from τ

to τ

and b is the distance tra-

versed from τ

to τ

. Then,

> τ

⇒ a > 0 ⇒b < d (35)

From equation 35, it is concluded that the time

that elapses from τ

to τ

is not enough to traverse

the whole arc < xy >. Therefore, vehicle 2 that

departs from node x at time instant τ

will arrive

at node y at τ

> τ

6 CONCLUSIONS

This work focused on the subject of shortest (in terms

of traveling time) path routing in transportation net-

works, where the speed in the network’s roads is a

function of the time interval. A procedure was pro-

posed that derives the road’s traversal time in these

networks, according to the time instant of departure,

under the assumption that the road’s speed is a linear

(possibly distinct) function of time inside each time

interval. This procedure can then be combined with

Dijkstra’s algorithm to obtain optimal shortest paths

for the networks under investigation. Further, it was

also proven that the approach utilized in this work for

modeling the investigated networks satisﬁes the FIFO

property for the general case of the speed being an

arbitrary function of time inside each time interval.

The proposed approach is more practical and readily

addresses the limitations of the conventional existing

approaches that assume that the speed is constant in-

side each time interval.

On going work focuses on the combination of

the proposed procedure with algorithms that provide

practically fast optimal shortest path routing in time-

independent networks, so as to develop their time-

dependent versions that will lead to optimal solutions

for the networks under investigation.

ACKNOWLEDGMENTS

This work was supported by the Cyprus Research

Promotion Foundation’s Framework Programme for

Research, Technological Development and Innova-

tion 2008 (DESMI 2008), co-funded by the Republic

of Cyprus and the European Regional Development

Fund, and speciﬁcally under Grant Project New In-

frastructure/Strategic/0308/26.

REFERENCES

Ahuja, R. K., Magnanti, T. L., and Orlin, J. B. (1993). Net-

work Flows: Theory, Algorithms, and Applications.

Prentice-Hall, Inc., Upper Saddle River, NJ, USA.

Batz, G. V., Geisberger, R., Sanders, P., and Vetter,

C. (2013). Minimum time-dependent travel times

with contraction hierarchies. J. Exp. Algorithmics,

18:1.4:1.1–1.4:1.43.

Chabini, I. (1998). Discrete dynamic shortest path prob-

lems in transportation applications: Complexity and

algorithms with optimal run time. Transportation Re-

search Records, 1645:170–175.

Chabini, I. and Lan, S. (2002). Adaptations of the A* algo-

rithm for the computation of fastest paths in determin-

istic discrete-time dynamic networks. Trans. Intell.

Transport. Sys., 3(1):60–74.

Cooke, K. and Halsey, E. (1966). The shortest route

through a network with time-dependent internodal

transit times. Journal of Mathematical Analysis and

Applications, 14(3):493–498.

Delling, D. (2011). Time-dependent sharc-routing. Algo-

rithmica, 60(1):60–94.

Delling, D. and Nannicini, G. (2012). Core routing on dy-

namic time-dependent road networks. INFORMS J.

on Computing, 24(2):187–201.

Delling, D., Sanders, P., Schultes, D., and Wagner, D.

(2009). Algorithmics of large and complex net-

works. chapter Engineering Route Planning Algo-

rithms, pages 117–139. Springer-Verlag, Berlin, Hei-

delberg.

Delling, D. and Wagner, D. (2009). Time-dependent route

planning. In Ahuja, R., Mhring, R., and Zaroliagis, C.,

editors, Robust and Online Large-Scale Optimization,

volume 5868 of Lecture Notes in Computer Science,

pages 207–230. Springer Berlin Heidelberg.

Dijkstra, E. W. (1959). A note on two problems in connex-

ion with graphs. Numerische Mathematik, 1:269–271.

10.1007/BF01386390.

Ding, B., Yu, J. X., and Qin, L. (2008). Finding time-

dependent shortest paths over large graphs. In Pro-

ceedings of the 11th International Conference on Ex-

tending Database Technology: Advances in Database

Technology, EDBT ’08, pages 205–216, New York,

NY, USA. ACM.

Fredman, M. L. and Tarjan, R. E. (1987). Fibonacci heaps

and their uses in improved network optimization algo-

rithms. J. ACM, 34(3):596–615.

May, A. D. (1990). Trafﬁc ﬂow fundamentals. Prentice Hall.

Nannicini, G., Delling, D., Schultes, D., and Liberti, L.

(2012). Bidirectional A* search on time-dependent

road networks. Networks, 59(2):240–251.

Sung, K., Bell, M. G., Seong, M., and Park, S. (2000).

Shortest paths in a network with time-dependent ﬂow

speeds. European Journal of Operational Research,

121(1):32–39.

VEHITS 2016 - International Conference on Vehicle Technology and Intelligent Transport Systems