Extended Shortest Path Problem

Generalized Dijkstra-Moore and Bellman-Ford Algorithms

Maher Helaoui

Higher Institute of Business Administration, University of Gafsa, Rades, Tunisia

Keywords:

Combinatorial Optimization, Valuation Structure, Extended Shortest Path Problem, Longest Path Problem,

Generalized Dijkstra-Moore Algorithm, Generalized Bellman-Ford Algorithm.

Abstract:

The shortest path problem is one of the classic problems in graph theory. The problem is to provide a solution

algorithm returning the optimum route, taking into account a valuation function, between two nodes of a graph

G. It is known that the classic shortest path solution is proved if the set of valuation is R or a subset of R and

the combining operator is the classic sum (+). However, many combinatorial problems can be solved by using

shortest path solution but use a set of valuation not a subset of R and/or a combining operator not equal to

the classic sum (+). For this reason, relations between particular valuation structure as the semiring and diod

structures with graphs and their combinatorial properties have been presented. On the other hand, if the set of

valuation is R or a subset of R and the combining operator is the classic sum (+), a longest path between two

given nodes s and t in a weighted graph G is the same thing as a shortest path in a graph −G derived from G

by changing every weight to its negation.

In this paper, in order to give a general model that can be used for any valuation structure we propose to

model both the valuations of a graph G and the combining operator by a valuation structure S. We discuss

the equivalence between longest path and shortest path problem given a valuation structure S. And we present

a generalization of the shortest path algorithms according to the properties of the graph G and the valuation

structure S.

1 INTRODUCTION

The shortest path problem is one of the classic prob-

lems in graph theory. The problem is to provide a so-

lution algorithm returning the optimum route, taking

into account a valuation function, between two nodes

of a graph G.

In (Shimbel, 1955; Ford and Lester, 1956; Bellman,

1958; Sedgewick and Wayne, 2011) the classic short-

est path solution is proved if

• the set of valuation is R or a subset of R.

• the combining operator is the classic sum (+)

Many combinatorial problems like Fuzzy, Weighted,

Probabilistic and Valued Constraint Satisfaction Prob-

lem (Schiex et al., 1995; Cooper, 2003; Cooper,

2004; Allouche et al., 2009) use a set of valuation

E not subset of R and a combining operator ⊕ 6= +

for weighted, fuzzy, probabilistic . . . valuations. In

(Cooper, 2003), the shortest path algorithm has been

used to solve Fuzzy and Valued Constraint Satisfac-

tion Problem.

In (Erickson, 2010), author observes that the classical

maximum ﬂow problem (Ford and Fulkerson, 1955;

Ford and Fulkerson, 1962) in any directed planar

graph G can be reformulated as a parametric short-

est path problem in the oriented dual graph G

∗

. In

(Cohen et al., 2004; Helaoui et al., 2013), a submod-

ular decompositions approach has been presented to

solve Valued Constraint Satisfaction Problem. This

solution use the maximum ﬂow algorithm.

Dijkstra-Moore and Bellman-Ford Algorithms are the

most known algorithmic solutions for the shortest

path problem.

• Since 1971, the Dijkstra-Moore Algorithm has

been used if the set of valuation is R

+

or a subset

of R

+

and the combining operator is the classic

sum (+).

• The Bellman-Ford Algorithm is the result of

(Shimbel, 1955; Ford and Lester, 1956; Bellman,

1958) works. It is used if the set of valuation is R

or a subset of R and the combining operator is the

classic sum (+).

As many combinatorial problems can be solved by us-

ing shortest path solution but use a set of valuation

306

Helaoui M.

Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms.

DOI: 10.5220/0006145303060313

In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 306-313

ISBN: 978-989-758-218-9

Copyright

c

2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

not a subset of R and/or a combining operator not

equal to the classic sum (+), then in (Gondran and

Minoux, 2008), authors present new models and al-

gorithms discussing relations between particular val-

uation structure: the semiring and diod structures with

graphs and their combinatorial properties.

In (Sedgewick and Wayne, 2011), a longest path be-

tween two given nodes s and t in a weighted graph

G is the same thing as a shortest path in a graph −G

derived from G by changing every weight to its nega-

tion. Therefore, if shortest paths can be found in −G,

then longest paths can also be found in G. This result

remains true if we have a valued graph G by a valua-

tion structure S?

In this paper, we provide an answer to this question

by discussing equivalence between longest path and

shortest path problem given a valuation structure S.

We present a generalization of Dijkstra-Moore Algo-

rithm for a graph G with a S

⊕

valuation structure. And

we present a generalization of Bellman-Ford Algo-

rithm with a more general valuation structure S.

We propose to model both the valuations of a graph G

and the combining operator by a valuation structure

S, in order to discuss the generalization of the short-

est path algorithms according to the properties of the

graph G and the valuation structure S:

• The valuation structure of G is S

⊕

.

• The graph G and the valuation structure S are ar-

bitrary.

The paper is organized as follows: the next Section

introduces deﬁnitions and notations needed in pre-

senting the generalization of the shortest path algo-

rithms. In Section 3 we study the extended Shortest

Path Notion and the equivalence between longest path

and shortest path problem. We propose a generalized

shortest path algorithms in Section 4. The paper is

concluded in Section 5.

2 DEFINITIONS AND

NOTATIONS

2.1 A Directed Digraph G

The peculiarity of the shortest path problem requires

to distinguish two directions between any two nodes.

In this case, the connection between two nodes x and

y can be deﬁned by the directed connection between

an original node for example x and a destination node

y.

Deﬁnition 1. A directed digraph G = (E

S

,E

~

A

) is de-

ﬁned by a set of nodes E

S

and a set of directed edges

E

~

A

, each edge (arc) is the connection between an

original node and a destination node.

If x and y are two nodes:

• the directed connection from x to y (denoted ~xy), if

it exists, is a directed connection (arc) of a graph

G.

• An arc ~xx: the directed connection from x to x is

known as a loop.

• A p-graph is a graph wherein there is never more

than p arcs ~xy between any two nodes.

• A Monograph is a graph wherein there is never

more than 1 arc ~xy between any two nodes and

there is never a loop.

2.2 A Valuation Structure

We assume that E the set of all possible valuations, is

a totally ordered set where ⊥ denotes its minimal ele-

ment and > its maximal element. In addition, we will

use a monotone binary operator ⊕. These elements

form a valuation structure deﬁned as follows

Deﬁnition 2. A valuation structure S of a graph G is

the triplet S = (E,⊕,) such as

• E is the set of possible valuations;

• is a total order on E;

• ⊕ is commutative, associative and monotone.

We deﬁne below a ﬁre and strictly monotone valua-

tion structure.

Deﬁnition 3. A valuation structure S is ﬁre if for each

pair of valuations α,β ∈ E, such as α β, there is a

maximum difference between β and α denoted β α.

An aggregation operator ⊕ is strictly monotonic if for

any α,β,γ in E such as α ≺ β and γ 6= >, we have

α ⊕ γ ≺ β ⊕ γ.

A valuation structure S is strictly monotonic if it has

an aggregation operator strictly monotonic.

In the remainder of this paper, we use only ﬁre and

strictly monotone valuation structures.

The ﬁre and strictly monotone valuation structures

satisfy the following two Lemmas, that has been

proved in (Cooper, 2004), (Lemma 7 and Theorem

38).

Lemma 1. Let S = (E,⊕,) a valuation structure

ﬁre and strictly monotone. Then for all α,β,γ ∈ E

such as γ β, we have (β γ) β and (α ⊕ γ) ⊕

(β γ) = α ⊕ β.

Lemma 2. Let S = (E,⊕,) a valuation structure

ﬁre and strictly monotone. Then for all α,β,γ ∈ E

such as γ β, we have (α ⊕ β) γ = α ⊕ (β γ).

Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms

307

Using both Lemmas (Lemma 1 and Lemma 2) pre-

sented above we can get Lemma 3:

Lemma 3. Let β ≺ α and γ ≺ α

α β ≺ α γ if and only if γ α ≺ β α.

Proof. (⇒ ) If we have α β ≺ α γ

then α β ⊕ (β α ⊕ γ α) ≺ α γ ⊕ (β

α ⊕ γ α)

then γ α ≺ β α

(⇐ ) If we have γ α ≺ β α

then γ α ⊕ (α β ⊕ α γ) ≺ β α ⊕ (α

β ⊕ α γ)

then α β ≺ α γ.

We deﬁne below a particular valuation structure,

widely used in practice, that we will note S

⊕

Deﬁnition 4. A valuation structure S

⊕

of a graph G

is the triplet S

⊕

= (E

⊕

,⊕,) such as:

• E

⊕

is the set of possible valuations such as for all

α,β,λ ∈ E

⊕

if α β then α β ⊕ λ;

• is a total order on E;

• ⊕ is commutative, associative and monotone.

3 SHORTEST PATH NOTION

3.1 Extended Shortest Path Problem

In the beginning of this paragraph we formally deﬁne

the shortest path between two nodes x and y of a graph

G.

For this way, we start by deﬁning the arc and path

valuations.

Deﬁnition 5. Let G = (E

S

,E

~

A

) a valued directed

graph. In each arc ~xy we associate a valuation func-

tion ϕ : E

S

×E

S

→ E such as ϕ(x,y) is the valuation of

~xy arc. A path between two nodes x and y is denoted

CH(x, y) from the node x to a node y.

For each path CH(x, y) we associate a valuation

Φ(CH(x, y)).

Φ(CH(x, y)) = [

M

~x

i

x

j

∈CH(x,y)

ϕ(x

i

,x

j

)]

Now we can deﬁne the shortest path:

Deﬁnition 6. Let G = (E

S

,E

~

A

) a valued directed

graph. The shortest path between x and y is the path

µ(x,y) started from a node x and ﬁnished at y such as:

Φ(µ(x,y)) = Min

CH(x,y)

[Φ(CH(x, y))]

3.2 Equivalent Problems of Shortest

Path Problem (SPP)

If we have to ﬁnd the Longest Path, is it possible to

use the Shortest Path solution?

Finding the Longest Path is it equivalent to ﬁnding the

Shortest Path?

Theorem 1. Given a ﬁre and strictly monotone valu-

ation structure S, ﬁnding the Longest Path Problem is

equivalent to ﬁnding the Shortest Path Problem.

Proof Theorem 1. (⇒) We start from a shortest path

problem. We replace each valuation (α β) by the

valuation (β α) and we prove that a shortest path

problem can be transformed in a longest path prob-

lem.

Let

Φ(µ(s,t)) = min Φ(CH(s,t))

Φ(µ(s,t)) = (α β

0

) Φ(CH

1

(s,t)) = (α β

1

)

. ..

Φ(CH

n

(s,t)) = (α β

n

)

If we replace each valuation (α β) by the valuation

(β α)

We get

Φ(µ(s,t)) = (β

0

α) Φ(CH

1

(s,t)) = (β

1

α)

. ..

Φ(CH

n

(s,t)) = (β

n

α)

then we get

Φ(µ(s,t)) = max Φ(CH(s,t)) = Φ(L (s,t))

(⇐) Now we start from a longest path problem. We

replace each valuation (α β) by the valuation (β

α) and we prove that a longest path problem can be

transformed in a shortest path problem.

Let

Φ(L (s,t)) = max Φ(CH(s,t))

Φ(L (s,t)) = (α β

0

) Φ(CH

1

(s,t)) = (α β

1

)

. ..

Φ(CH

n

(s,t)) = (α β

n

)

If we replace each valuation (α β) by the valuation

(β α)

We get by Lemma 3

Φ(L (s,t)) = (β

0

α) Φ(CH

1

(s,t)) = (β

1

α)

. ..

Φ(CH

n

(s,t)) = (β

n

α)

Then we get

Φ(L (s,t)) = min Φ(CH(s,t)) = Φ(µ(s,t))

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

308

Example 1. In February 2017, a large multinational

X in Porto wishes to invest the sum of 3.000.000 e

in a new project Π. To do this, each year X has d in-

vestment choices. One study allowed him to estimate

the certainty of acceptable proﬁtability (probabilities)

according to the various decisions taken. In order to

maximize the certainty of an acceptable overall yield

of Π, X wishes to ﬁnd the longest path in the mono-

graph G

1

, such that, S

1

= (]0,1[, ×,≤).

b d

c e

0.9

a f

0.7

0.5

0.1

0.98

0.2

0.97

0.99

0.5

0.6

0.95

Figure 1: A directed monograph G

1

.

By Theorem 1 and in order to maximize the cer-

tainty probabilities of an acceptable overall yield of

Π, X can ﬁnd the shortest path in the monograph

G

2

minimize the uncertainty probabilities. Such that

S

2

= S

1

. For each valuation α in G

1

we associate a

valuation β in G

2

such that β = α = 1 − α

α ⊕ β = α × (1 − α) ⇒ S

1

and S

2

are different

from the semiring or diod structures.

b d

c e

0.1

a f

0.3

0.5

0.9

0.02

0.8

0.03

0.01

0.5

0.4

0.05

Figure 2: A directed monograph G

2

.

3.3 Optimality Notion

The dynamic programming repose on the fundamen-

tal principle of optimality:

Given a directed graph G, a sub-path of a shortest path

µ ∈ G is a shortest path in G

s

a sub-graph of G.

Theorem 2. Let:

• A directed graph G = (E

S

,E

~

A

),

• A valuation function of G ϕ : E

S

× E

S

→ E with a

ﬁre and strictly monotone valuation structure S,

• a shortest path µ(x

1

,x

k

) from x

1

∈ E

S

to x

k

∈ E

S

µ(x

1

,x

k

) = x

1

→ x

2

→ .. . → x

k

= x

1

∗

→ x

k

,

• A sub-path CH(x

i

,x

j

) of µ(x

1

,x

k

) from x

i

to x

j

:

CH(x

i

,x

j

) = x

i

→ .. . → x

j

such as x

i

∗

→ x

j

1

.

Then CH(x

i

,x

j

) = µ(x

i

,x

j

) is the shortest path from

x

i

∗

→ x

j

.

Proof Theorem 2. We proceed by absurd reasoning.

Assume that:

1. µ(x

1

,x

k

) is a shortest path

2. CH(x

i

,x

j

) is a sub path of µ(x

1

,x

k

)

3. It ∃ CH’(x

i

,x

j

) of G such as Φ(CH’(x

i

,x

j

)) ≺

Φ(CH(x

i

,x

j

)).

We proceed to get a contradiction.

Decompose µ(x

1

,x

k

) in CH(x

1

,x

i

), CH(x

i

,x

j

) and

CH(x

j

,x

k

)

then

Φ(µ(x

1

,x

k

)) = Φ(CH(x

1

,x

i

)) ⊕ Φ(CH(x

i

,x

j

)) ⊕

Φ(CH(x

j

,x

k

)).

As it ∃ CH’(x

i

,x

j

) such as Φ(CH’(x

i

,x

j

)) ≺

Φ(CH(x

i

,x

j

)) and given that ⊕ is strictly monotone

then

Φ(µ(x

1

,x

k

)) Φ(CH(x

1

,x

i

)) ⊕ Φ(CH’(x

i

,x

j

)) ⊕

Φ(CH(x

j

,x

k

)), which is a contradiction by the

fact that the path µ(x

1

,x

k

) is a shortest path then

CH(x

i

,x

j

) = µ(x

i

,x

j

).

4 GENERALIZATION OF

DIJKSTRA-MOORE AND

BELLMAN-FORD

ALGORITHMS

4.1 Common Functions

The generalized Dijkstra-Moore and Bellman-Ford

algorithms presented in this paper use

1. Algorithm 1 is an initialization marker algorithm

of all nodes of G that we will denote

INITMARK(G,s);

2. Algorithm 2 is an initialization ﬁnding shortest

path algorithm that we will denote

INITSPP(G, s);

1

– x

i

is an immediate predecessor or a non immediate of x

j

(it

exists a path from x

i

to x

j

).

– x

i

is an immediate successor or a non immediate of x

1

or x

i

= x

1

– x

j

is an immediate predecessor or a non immediate of x

k

or

x

j

= x

k

Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms

309

3. The updating shortest path Algorithm, denoted

UPDATESPP(x

i

,x

j

,ϕ).

Updating the shortest path between two nodes x

i

and x

j

consists in updating the valuation of one

of the arcs ~x

i

x

j

:

(a) The valuation spv[x

j

] of the shortest path until

x

j

;

(b) The predecessor of x

j

predspp[x

j

] in the short-

est path until x

j

.

Algorithm 1: InitMark(G, s):Mark.

for (x

i

= s Nbr

Sommets

G

) do

Mark[x

i

] ← 0;

Mark[s] ← 2;

Algorithm 2: InitSpp(G, s):spv,predspp.

for (x

i

= s Nbr

Sommets

G

) do

spv[x

i

] ← >;

predspp[x

i

] ← 0;

spv[s] ← α

0

;

The updating process is based on the Theorem 2

where each sub-path of the shortest path is a shortest

path in the sub-graph involving this sub-path.

The UpdateSpp(x

i

,x

j

,ϕ) function update the shortest

path from one origin node to all other one if a shortest

path is detected.

Algorithm 3: UpdateSpp(x

i

,x

j

,ϕ):spv,predspp.

if (spv[x

j

] spv[x

i

] ⊕ ϕ[x

i

][x

j

]) then

spv[x

j

] ← spv[x

i

] ⊕ ϕ[x

i

][x

j

];

predspp[x

j

] ← x

i

;

In the DIJKSTRA-MOORE Algorithm 4, each arc

is updated exactly one way. In BELLMAN-FORD Al-

gorithm, each arc can be updated many way.

4.2 Generalization of the

Dijkstra-Moore Algorithm

If the arcs valuation, is in R, can model for example

• A distance (kilometers)

• A cost (e)

In this case, the classic Dijkstra-Moore Algorithm can

be used.

In this paper, we present a generalization of Dijkstra-

Moore Algorithm 4 for a graph G with a S

⊕

valuation

structure.

Let G a valued directed graph given a valuation struc-

ture S

⊕

. We denote by s the origin node of G and x

i

the destination node. For each node x

i

of G, the Al-

gorithm 4 associate

• The valuation spv[x

j

] for the shortest sub-path un-

til x

i

;

• The predecessor of x

i

predspp[x

i

] in the shortest

sub-path until x

i

.

• The marker of x

i

denoted Mark[x

i

] verifying if the

distance from s to x

i

has been updated.

Principe of the Algorithm:

1. initialization:

For the node s

• Mark[s] ← 2

• predspp[s] ← 0

• spv[s] ← α

0

for all other nodes x

i

• Mark[x

i

] ← 0

• predspp[x

i

] ← 0

• spv[x

i

] ← >

2. Let X = the set of non marked nodes;

Do

• For each non marked node i successor of y

– UpdateSpp(y,i, ϕ)

• Mark the node y if spv[y] = min

x∈X

pcc[x].

While X 6=

/

0

b d

c e

β

a f

λ

λ β

T

β

λ β

α

α

α

λ λ

λ

β

Figure 3: A directed monograph G.

Example 2. Let the directed monograph G presented

by Figure 3. And let a valuation structure S

⊕

(can be

6= to the semiring or diod structures) such that

• {α, β, λ, γ, >} ⊂ E

⊕

• α ≺ β ≺ λ

• β = α ⊕ α

• λ = α ⊕ β

• if γ λ ⊕ λ ⊕ λ ⊕ β then γ = >

1. We apply the principle of Generalized Dijkstra al-

gorithm on G: Figure 4

2. We present an algorithmic solution: Algorithm 4.

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

310

b d

c e

β

a f

λ

λ β

T

β

λ β

α

α

α

λ λ

λ

β

M|pred|pcc

a: 2|0|

b: 0|0|T

0|a|λ

c: 0|0|T

0|a|α

d: 0|0|T

0|a|λ β

e: 0|0|T

f: 0|0|T

b d

c e

β

a f

λ

λ β

T

β

λ β

α

α

α

λ λ

λ

β

M|pred|pcc

a: 2|0|

b: 0|0|T

0|a|λ

c: 0|0|T

0|a|α

2|a|α

d: 0|0|T

0|a|λ β

e: 0|0|T

f: 0|0|T

b d

c e

β

a f

λ

λ β

T

β

λ β

α

α

α

λ λ

λ

β

M|pred|pcc

a: 2|0|

b: 0|0|T

0|a|λ

0|c|α α

2|c|α α

c: 0|0|T

0|a|α

2|a|α

d: 0|0|T

0|a|λ β

e: 0|0|T

0|c|α λ λ

f: 0|0|T

b d

c e

β

a f

λ

λ β

T

β

λ β

α

α

α

λ λ

λ

β

M|pred|pcc

a: 2|0|

b: 0|0|T

0|a|λ

0|c|α α

2|c|α α

c: 0|0|T

0|a|α

2|a|α

d: 0|0|T

0|a|λ β

0|b| α α β

2|b| α α β

e: 0|0|T

0|c| α λ λ

f: 0|0|T

b d

c e

β

a f

λ

λ β

T

β

λ β

α

α

α

λ λ

λ

β

M|pred|pcc

a: 2|0|

b: 0|0|T

0|a|λ

0|c|α α

2|c|α α

c: 0|0|T

0|a|α

2|a|α

d: 0|0|T

0|a|λ β

0|b| α α β

2|b| α α β

e: 0|0|T

0|c|α λ λ

0|d| α α β β

2|d| α α β β

f: 0|0|T

0|d|α α β λ β

b d

c e

β

a f

λ

λ β

T

β

λ β

α

α

α

λ λ

λ

β

M|pred|pcc

a: 2|0|

b: 0|0|T

0|a|λ

0|c|α α

2|c|α α

c: 0|0|T

0|a|α

2|a|α

d: 0|0|T

0|a|λ β

0|b|α α β

2|b|α α β

e: 0|0|T

0|c|α λ λ

0|d|α α β β

2|d|α α β β

f: 0|0|T

0|d|α α β λ β

0|e|α α β β β

2|e|α α β β β

Figure 4: Applying Generalized-Dijkstra-Moore on G.

Algorithm 4: Generalized-Dijkstra-Moore(G, ϕ,s,t):

spv,predspp.

InitMark(G,s);

InitSpp(G,s);

recent ← s;

t ← f ;

while (Mark[t] = 0) do

j ← 0;

while (succ[recent][ j]) do

if (Mark[ j] = 0) then

UpdateSpp(recent, j, ϕ);

j ← j + 1;

y ← min(spv);

Mark[y] ← 2;

recent ← y;

Theorem 3. Given a directed monograph G with n

nodes and a valuation structure S

⊕

, the shortest path

from one started node to all others can be done in

O(n

2

).

Proof Theorem 3. Given a directed monograph G

with n nodes and a valuation structure S

⊕

, and re-

ferred to Theorem 2, the shortest path from one

started node to all others can be done by applying

Algorithm 4 to G. And the Algorithm 4 run in O(n

2

).

4.3 Generalization of Bellman-Ford

Algorithm

Given a ﬁre and strictly monotone valuation structure

S we can model as example earnings and bounded

costs!

Unfortunately, as nodes may be marked only once,

the DIJKSTRA-MOORE algorithm does not guarantee

the optimal solution if we consider the valuation

structure not a subset of S

⊕

(For example the bounded

negative arcs). In fact, once the node is marked we

cannot change the marking in subsequent iterations.

Fortunately, we can present an algorithm that ensures

marking update until the program is not determined:

Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms

311

b d

c e

β

a f

λ

λ β

T

β

λ β

α β

α

α

λ λ

λ

α λ

Figure 5: A directed monograph G

0

.

a generalization of the BELLMAN-FORD Algorithm

can be used for a ﬁre and strictly monotone valuation

structure S.

Theorem 4. Given a ﬁre and strictly monotone

valuation structure S, the ﬁnal values of the shortest

paths are obtained by at most n − 1 iterations.

Proof Theorem 4. In the absence of absorbing cir-

cuitry, a shortest path from s to all other nodes is an

element path, that is to say a path of at most n − 1

arcs. By consulting the predecessors of all nodes Al-

gorithm 5 must obtain the ﬁnal values of the shortest

paths by at most n − 1 iterations.

Corollary 1. If after n iterations, the values spv[i]

continue to be modiﬁed, is that the graph has an

absorbent circuitry.

Based on the results of Theorem 4 and Corollary 1 we

can introduce the principle of the BELLMAN-FORD

generalization algorithm.

Principle of the BELLMAN-FORD generalization

algorithm:

1. InitSpp(G, s)

2. Do

UpdateSpp(i, j,ϕ)

While there is an edge to decrease spv[i].

Algorithm 5 presents an algorithmic solution for the

generalized BELLMAN-FORD algorithm.

Example 3. Let the directed monograph G

0

given by

Figure 5. And let a valuation structure S (can be 6= to

the semiring or diod structures) such that

• {α, β, λ, γ, >} ⊂ E

• α ≺ β ≺ λ

• β = α ⊕ α

• λ = α ⊕ β

• if γ λ ⊕ λ ⊕ λ ⊕ β then γ = >

1. Present an algorithmic solution: Algorithm 5.

Algorithm 5: Generalized-Bellman-Ford(G,ϕ, s):

spv,predspp.

InitSpp(G,s);

k ← 0;

t ← 1;

while (t ou k > N

Sommet

G

− 1) do

t ← 0;

for (i de 1 N

Sommet

G

) do

j ← 0;

while (predspp[i][ j]) do

UpdateSpp( j,i,ϕ);

if (spv[i] spv[ j] ⊕ ϕ[ j][i]) then

t ← 1;

j ← j + 1;

k ← k + 1;

Theorem 5. Given a directed monograph G with n

nodes and a ﬁre and strictly valuation structure S, the

shortest path from one started node to all others can

be done in O(n

3

).

Proof Theorem 5. Given a directed monograph G

with n nodes and a ﬁre and strictly valuation structure

S, and referred to Theorem 2, Theorem 4 and Corol-

lary 1, the shortest path from one started node to all

others can be done by applying Algorithm 5 to G. And

the Algorithm 5 run in O(n

3

).

5 CONCLUSION

This paper addressed combinatorial problems that

can be expressed as shortest path solution but use a

set of valuation not a subset of R and/or a combining

operator not equal to the classic sum (+).

Firstly, we have modeled the valuations of a graph G

by using a general valuation structure S.

Secondly, given a general valuation structure S, we

have discussed the equivalence between longest path

and shortest path problem.

And ﬁnally, we have discussed the generalization

of the shortest path algorithms according to the

properties of the graph G and the valuation structure

S:

1. The valuation structure of G is S

⊕

.

2. The graph G and the valuation structure S are ar-

bitrary.

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

312

REFERENCES

Allouche, D., de Givry, S., Sanchez, M., and Schiex, T.

(2009). Tagsnp selection using weighted csp and rus-

sian doll search with tree decomposition 1. In In Pro-

ceedings of the WCB09. Lisbon Portugal, 1–8.

Bellman, R. (1958). On a routing problem. In Quarterly of

Applied Mathematics. 16: 87–90.

Cohen, D. A., Cooper, M. C., Jeavons, P. G., and Krokhin,

A. A. (2004). A maximal tractable class of soft con-

straint satisfaction. In Journal of Artiﬁcial Intelligence

Research 22, 1–22.

Cooper, M. C. (2003). Reduction operations in fuzzy or val-

ued constraint satisfaction. In Fuzzy Sets and Systems

134 311–342.

Cooper, M. C. (2004). Arc consistency for soft constraints.

In Artiﬁcial Intelligence 154, 199–227.

Erickson, J. (2010). Maximum ﬂows and parametric short-

est paths in planar graphs. In In Proceedings of the

21st Annual ACM-SIAM Symposium on Discrete Al-

gorithms, 794–804.

Ford, J. and Lester, R. (1956). Network ﬂow theory. In

RAND Corporation, 923, Santa Monica, California.

Ford, L. R. and Fulkerson, D. R. (1955). A simple algorithm

for ﬁnding maximal network ﬂows and an application

to the hitchock problem. In In Rand Report Rand Cor-

poration, Santa Monica, California.

Ford, L. R. and Fulkerson, D. R. (1962). Flows in networks.

In In Princeton University Press.

Gondran, M. and Minoux, M. (2008). Graphs, dioids

and semirings: New models and algorithms (opera-

tions research/computer science interfaces series). In

Springer Publishing Company.

Helaoui, M., Naanaa, W., and Ayeb, B. (2013).

Submodularity-based decomposing for valued csp. In

International Journal on Artiﬁcial Intelligence Tools.

Schiex, T., Fargier, H., and Verfaillie, G. (1995). Valued

constraint satisfaction problems: hard and easy prob-

lems. In In Proceedings of the 14

th

IJCAI, 631–637,

Montr

´

eal, Canada.

Sedgewick, R. and Wayne, K. D. (2011). Algorithms (4th

ed.). In Addison-Wesley Professional, 661–666.

Shimbel, A. (1955). Structure in communication nets. In

Proceedings of the Symposium on Information Net-

works, 199–203, New York, NY: Polytechnic Press of

the Polytechnic Institute of Brooklyn.

Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms

313