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 maximum ﬂow problem (Ford and Fulkerson, 1955; Ford and Fulkerson, 1962) in any directed planar The shortest path problem is one of the classic prob- graph G can be reformulated as a parametric short- lems in graph theory. The problem is to provide a so- est path problem in the oriented dual graph G∗. In lution algorithm returning the optimum route, taking (Cohen et al., 2004; Helaoui et al., 2013), a submod- into account a valuation function, between two nodes ular decompositions approach has been presented to of a graph G. solve Valued Constraint Satisfaction Problem. This In (Shimbel, 1955; Ford and Lester, 1956; Bellman, solution use the maximum ﬂow algorithm. 1958; Sedgewick and Wayne, 2011) the classic short- Dijkstra-Moore and Bellman-Ford Algorithms are the est path solution is proved if most known algorithmic solutions for the shortest path problem. the set of valuation is R or a subset of R. • Since 1971, the Dijkstra-Moore Algorithm has the combining operator is the classic sum (+) • • been used if the set of valuation is R+ or a subset Many combinatorial problems like Fuzzy, Weighted, + of R and the combining operator is the classic Probabilistic and Valued Constraint Satisfaction Prob- sum (+). lem (Schiex et al., 1995; Cooper, 2003; Cooper, 2004; Allouche et al., 2009) use a set of valuation The Bellman-Ford Algorithm is the result of • (Shimbel, 1955; Ford and Lester, 1956; Bellman, E not subset of R and a combining operator = + for weighted, fuzzy, probabilistic . . . valuations.⊕ 6 In 1958) works. It is used if the set of valuation is R (Cooper, 2003), the shortest path algorithm has been or a subset of R and the combining operator is the used to solve Fuzzy and Valued Constraint Satisfac- classic sum (+). tion Problem. As many combinatorial problems can be solved by us- In (Erickson, 2010), author observes that the classical ing shortest path solution but use a set of valuation

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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

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

not a subset of R and/or a combining operator not E~A, each edge (arc) is the connection between an equal to the classic sum (+), then in (Gondran and original node and a destination node. Minoux, 2008), authors present new models and al- If x and y are two nodes: gorithms discussing relations between particular val- the directed connection from x to y (denoted xy),~ if uation structure: the semiring and diod structures with • it exists, is a directed connection (arc) of a graph graphs and their combinatorial properties. G. In (Sedgewick and Wayne, 2011), a longest path be- An arc xx:~ the directed connection from x to x is tween two given nodes s and t in a weighted graph • known as a loop. G is the same thing as a shortest path in a graph G − A p-graph is a graph wherein there is never more derived from G by changing every weight to its nega- • tion. Therefore, if shortest paths can be found in G, than p arcs xy~ between any two nodes. then longest paths can also be found in G. This result− A Monograph is a graph wherein there is never • remains true if we have a valued graph G by a valua- more than 1 arc xy~ between any two nodes and tion structure S? there is never a loop. In this paper, we provide an answer to this question by discussing equivalence between longest path and 2.2 A Valuation Structure shortest path problem given a valuation structure S. We present a generalization of Dijkstra-Moore Algo- We assume that E the set of all possible valuations, is rithm for a graph G with a S⊕ valuation structure. And a totally ordered set where denotes its minimal ele- we present a generalization of Bellman-Ford Algo- ment and its maximal element.⊥ In addition, we will rithm with a more general valuation structure S. use a monotone> binary operator . These elements We propose to model both the valuations of a graph G form a valuation structure defined⊕ as follows and the combining operator by a valuation structure Definition 2. A valuation structure S of a graph G is S, in order to discuss the generalization of the short- the triplet S = (E, , ) such as est path algorithms according to the properties of the ⊕ E is the set of possible valuations; graph G and the valuation structure S: • is a total order on E; The valuation structure of G is S⊕. • • is commutative, associative and monotone. The graph G and the valuation structure S are ar- •⊕ • bitrary. We define below a fire and strictly monotone valuation structure. The paper is organized as follows: the next Section introduces definitions and notations needed in pre- Definition 3. A valuation structure S is fire if for each , senting the generalization of the shortest path algo- pair of valuations α β E, such as α β, there is a maximum difference between∈ β and α denoted β α. rithms. In Section 3 we study the extended Shortest Path Notion and the equivalence between longest path An aggregation operator is strictly monotonic if for , , ⊕ = and shortest path problem. We propose a generalized any α β γ in E such as α β and γ , we have α γ β γ. ≺ 6 > shortest path algorithms in Section 4. The paper is ⊕ ≺ ⊕ concluded in Section 5. 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. 2 DEFINITIONS AND The fire and strictly monotone valuation structures NOTATIONS satisfy the following two Lemmas, that has been proved in (Cooper, 2004), (Lemma 7 and Theorem 2.1 A Directed Digraph G 38). Lemma 1. Let S = (E, , ) a valuation structure The peculiarity of the shortest path problem requires fire and strictly monotone.⊕ Then for all α,β,γ E to distinguish two directions between any two nodes. such as γ β, we have (β γ) β and (α γ∈) In this case, the connection between two nodes x and (β γ) =α β. ⊕ ⊕ ⊕ y can be defined by the directed connection between Lemma 2. Let S = (E, , ) a valuation structure an original node for example x and a destination node fire and strictly monotone.⊕ Then for all α,β,γ E y. such as γ β, we have (α β) γ = α (β ∈γ). ⊕ ⊕ Definition 1. A directed digraph G = (ES,E~A) is defined by a set of nodes ES and a set of directed edges

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Using both Lemmas (Lemma 1 and Lemma 2) pre- 3.2 Equivalent Problems of Shortest sented above we can get Lemma 3: Path Problem (SPP) Lemma 3. Let β α and γ α α β α γ if≺ and only if ≺γ α β α. 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 Proof. ( ) If we have α β α γ Shortest Path? then α ⇒β (β α γ α≺) α γ (β α γ α)⊕ ⊕ ≺ ⊕ Theorem 1. Given a fire and strictly monotone valu- then⊕ γ α β α ation structure S, finding the Longest Path Problem is ( ) If we have≺ γ α β α equivalent to finding the Shortest Path Problem. ⇐ ≺ then γ α (α β α γ) β α (α Proof Theorem 1. ( ) We start from a shortest path ⊕ ⊕ ≺ ⊕ β α γ) problem. We replace⇒ each valuation (α β) by the ⊕ then α β α γ. valuation (β α) and we prove that a shortest path ≺ problem can be transformed in a longest path prob- We define below a particular valuation structure, lem. widely used in practice, that we will note S⊕ Let Φ(µ(s,t)) = minΦ(CH(s,t)) Definition 4. A valuation structure S⊕ of a graph G Φ(µ(s,t)) = (α β0) Φ(CH1(s,t)) = (α β1) is the triplet S⊕ = (E⊕, , ) such as: ⊕ ... E⊕ is the set of possible valuations such as for all • Φ(CHn(s,t)) = (α βn) α,β,λ E⊕ if α β then α β λ; ∈ ⊕ If we replace each valuation (α β) by the valuation is a total order on E; • (β α) is commutative, associative and monotone. We get •⊕ Φ(µ(s,t)) = (β α) Φ(CH (s,t)) = (β α) 0 1 1 ... 3 SHORTEST PATH NOTION Φ(CH (s,t)) = (β α) n n 3.1 Extended Shortest Path Problem then we get Φ(µ(s,t)) = maxΦ(CH(s,t)) = Φ(L(s,t)) In the beginning of this paragraph we formally define the shortest path between two nodes x and y of a graph ( ) Now we start from a longest path problem. We replace⇐ each valuation (α β) by the valuation (β G. For this way, we start by defining the arc and path α) and we prove that a longest path problem can be valuations. transformed in a shortest path problem. Let Definition 5. Let G = (ES,E~A) a valued directed Φ(L(s,t)) = maxΦ(CH(s,t)) graph. In each arc xy~ we associate a valuation func- Φ(L(s,t)) = (α β0) Φ(CH1(s,t)) = (α β1) tion ϕ : E E E such as ϕ(x,y) is the valuation of S × S → ... xy~ arc. A path between two nodes x and y is denoted ( , ) Φ(CHn(s,t)) = (α βn) CH x y from the node x to a node y. For each path CH(x,y) we associate a valuation If we replace each valuation (α β) by the valuation Φ(CH(x,y)). (β α) We get by Lemma 3 Φ(CH(x,y)) = [ ϕ(xi,x j)] x~ix j CH(x,y) Φ(L(s,t)) = (β α) Φ(CH (s,t)) = (β α) ∈M 0 1 1 ... Now we can define the shortest path: Φ(CHn(s,t)) = (βn α) Definition 6. Let G = (E ,E ) a valued directed S ~A Then we get 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(L(s,t)) = minΦ(CH(s,t)) = Φ(µ(s,t))

Φ(µ(x,y)) = MinCH(x,y)[Φ(CH(x,y))]

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

Example 1. In February 2017, a large multinational A sub-path CH(xi,x j) of µ(x1,xk) from xi to x j: e • X in Porto wishes to invest the sum of 3.000.000 CH(x ,x ) = x ... x such as x ∗ x 1. in a new project Π. To do this, each year X has d in- i j i → → j i → j vestment choices. One study allowed him to estimate Then CH(xi,x j) = µ(xi,x j) is the shortest path from ∗ the certainty of acceptable profitability (probabilities) xi x j. according to the various decisions taken. In order to → Proof Theorem 2. We proceed by absurd reasoning. maximize the certainty of an acceptable overall yield Assume that: of Π, X wishes to find the longest path in the monograph G , such that, S = (]0,1[, , ). 1. µ(x1,xk) is a shortest path 1 1 × ≤ 2.CH (xi,x j) is a sub path of µ(x1,xk) 0.9 3. It CH’(xi,x j) of G such as Φ(CH’(xi,x j)) 0.7 b d ∃ ≺ 0.2 Φ(CH(x ,x )). 0.99 0.5 i j 0.95 a 0.6 0.1 f We proceed to get a contradiction. Decompose µ(x1,xk) in CH(x1,xi), CH(xi,x j) and 0.98 CH(x ,x ) 0.97 c e j k then 0.5 Φ(µ(x ,x )) = Φ(CH(x ,x )) Φ(CH(x ,x )) 1 k 1 i ⊕ i j ⊕ Figure 1: A directed monograph G1. Φ(CH(x j,xk)). As it CH’(x ,x ) such as Φ(CH’(x ,x )) By Theorem 1 and in order to maximize the cer- ∃ i j i j ≺ Φ(CH(xi,x j)) and given that is strictly monotone tainty probabilities of an acceptable overall yield of then ⊕ Π, X can find the shortest path in the monograph Φ(µ(x1,xk)) Φ(CH(x1,xi)) Φ(CH’(xi,x j)) G2 minimize the uncertainty probabilities. Such that ⊕ ⊕ Φ(CH(x j,xk)), which is a contradiction by the S2 = S1. For each valuation α in G1 we associate a fact that the path µ(x1,xk) is a shortest path then valuation β in G2 such that β = α = 1 α − CH(xi,x j) = µ(xi,x j). α β = α (1 α) S and S are different ⊕ × − ⇒ 1 2 from the semiring or diod structures.

0.1 0.3 b 0.8 d 4 GENERALIZATION OF 0.01 0.5 0.05 DIJKSTRA-MOORE AND a 0.4 0.9 f BELLMAN-FORD 0.02 0.03 c e ALGORITHMS

0.5

Figure 2: A directed monograph G2. 4.1 Common Functions

The generalized Dijkstra-Moore and Bellman-Ford 3.3 Optimality Notion algorithms presented in this paper use The dynamic programming repose on the fundamen- 1. Algorithm 1 is an initialization marker algorithm tal principle of optimality: of all nodes of G that we will denote Given a directed graph G, a sub-path of a shortest path INITMARK(G,s); µ G is a shortest path in Gs a sub-graph of G. 2. Algorithm 2 is an initialization ﬁnding shortest ∈ Theorem 2. Let: path algorithm that we will denote INITSPP(G,s); A directed graph G = (E ,E ), • S ~A A valuation function of G ϕ : E E E with a • S × S → ﬁre and strictly monotone valuation structure S, 1 x is an immediate predecessor or a non immediate of x (it a shortest path µ(x1,xk) from x1 ES to xk ES – i j • ∈ ∈ exists a path from x to x ). ∗ i j µ(x1,xk) = x1 x2 ... xk = x1 xk, → → → → – xi is an immediate successor or a non immediate of x1 or xi = x1 – x j is an immediate predecessor or a non immediate of xk or x j = xk

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3. The updating shortest path Algorithm, denoted Let G a valued directed graph given a valuation struc- UPDATESPP(xi,x j,ϕ). ture S⊕. We denote by s the origin node of G and xi Updating the shortest path between two nodes xi the destination node. For each node xi of G, the Al- and x j consists in updating the valuation of one gorithm 4 associate x~x of the arcs i j: The valuation spv[x ] for the shortest sub-path un- • j til xi; (a) The valuation spv[x j] of the shortest path until The predecessor of xi predspp[xi] in the shortest x j; • sub-path until xi. (b) The predecessor of x j predspp[x j] in the short- The marker of xi denoted Mark[xi] verifying if the est path until x j. • distance from s to xi has been updated. Algorithm 1: InitMark(G,s):Mark. Principe of the Algorithm: for (xi = s NbrSommetsG ) do Mark[x ] 0; 1. initialization: i ← Mark[s] 2; For the node s ← Mark[s] 2 • ← predspp[s] 0 Algorithm 2: InitSpp(G,s):spv,predspp. • ← spv[s] α0 for (x = s Nbr ) do • ← i SommetsG x spv[x ] ; for all other nodes i i ← > predspp[x ] 0; Mark[xi] 0 i ← • ← predspp[x ] 0 spv[s] α0; • i ← ← spv[x ] • i ← > The updating process is based on the Theorem 2 2. Let X = the set of non marked nodes; where each sub-path of the shortest path is a shortest Do path in the sub-graph involving this sub-path. For each non marked node i successor of y The UpdateSpp(xi,x j,ϕ) function update the shortest • path from one origin node to all other one if a shortest – UpdateSpp(y,i,ϕ) path is detected. Mark the node y if spv[y] = minx X pcc[x]. • ∈ While X = 0/ 6 Algorithm 3: UpdateSpp(xi,x j,ϕ):spv,predspp. β

if (spv[x j] spv[xi] ϕ[xi][x j]) then λ b λ β d ⊕ λ β spv[x ] spv[x ] ϕ[x ][x ]; α λ j ← i ⊕ i j predspp[x j] xi; a T f ← α β β α c e In the DIJKSTRA-MOORE Algorithm 4, each arc is updated exactly one way. In BELLMAN-FORD Al- λ λ gorithm, each arc can be updated many way. Figure 3: A directed monograph G.

4.2 Generalization of the Example 2. Let the directed monograph G presented by Figure 3. And let a valuation structure S⊕ (can be Dijkstra-Moore Algorithm = to the semiring or diod structures) such that 6 α, β, λ, γ, E If the arcs valuation, is in R, can model for example •{ >} ⊂ ⊕ α β λ A distance (kilometers) • ≺ ≺ • β = α α A cost (e) • ⊕ • λ = α β • ⊕ In this case, the classic Dijkstra-Moore Algorithm can if γ λ λ λ β then γ = be used. • ⊕ ⊕ ⊕ > In this paper, we present a generalization of Dijkstra- 1. We apply the principle of Generalized Dijkstra algorithm on G: Figure 4 Moore Algorithm 4 for a graph G with a S⊕ valuation structure. 2. We present an algorithmic solution: Algorithm 4.

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

b: 0|0|T d: 0|0|T b: 0|0|T d: 0|0|T 0|a|λ 0|a|λ β 0|a|λ 0|a|λ β β β

λ b λ β d f: 0|0|T λ b λ β d f: 0|0|T

α λ λ β α λ λ β

a T f a T f

c: 0|0|T c: 0|0|T λ λ e: 0|0|T 0|a|α λ λ e: 0|0|T 0|a|α 2|a|α

b: 0|0|T b: 0|0|T d: 0|0|T 0|a|λ d: 0|0|T 0|a|λ 0|a|λ β 0|c|α α 0|a|λ β 0|c|α α 0|b| α α β 2|c|α α β 2|c|α α β 2|b| α α β f: 0|0|T f: 0|0|T λ b λ β d λ b λ β d λ β λ β α λ α λ

a T f a T f

c: 0|0|T c: 0|0|T e: 0|0|T e: 0|0|T 0|a|α λ λ 0|a|α λ λ 0|c|α λ λ 0|c| α λ λ 2|a|α 2|a|α

d: 0|0|T d: 0|0|T b: 0|0|T 0|a|λ β b: 0|0|T 0|a|λ β 0|a|λ 0|b| α α β 0|a|λ 0|b|α α β f: 0|0|T 0|c|α α 2|b| α α β 0|c|α α 2|b|α α β 0|d|α α β λ β 2|c|α α β f: 0|0|T 2|c|α α β 0|d|α α β λ β 0|e|α α β β β 2|e|α α β β β λ b λ β d λ b λ β d

α λ λ β α λ λ β a T f a T f

c: 0|0|T e: 0|0|T c: 0|0|T e: 0|0|T λ λ λ λ 0|a|α 0|c|α λ λ 0|a|α 0|c|α λ λ

2|a|α 0|d| α α β β 2|a|α 0|d|α α β β 2|d| α α β β 2|d|α α β β Figure 4: Applying Generalized-Dijkstra-Moore on G.

Algorithm 4: Generalized-Dijkstra-Moore(G,ϕ,s,t): Proof Theorem 3. Given a directed monograph G spv,predspp. with n nodes and a valuation structure S⊕, and referred to Theorem 2, the shortest path from one InitMark(G,s); started node to all others can be done by applying InitSpp(G,s); Algorithm 4 to G. And the Algorithm 4 run in O(n2). recent s; t f←; ← while (Mark[t] = 0) do j 0; 4.3 Generalization of Bellman-Ford ← while (succ[recent][ j]) do Algorithm if (Mark[ j] = 0) then UpdateSpp(recent, j,ϕ); Given a ﬁre and strictly monotone valuation structure S we can model as example earnings and bounded j j + 1; ← costs! y min(spv); Unfortunately, as nodes may be marked only once, ← Mark[y] 2; the DIJKSTRA-MOORE algorithm does not guarantee ← recent y; the optimal solution if we consider the valuation ← structure not a subset of S⊕ (For example the bounded Theorem 3. Given a directed monograph G with n negative arcs). In fact, once the node is marked we nodes and a valuation structure S⊕, the shortest path cannot change the marking in subsequent iterations. from one started node to all others can be done in Fortunately, we can present an algorithm that ensures O(n2). marking update until the program is not determined:

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β Algorithm 5: Generalized-Bellman-Ford(G,ϕ,s): λ b λ β d spv,predspp. α λ λ β InitSpp(G,s); a T f α β α λ k 0; β t ← 1; α c e ← while (t ou k > N 1) do SommetG − λ λ t 0; ← Figure 5: A directed monograph G0 . for (i de 1 NSommetG ) do j 0; ← a generalization of the BELLMAN-FORD Algorithm while (predspp[i][ j]) do can be used for a fire and strictly monotone valuation UpdateSpp( j,i,ϕ); structure S. if (spv[i] spv[ j] ϕ[ j][i]) then t 1; ⊕ ← j j + 1; Theorem 4. Given a fire and strictly monotone ← valuation structure S, the final values of the shortest k k + 1; paths are obtained by at most n 1 iterations. ← −

Proof Theorem 4. In the absence of absorbing circuitry, a shortest path from s to all other nodes is an Theorem 5. Given a directed monograph G with n element path, that is to say a path of at most n 1 nodes and a fire and strictly valuation structure S, the arcs. By consulting the predecessors of all nodes− Al- shortest path from one started node to all others can 3 gorithm 5 must obtain the final values of the shortest be done in O(n ). paths by at most n 1 iterations. Given a directed monograph G − Proof Theorem 5. with n nodes and a fire and strictly valuation structure Corollary 1. If after n iterations, the values spv[i] S, and referred to Theorem 2, Theorem 4 and Corol- continue to be modified, is that the graph has an lary 1, the shortest path from one started node to all absorbent circuitry. others can be done by applying Algorithm 5 to G. And the Algorithm 5 run in O(n3). Based on the results of Theorem 4 and Corollary 1 we can introduce the principle of the BELLMAN-FORD generalization algorithm. 5 CONCLUSION Principle of the BELLMAN-FORD generalization algorithm: This paper addressed combinatorial problems that can be expressed as shortest path solution but use a 1. InitSpp(G,s) set of valuation not a subset of R and/or a combining 2. Do operator not equal to the classic sum (+). UpdateSpp(i, j,ϕ) Firstly, we have modeled the valuations of a graph G While there is an edge to decrease spv[i]. by using a general valuation structure S. Algorithm 5 presents an algorithmic solution for the Secondly, given a general valuation structure S, we generalized BELLMAN-FORD algorithm. have discussed the equivalence between longest path and shortest path problem. Example 3. Let the directed monograph G0 given by And finally, we have discussed the generalization Figure 5. And let a valuation structure S (can be = to 6 of the shortest path algorithms according to the the semiring or diod structures) such that properties of the graph G and the valuation structure α, β, λ, γ, E S: •{ >} ⊂ α β λ • ≺ ≺ β = α α 1. The valuation structure of G is S⊕. • ⊕ λ = α β 2. The graph G and the valuation structure S are ar- • ⊕ if γ λ λ λ β then γ = bitrary. • ⊕ ⊕ ⊕ > 1. Present an algorithmic solution: Algorithm 5.

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