- Email: [email protected]
Detecting flows congesting a target network link
Electronic Notes in Discrete Mathematics 19 (2005) 233–239 www.elsevier.com/locate/endm
Detecting flows congesting a target network link D. Barth a , P. Berthom´e b , M. Diallo c a
b
c
Laboratoire PRiSM, UMR 8144, CNRS, Universit´e de Versailles, 45 Av. des Etats-Unis, 78035 Versailles-CEDEX, FRANCE
Laboratoire de Recherche en Informatique (LRI), UMR 8623, CNRS, Universit´e Paris-Sud, 91405, Orsay-CEDEX, FRANCE
CNRS, LIMOS UMR 6158, Universit´e Clermont 2, ISIMA, Campus des C´ezeaux - BP 10125, 63173 Aubi`ere CEDEX, FRANCE, [email protected]
Abstract In this paper, we deal with the analysis of network links saturation. Given a network, we target one of its links and provide an interesting analysis that allows to detect all vertex pairs for which any maximum flow always saturates the targeted link. The whole analysis complexity remains around O(n) maximum–flows/minimum– cuts computations using Gomory and Hu cut-trees. Keywords: link saturation, network congestion, multi-terminal flows, network analysis, parametric capacity
1
Introduction
In the current telecommunications and transports networks, congestion is one of the problems that merit great attention of the operators. From a prevention
1571-0653/2005 Published by Elsevier B.V. doi:10.1016/j.endm.2005.05.032
234
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
point of view, we are interested in deeply analyzing the saturation (congestion) of network links. When a flow goes from its source to its sink, the saturated links are bottlenecks for this flow. In some practical problems, determining bottlenecks is relevant. For instance, the building evacuation problem [5] is a particular case of the dynamic flow problem. Given the minimum evacuation time, this problem includes the detection of all the bottlenecks that may cause delay. Such bottlenecks correspond exactly to links belonging to minimum cuts (saturated by maximum flows). Thus, our objective in this paper is, given a link e in an undirected network, to show a way to detect the set of all vertex pairs for which any maximum flow always saturates e. This problem will be referred as ASF problem, where ASF stands for all pairs Always Saturating Flows. The searched set, with respect to link e, will be denoted ASF [e]. Such a set may help decision makers to know, for instance, in telecommunications, the communications that would take benefit from a Quality of Service improvement on the link e or that would surely suffer from a failure on e. Note that in both situations flows that uses e but do not saturate it will also be concerned with profit or damage on e. In this first work, we are not interested in such flows. For more applications, we refer to [2]. The remainder of the paper is devoted to the novel analysis we provide in order to solve the ASF problem. Conclusion and perspectives end the paper.
2
The ASF Problem
Let G = (V, E) be a network with a vertex set V and a link set E such that n = |V | and m = |E|. Each link e ∈ E has a capacity c(e) > 0. A flow f between the source s ∈ V and the sink t ∈ V is as defined by Ford and Fulkerson [3]. But this definition is given for directed networks. For undirected ones, we should consider the equivalent symmetric directed network (as detailed in [2]). In this paper, we only deal with undirected network and will note: →
f s,t : maximum flow between the source s and the sink t ((s-t)-maximum flow); →
fs,t : the value of f s,t ; →
→
→
→
f s,t (e): the flow on link e induced by f s,t , and | f s,t (e)|: the value of f s,t (e).
Furthermore, since our networks are symmetric, we have fs,t = ft,s .
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
235
A proper subset X of V (∅ X V ) is called a cut separating two vertices s and t if s ∈ X and t ∈ V \ X (or vice versa). The capacity of a cut X is c[x, y]. A minimum cut separating vertices defined as c(X) = x∈X,y∈V \X,[x,y]∈E
s and t, denoted hereafter Cs,t or (s-t)-minimum cut, is a cut with minimal capacity among all the cuts separating s and t. A link l belongs to a cut X if X separates its extremities. Now, we can formally describe the targeted set as follows. Let e be a link of G. The set ASF [e] for e is the set of vertex pairs {s, t} for which any maximum →
flow f s,t saturates e. Thus, → → ASF[e] = {s, t} ∈ V 2 | ∀ f st , | f st (e)| = c(e) .
2.1 On ASF [e] approximations In order to determine ASF [e], we need to check all maximum flows between all vertex pairs. This is a complex task if one does not use the concept of cut-tree introduced in 1961 by Gomory and Hu [4]. Given a network G = (V, E), a (Gomory-Hu) cut-tree T obtained from G is a weighted tree with vertex set V such that: — ∀ {s, t} ∈ G, fs,t in G equals to fs,t in T , i.e., the minimum weight on the unique path between s and t in T ; — A minimum cut Cs,t in T is also a (s-t)-minimum cut in G. Thus, with only (n-1) minimum–cut/maximum–flow computations, a cuttree provides all the n(n−1) maximum flow values and a minimum cut for each 2 vertex pair [4]. We notice that cut-trees are generally not unique. In the following, our results illustrate the use of cut-trees to solve the ASF problem. Let e = [i, j] ∈ E be the target link, and T a cut-tree of G. With respect to i and j, let Pi,j be the unique path with end points i and j in T . From the definition of a cut-tree, the removal of any link of Pi,j separates i and j in T , and thus, creates a vertex partition corresponding to a minimum cut in G that contains e. Lemma 2.1 states the Pi,j and ASF [e] relationship. Lemma 2.1 Let G = (V, E) be a network and e = [i, j] ∈ E. Consider a cut-tree T of G and the path Pi,j . Then, Pi,j × Pi,j ⊂ ASF [e]. Proof. This proof is illustrated in Figure 1. Consider any two vertices p and q in Pi,j . Let [x, y] be a link labeled with the minimum weight between p
236
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
and q. By definition of cut-tree, removing [x, y] in T creates a (p-q)-minimum ¯ in G with p ∈ X and q ∈ X. ¯ Thus, i belonging to X and j to cut (X, X) ¯ ¯ X implies e = [i, j] ∈ (X, X). By the Max-Flow/Min-Cut theorem [3], e is saturated by any (s-t)-maximum flow. Thus, {s, t} ∈ ASF [e]. 2 s Ps,t i
X
j
p
Pi,j
x
q
y
¯ X
t
Fig. 1. Illustration of Lemma 2.1 and Lemma 2.2.
Lemma 2.2 Extension of Lemma 2.1. Let e = [i, j] ∈ E be a link of G = (V, E), T a cut-tree of G and Pi,j as before. For any {s, t} in G, if a link of minimum weight in Ps,t in T belongs also to Pi,j , then {s, t} belongs to ASF [e]. Proof. Assume that a link [x, y] with the minimum weight on Ps,t lies on Ps,t ∩ Pi,j as shown in Figure 1. Removing [x, y] in T gives a (s-t)-minimum cut (X) in G that contains e = [i, j]. From Lemma 2.1, {s, t} ∈ ASF [e]. 2 The above results are not sufficient to get ASF [e]. For instance, in Figure 2, the top simple weighted path of length 3 is an arbitrary cut-tree that may be obtained from both the networks it points out: either from (T ) being the path itself or from the cycle (S). T
1
2
2
2
3
2
4 1
1
2
2
2
3
2
4
4
S
1 T
1
2
1
1
3
Fig. 2. Approximations of ASF [e]
In both S and T , let e = [3, 4] be the target link. As for ASF [e], we have: (i) in T , ASF [e] = {(1, 4), (2, 4), (3, 4)}, and (ii) in S, ASF [e] = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. But, if only the information of Lemma 2.2 is used, for both networks (S and T ), one gets the same set {{1, 4}, {2, 4}, {3, 4}}. This set corresponds exactly to ASF [e] in T while it is different from the ASF [e] in S.
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
237
In seeking to get set ASF [e] in the network S, we arise to Lemma 2.3. Lemma 2.3 Let G = (V, E) be a network, e = [i, j] ∈ E and T a cut-tree of G. Let w be the minimum weight on Pi,j in T . If all edges of T labeled with a weight strictly less than w are deleted, then a subtree K of T containing i and j is created and ASF [e] ⊆ V (K) × V (K); V (K) is the vertex set in K. Proof. Assume ∃{s, t} ∈ ASF [e] such that {s, t} ∈ / V (K) × V (K). Then, a (s-t)-minimum cut will be such that i and j belongs to its same component. →
Thus, link e is in no (s-t)-minimum cut. Then, there exists a f s,t such that →
| f s,t (e)| < c(e) : contradiction with hypothesis and definition of ASF [e].
2
Lemma 2.2 and 2.3 give bounds for ASF [e]. However, it can be shown that they do not always describe the requested set ASF [e] (illustration skipped for the sake of space but is in [2]). Until now, we showed that, given a network G and a target link e of G, a single cut-tree T of G on its own does not allow to solve the ASF problem. Next, we turn to show the impact of a slight perturbation on c(e) on the resolution of the problem. 2.2 Characterization of ASF [e] Given a network G, notice that a link e of G with c(e) = c0 that is saturated by a maximum flow remains saturated even if c(e) is slightly decreased; c(e) = µ0 < c0 , where µ0 > 0 is fixed. Moreover, for a pair {s, t}, if e is in a (s-t)minimum cut, then c(e) = µ0 implies that fs,t will also linearly decrease by the quantity (c0 − µ0 ) (proved in [1]). Intuitively, the maximum flows, for which a decrease on c(e) implies a similar decrease on their values, saturate e. Theorem 2.4 Let G = (V, E) be a network, e = [i, j] ∈ E with c(e) = c0 , and T a cut-tree of G. Consider, a fixed slight decrease δ0 > 0 on the capacity c(e), i.e., c(e) = c0 − δ0 > 0. Let Gδ0 be the new network and let Tδ0 be a T T cut-tree of Gδ0 . Let fs,t and fs,tδ0 denote respectively the value of fs,t in T and T
T Tδ0 . Then, ∀ s, t ∈ V, {s, t} ∈ ASF [e] ⇔ fs,tδ0 = fs,t − δ0 .
Proof. Let G = (V, E) be a network and e ∈ E. Assume an arbitrary {s, t} →
in ASF [e]. Then any f s,t saturates e that belongs to any (s-t)-minimum cut. Thus, any decrease on c(e) would similarly apply for the value fs,t , since such a decrease does not avoid e to belong to a (s-t)-minimum cut. Reversely, if fs,t decreases when c(e) is decreased, then e belongs to a (s-t)-minimum cut. →
Thus, f s,t saturates e implying that {s, t} ∈ ASF [e].
2
238
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
Corollary 2.5 If all capacities are integral, the ASF problem is solved with a single cut-tree computation Tδ0 on Gδ0 with 0 < δo < 1. Proof. With Tδ0 , ∀{s, t} ∈ V, If fs,t not integral, Then {s, t} ∈ ASF [e].
2
In Figure 3, we illustrate how Theorem 2.4 derives the example of Figure 2 in two cases. Decrease c(e) of e = [3, 4] by δ0 = 0.5 in the former networks S and T , and let Tδ0 and Sδ0 be the new networks pointing a respective cut-tree T. 1 1 T δ0
1
1
2
2
2
2
2
2
1.5
3
1.5
3
4
2
T
4
1.5
1
1
4
S δ0
2
1 1.5
0.5 3
3
1.5
4
Fig. 3. Cut-trees with respect to δ0
The application of Theorem 2.4 to each of the networks provides the respective sets ASF [3, 4] of the previous Points (i) and (ii) in Section 2.1.
3
Conclusion and perspectives
We have provided a novel analysis to detect flows that always saturate a target network link. We showed that two cut-trees construction were sufficient for such an analysis. This results in a complexity of O(n) maximum flow computations. Nevertheless, for a pair {s, t}, it may happen that it exists a →
f s,t that saturates the target link without this latter being in a (s-t)-minimum cut. Thus in order to complete the analysis we will turn to search for a larger set than ASF [e], i.e, the denoted ESF [e] set, standing for the set of all pairs Existing Saturating Flows for e. Given a network G and a target link e of G, the ESF Problem consists in identifying the set of all vertex pairs for which it exists a maximum flow that saturates e. For instance, in Figure 4, let →
→
e = [a, b] be the target link. A f s,t through path {s, a, b, t} with | f s,t (e)| = 1 saturates e without e being in a (s-t)-minimum cut. Moreover {s, t} ∈ ESF [e] but {s, t} ∈ / ASF [e]. s
1
1
b
1
1
c
1
a
t
Fig. 4. Example of {s, t} belonging to ESF [e] and not to ASF [e]
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
239
References [1] P. Berthom´e, M. Diallo, and A. Ferreira. Generalized parametric multi-terminal flows problem. In H.L. Bodlaender, editor, Graph-theoretic concepts in computer science, volume 2880 of Lecture Notes in Computer Science, pages 71–80. Springer Verlag, 2003. [2] M. Diallo. R´eseaux de Flots : Flots Param´etr´es et Tarification. PhD thesis, Universit´e de Versailles, France, December 2003. (In French) Available at http://www.prism.uvsq.fr/~diallo. [3] L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, Princeton, NJ, 1973. [4] R. E. Gomory and T. C. Hu. Multi-terminal network flows. SIAM Journal of Computing, 9(4):551–570, December 1961. [5] H. W. Hamacher and S. Tjandra. Mathematical modelling of evacuation problems. In M. Schreckenberg and S. D. Sharma, editors, Pedestrian and Evacuation Dynamics, pages 227–266. Springer, 2002.
Detecting flows congesting a target network link D. Barth a , P. Berthom´e b , M. Diallo c a
b
c
Laboratoire PRiSM, UMR 8144, CNRS, Universit´e de Versailles, 45 Av. des Etats-Unis, 78035 Versailles-CEDEX, FRANCE
Laboratoire de Recherche en Informatique (LRI), UMR 8623, CNRS, Universit´e Paris-Sud, 91405, Orsay-CEDEX, FRANCE
CNRS, LIMOS UMR 6158, Universit´e Clermont 2, ISIMA, Campus des C´ezeaux - BP 10125, 63173 Aubi`ere CEDEX, FRANCE, [email protected]
Abstract In this paper, we deal with the analysis of network links saturation. Given a network, we target one of its links and provide an interesting analysis that allows to detect all vertex pairs for which any maximum flow always saturates the targeted link. The whole analysis complexity remains around O(n) maximum–flows/minimum– cuts computations using Gomory and Hu cut-trees. Keywords: link saturation, network congestion, multi-terminal flows, network analysis, parametric capacity
1
Introduction
In the current telecommunications and transports networks, congestion is one of the problems that merit great attention of the operators. From a prevention
1571-0653/2005 Published by Elsevier B.V. doi:10.1016/j.endm.2005.05.032
234
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
point of view, we are interested in deeply analyzing the saturation (congestion) of network links. When a flow goes from its source to its sink, the saturated links are bottlenecks for this flow. In some practical problems, determining bottlenecks is relevant. For instance, the building evacuation problem [5] is a particular case of the dynamic flow problem. Given the minimum evacuation time, this problem includes the detection of all the bottlenecks that may cause delay. Such bottlenecks correspond exactly to links belonging to minimum cuts (saturated by maximum flows). Thus, our objective in this paper is, given a link e in an undirected network, to show a way to detect the set of all vertex pairs for which any maximum flow always saturates e. This problem will be referred as ASF problem, where ASF stands for all pairs Always Saturating Flows. The searched set, with respect to link e, will be denoted ASF [e]. Such a set may help decision makers to know, for instance, in telecommunications, the communications that would take benefit from a Quality of Service improvement on the link e or that would surely suffer from a failure on e. Note that in both situations flows that uses e but do not saturate it will also be concerned with profit or damage on e. In this first work, we are not interested in such flows. For more applications, we refer to [2]. The remainder of the paper is devoted to the novel analysis we provide in order to solve the ASF problem. Conclusion and perspectives end the paper.
2
The ASF Problem
Let G = (V, E) be a network with a vertex set V and a link set E such that n = |V | and m = |E|. Each link e ∈ E has a capacity c(e) > 0. A flow f between the source s ∈ V and the sink t ∈ V is as defined by Ford and Fulkerson [3]. But this definition is given for directed networks. For undirected ones, we should consider the equivalent symmetric directed network (as detailed in [2]). In this paper, we only deal with undirected network and will note: →
f s,t : maximum flow between the source s and the sink t ((s-t)-maximum flow); →
fs,t : the value of f s,t ; →
→
→
→
f s,t (e): the flow on link e induced by f s,t , and | f s,t (e)|: the value of f s,t (e).
Furthermore, since our networks are symmetric, we have fs,t = ft,s .
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
235
A proper subset X of V (∅ X V ) is called a cut separating two vertices s and t if s ∈ X and t ∈ V \ X (or vice versa). The capacity of a cut X is c[x, y]. A minimum cut separating vertices defined as c(X) = x∈X,y∈V \X,[x,y]∈E
s and t, denoted hereafter Cs,t or (s-t)-minimum cut, is a cut with minimal capacity among all the cuts separating s and t. A link l belongs to a cut X if X separates its extremities. Now, we can formally describe the targeted set as follows. Let e be a link of G. The set ASF [e] for e is the set of vertex pairs {s, t} for which any maximum →
flow f s,t saturates e. Thus, → → ASF[e] = {s, t} ∈ V 2 | ∀ f st , | f st (e)| = c(e) .
2.1 On ASF [e] approximations In order to determine ASF [e], we need to check all maximum flows between all vertex pairs. This is a complex task if one does not use the concept of cut-tree introduced in 1961 by Gomory and Hu [4]. Given a network G = (V, E), a (Gomory-Hu) cut-tree T obtained from G is a weighted tree with vertex set V such that: — ∀ {s, t} ∈ G, fs,t in G equals to fs,t in T , i.e., the minimum weight on the unique path between s and t in T ; — A minimum cut Cs,t in T is also a (s-t)-minimum cut in G. Thus, with only (n-1) minimum–cut/maximum–flow computations, a cuttree provides all the n(n−1) maximum flow values and a minimum cut for each 2 vertex pair [4]. We notice that cut-trees are generally not unique. In the following, our results illustrate the use of cut-trees to solve the ASF problem. Let e = [i, j] ∈ E be the target link, and T a cut-tree of G. With respect to i and j, let Pi,j be the unique path with end points i and j in T . From the definition of a cut-tree, the removal of any link of Pi,j separates i and j in T , and thus, creates a vertex partition corresponding to a minimum cut in G that contains e. Lemma 2.1 states the Pi,j and ASF [e] relationship. Lemma 2.1 Let G = (V, E) be a network and e = [i, j] ∈ E. Consider a cut-tree T of G and the path Pi,j . Then, Pi,j × Pi,j ⊂ ASF [e]. Proof. This proof is illustrated in Figure 1. Consider any two vertices p and q in Pi,j . Let [x, y] be a link labeled with the minimum weight between p
236
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
and q. By definition of cut-tree, removing [x, y] in T creates a (p-q)-minimum ¯ in G with p ∈ X and q ∈ X. ¯ Thus, i belonging to X and j to cut (X, X) ¯ ¯ X implies e = [i, j] ∈ (X, X). By the Max-Flow/Min-Cut theorem [3], e is saturated by any (s-t)-maximum flow. Thus, {s, t} ∈ ASF [e]. 2 s Ps,t i
X
j
p
Pi,j
x
q
y
¯ X
t
Fig. 1. Illustration of Lemma 2.1 and Lemma 2.2.
Lemma 2.2 Extension of Lemma 2.1. Let e = [i, j] ∈ E be a link of G = (V, E), T a cut-tree of G and Pi,j as before. For any {s, t} in G, if a link of minimum weight in Ps,t in T belongs also to Pi,j , then {s, t} belongs to ASF [e]. Proof. Assume that a link [x, y] with the minimum weight on Ps,t lies on Ps,t ∩ Pi,j as shown in Figure 1. Removing [x, y] in T gives a (s-t)-minimum cut (X) in G that contains e = [i, j]. From Lemma 2.1, {s, t} ∈ ASF [e]. 2 The above results are not sufficient to get ASF [e]. For instance, in Figure 2, the top simple weighted path of length 3 is an arbitrary cut-tree that may be obtained from both the networks it points out: either from (T ) being the path itself or from the cycle (S). T
1
2
2
2
3
2
4 1
1
2
2
2
3
2
4
4
S
1 T
1
2
1
1
3
Fig. 2. Approximations of ASF [e]
In both S and T , let e = [3, 4] be the target link. As for ASF [e], we have: (i) in T , ASF [e] = {(1, 4), (2, 4), (3, 4)}, and (ii) in S, ASF [e] = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. But, if only the information of Lemma 2.2 is used, for both networks (S and T ), one gets the same set {{1, 4}, {2, 4}, {3, 4}}. This set corresponds exactly to ASF [e] in T while it is different from the ASF [e] in S.
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
237
In seeking to get set ASF [e] in the network S, we arise to Lemma 2.3. Lemma 2.3 Let G = (V, E) be a network, e = [i, j] ∈ E and T a cut-tree of G. Let w be the minimum weight on Pi,j in T . If all edges of T labeled with a weight strictly less than w are deleted, then a subtree K of T containing i and j is created and ASF [e] ⊆ V (K) × V (K); V (K) is the vertex set in K. Proof. Assume ∃{s, t} ∈ ASF [e] such that {s, t} ∈ / V (K) × V (K). Then, a (s-t)-minimum cut will be such that i and j belongs to its same component. →
Thus, link e is in no (s-t)-minimum cut. Then, there exists a f s,t such that →
| f s,t (e)| < c(e) : contradiction with hypothesis and definition of ASF [e].
2
Lemma 2.2 and 2.3 give bounds for ASF [e]. However, it can be shown that they do not always describe the requested set ASF [e] (illustration skipped for the sake of space but is in [2]). Until now, we showed that, given a network G and a target link e of G, a single cut-tree T of G on its own does not allow to solve the ASF problem. Next, we turn to show the impact of a slight perturbation on c(e) on the resolution of the problem. 2.2 Characterization of ASF [e] Given a network G, notice that a link e of G with c(e) = c0 that is saturated by a maximum flow remains saturated even if c(e) is slightly decreased; c(e) = µ0 < c0 , where µ0 > 0 is fixed. Moreover, for a pair {s, t}, if e is in a (s-t)minimum cut, then c(e) = µ0 implies that fs,t will also linearly decrease by the quantity (c0 − µ0 ) (proved in [1]). Intuitively, the maximum flows, for which a decrease on c(e) implies a similar decrease on their values, saturate e. Theorem 2.4 Let G = (V, E) be a network, e = [i, j] ∈ E with c(e) = c0 , and T a cut-tree of G. Consider, a fixed slight decrease δ0 > 0 on the capacity c(e), i.e., c(e) = c0 − δ0 > 0. Let Gδ0 be the new network and let Tδ0 be a T T cut-tree of Gδ0 . Let fs,t and fs,tδ0 denote respectively the value of fs,t in T and T
T Tδ0 . Then, ∀ s, t ∈ V, {s, t} ∈ ASF [e] ⇔ fs,tδ0 = fs,t − δ0 .
Proof. Let G = (V, E) be a network and e ∈ E. Assume an arbitrary {s, t} →
in ASF [e]. Then any f s,t saturates e that belongs to any (s-t)-minimum cut. Thus, any decrease on c(e) would similarly apply for the value fs,t , since such a decrease does not avoid e to belong to a (s-t)-minimum cut. Reversely, if fs,t decreases when c(e) is decreased, then e belongs to a (s-t)-minimum cut. →
Thus, f s,t saturates e implying that {s, t} ∈ ASF [e].
2
238
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
Corollary 2.5 If all capacities are integral, the ASF problem is solved with a single cut-tree computation Tδ0 on Gδ0 with 0 < δo < 1. Proof. With Tδ0 , ∀{s, t} ∈ V, If fs,t not integral, Then {s, t} ∈ ASF [e].
2
In Figure 3, we illustrate how Theorem 2.4 derives the example of Figure 2 in two cases. Decrease c(e) of e = [3, 4] by δ0 = 0.5 in the former networks S and T , and let Tδ0 and Sδ0 be the new networks pointing a respective cut-tree T. 1 1 T δ0
1
1
2
2
2
2
2
2
1.5
3
1.5
3
4
2
T
4
1.5
1
1
4
S δ0
2
1 1.5
0.5 3
3
1.5
4
Fig. 3. Cut-trees with respect to δ0
The application of Theorem 2.4 to each of the networks provides the respective sets ASF [3, 4] of the previous Points (i) and (ii) in Section 2.1.
3
Conclusion and perspectives
We have provided a novel analysis to detect flows that always saturate a target network link. We showed that two cut-trees construction were sufficient for such an analysis. This results in a complexity of O(n) maximum flow computations. Nevertheless, for a pair {s, t}, it may happen that it exists a →
f s,t that saturates the target link without this latter being in a (s-t)-minimum cut. Thus in order to complete the analysis we will turn to search for a larger set than ASF [e], i.e, the denoted ESF [e] set, standing for the set of all pairs Existing Saturating Flows for e. Given a network G and a target link e of G, the ESF Problem consists in identifying the set of all vertex pairs for which it exists a maximum flow that saturates e. For instance, in Figure 4, let →
→
e = [a, b] be the target link. A f s,t through path {s, a, b, t} with | f s,t (e)| = 1 saturates e without e being in a (s-t)-minimum cut. Moreover {s, t} ∈ ESF [e] but {s, t} ∈ / ASF [e]. s
1
1
b
1
1
c
1
a
t
Fig. 4. Example of {s, t} belonging to ESF [e] and not to ASF [e]
D. Barth et al. / Electronic Notes in Discrete Mathematics 19 (2005) 233–239
239
References [1] P. Berthom´e, M. Diallo, and A. Ferreira. Generalized parametric multi-terminal flows problem. In H.L. Bodlaender, editor, Graph-theoretic concepts in computer science, volume 2880 of Lecture Notes in Computer Science, pages 71–80. Springer Verlag, 2003. [2] M. Diallo. R´eseaux de Flots : Flots Param´etr´es et Tarification. PhD thesis, Universit´e de Versailles, France, December 2003. (In French) Available at http://www.prism.uvsq.fr/~diallo. [3] L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, Princeton, NJ, 1973. [4] R. E. Gomory and T. C. Hu. Multi-terminal network flows. SIAM Journal of Computing, 9(4):551–570, December 1961. [5] H. W. Hamacher and S. Tjandra. Mathematical modelling of evacuation problems. In M. Schreckenberg and S. D. Sharma, editors, Pedestrian and Evacuation Dynamics, pages 227–266. Springer, 2002.