On the Broadcast Routing Problem

02/25/2018
by   Brahim Chaourar, et al.
0

Given an undirected graph G = (V, E), and a vertex r∈ V, an r-acyclic orientation of G is an orientation OE of the edges of G such that the digraph OG = (V, OE) is acyclic and r is the unique vertex with indegree equal to 0. For w∈^E_+, k(G, w) is the w-maximum packing of r-arborescences for all r∈ V and all r-acyclic orientations OE of G. In this case, the Broadcast Routing (in Computers Networks) Problem (BRP) is to compute k(G, w), by finding an optimal r and an optimal r-acyclic orientation. BRP is a mathematical formulation of multipath broadcast routing in computer networks. In this paper, we provide a polynomial time algorithm to solve BRP in outerplanar graphs. Outerplanar graphs are encountered in many applications such computational geometry, robotics, etc.

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1. Introduction

Sets and their characteristic vectors will not be distinguished.

We refer to Bondy and Murty [6] about graph theory terminology and facts.
In computer networking, broadcasting refers to sending a packet to every destination simultaneously [34]. Broadcast is a communication function that a node, called the source, sends messages to all the other nodes in the network. It is an important and complex function for implementation in optimal network routing with Quality of Service (QoS) requirement [10, 12].
On the inverse of unicast routing, that is sending packets to one single destination, where several efficient multipath algorithms have been proposed [8, 3, 18, 24, 28, 29, 30], all known algorithms for broadcast routing use a unique path to reach the destination, a spanning tree (or a rooted arborescence). In this paper, we will propose a novel and efficient multipath algorithm for broadcast routing in outerplanar networks.
Multipath routing is proposed as an alternative to single path routing to take advantage of network redundancy, distribute load [33], improve packet delivery reliability [23], ease congestion on a network [2, 20], improve robustness [35], increase network security [4] and address QoS issues [5].
Instead of finding the best spanning tree (or rooted arborescence) when using a single path, we should find rooted arborescences, i.e., directed rooted spanning trees when using multipath routing because we should avoid corruption and redundancy, i.e., sending packets in two inverse directions at the same time.
An outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. The outerplanar graphs are a subclass of the planar graphs. It follows that any simple and 2-connected outerplanar graph is formed by a circuit of the outer face and single edges linking between vertices of this circuit in a manner that preserves planarity. We call this circuit an outer circuit of , these edges the chords of regarding this outer circuit, and the common vertices between the outer circuit and the chords the join vertices. We call the paths of the outer circuit joining successive join vertices the outer chords. Finally, we call the outer chords containing at least one vertex of degree 2, the series outer chords. The outer chords which are not series ones are single edges. Two chords (outer or not) are adjacent if they have one common join vertex.

Many Combinatorial Optimization problems have been studied in outerplanar graphs/networks

[1, 7, 9, 11, 13, 15, 16, 17, 19, 21, 22, 25, 26, 27, 31, 36, 37] and most of them have been proved polynomial in this class of graphs. Another motivation is that ”outerplanar graphs constitute an important class of graphs, often encountered in various applications, e.g., computational geometry, robotics, etc.” [22]
On the other hand, studying the broadcast routing problem in these outerplanar networks is motivated by the difficulty of sending packets through parallel rooted arborescences in general graphs/networks without avoiding corruption and redundancy.
In this paper, we solve this problem in the special case of outerplanar networks by using combinatorial techniques.
The remainder of the paper is organized as follows: in section 2, we give a mathematical formulation of the problem, then, in section 3, we solve it in outerplanar graphs. And we conclude in section 4.

2. Mathematical Formulation

Given an undirected graph , and a vertex , an acyclic orientation of rooted at (or a rooted acyclic orientation, or an -acyclic orientation) is an orientation of the edges of G such that the digraph is acyclic and is the unique vertex with indegree equal to 0.
Given a graph , and a nonnegative real weight function defined on , we denote by the -maximum packing of spanning trees in . Given a vertex , and an -acyclic orientation of the edges of , we denote by the -maximum packing of -arborescences in , an -acyclic orientation of }, and .
Now we can define the Broadcast Routing Problem (BRP) as follows. Given a graph , and a nonnegative real weight function defined on . The question is to compute by finding appropriate optimal root and -acyclic orientation , and the corresponding -arborescences involved in the maximum packing.
should be acyclic to avoid corruption and redundancy, and the root represents the source node in the broadcast routing. By routing data through , we will maximize the used bandwidth for each directed edge of and, virtually, we will have a multipath broadcast routing which is given by the involved -arborescences in the maximum packing.
We need also to define a more general problem rBRP as follows. Given a graph , a vertex , and a nonnegative real weight function defined on . The question is to compute by finding an appropriate optimal -acyclic orientation , and the corresponding -arborescences involved in the maximum packing. It is clear that BRP can be reduced to rBRP in , where .
Since we are dealing with maximum packing of rooted arborescences, we need the following theorem due to Edmonds [14].

Theorem 2.1.

Given a digraph , a vertex , and a weight function , then the -maximum weight packing of -arborescences equals the -minimum weight of an -cut.

Note that and are not equal in general, even for (circuits and) outerplanar graphs as shown in the minimal example here below.
Example: Let be a circuit on 4 vertices such that , , and .
It is not difficult to see that and because for any chosen root and any -acyclic orientation of , there is a trivial -cut with weight 1, and, according to Theorem 2.1, the maximum packing of -arborescences equals the minimum weight of an -cut.
We give here a result about the hardness of getting a solution by enumeration.

Proposition 2.2.

The number of rooted acyclic orientations of () is .

Proof.

Since every acyclic orientation gives a topological order of the vertices and vice-versa, and there is a unique acyclic orientation if we fix an order for the vertices, then the number of rooted acyclic orientations of is equal to the number of orders on its vertices which is . ∎

The main result of this paper is to solve BRP in outerplanar graphs.

3. BRP is polynomial in outerplanar networks

First we can reduce BRP and rBRP to simple graphs by using the following proposition.

Proposition 3.1.

Consider the two following cases:
(1) Let be an undirected graph, and are parallel edges of , and . Let and such that and for .
(2) Let be an undirected graph, is a loop of , and . Let and such that for .
Then for both cases (1) and (2), we have: and for any .

Proof.

Direct from the fact that parallel edges in acyclic digraphs should be oriented in the same direction, and removing loops do not affect any packing of rooted arborescences because arborescences do not contain loops. ∎

We can also reduce BRP and rBRP to 2-connected graphs by using the following proposition.

Proposition 3.2.

Let be an undirected graph, such that , , , , , and , , the projection of on , .
Then = min
{, }.

Proof.

Direct from the fact that the restriction of any -acyclic orientation of to , , gives a -acyclic orientation of and a -acyclic orientation of , respectively. ∎

We need the following lemma for solving BRP and rBRP in any kind of graphs.

Lemma 3.3.

Given an acyclic digraph , a vertex , and a weight function , then the -minimum -cut is reached at a (single vertex) trivial -cut.

Proof.

Let such that and is not empty. Then the subgraph induced by is also acyclic and contains at least one (root) vertex such that the indegree of in is zero. It follows that and because the weights are nonegative. We can conclude by taking the minimum of both sides of the later inequality. ∎

In this case, we denote by the -minimum trivial -cut for a given weight function . We can deduce here below how to compute the maximum packing of arborescences in acyclic digraphs.

Corollary 3.4.

Given an acyclic digraph rooted at the vertex , and a weight function , then the -maximum packing of -arborescences can be computed in where .

Proof.

We give here an algorithm for this purpose.
Input: An acyclic digraph rooted at , and .
Output: Pairs of positive reals and -arborescences such that , for any edge and is maximum.
(1) Set .
(2) Find such that .
(3) If then do:
(3.1) set ,
(3.2.1) if we can complete to an -arborescence by picking one edge for all such that , then ,
(3.2.2) else: stop.
(3.3) set for all , and set ,
(3.4) go to (2).
(4) End of algorithm.
Since, in each iteration, there is at least one edge whose weight becomes zero (step 3.3), then we need at most iterations to get the output. In each iteration, we need to get the minimum weight edge in (step 2), to complete one edge to a rooted arborescence in (step 3.2.1), where , and to update the weight function in (step 3.3). So the whole running time is .
Now, for the correctness of the algorithm, we will prove it by induction on the cardinality of support() = such that .
Let the weight function obtained from after the first iteration of the algorithm, i.e., after step (3.3) for the first time. It is clear that . It follows that is a -maximum packing of -arborescences in . According to Theorem 2.1 and Lemma 3.3, . Now, for any , . By taking the minimum of both sides, we have , which means that our packing is maximum. ∎

First, we solve BCNP and rBCNP in circuits.

Lemma 3.5.

BCNP and rBCNP can be solved in , the circuit on vertices, with , in linear time ().

Proof.

Let be a circuit on vertices with , and a weight function . First we find the three -minimum edges , for any .
Since the case is trivial then we can suppose that . Given , it is not difficult to see that, in any -acyclic orientation of , there is exactly one sink , i.e., a vertex with outdegree equal to zero. According to Theorem 2.1 and Lemma 3.3, the -maximum packing of -arborescences equals the -minimum trivial -cut. Thus, should be incident to in any optimal -acyclic orientation of because, otherwise, . We have then the following cases:
Case 1: If is adjacent to then the optimal sink should be incident to both edges, and any vertex distinct from can be an optimal root.
Case 2: If is not adjacent to then let and be the two edges adjacent to such that . Thus, the optimal sink should be incident to and , and any vertex distinct from can be an optimal root. ∎

For simple and 2-connected outerplanar graphs, we need the following results.

Lemma 3.6.

Let be a simple and 2-connected outerplanar graph with chords. If is the maximum number of sinks with degree 2, for all and all -acyclic orientations of , then .

Proof.

Let and be root and acyclic orientation for which the number of sinks with degree 2 is maximum, i.e., equals . We will prove that by induction on .
If then is a circuit and trivially any acyclic orientation contains exactly one sink.
Suppose now that .
Note that all chords of do not contain sinks of degree 2 in .
Let be an outer circuit of and a sink of degree 2 in . It follows that there is a directed -path in passing through at least one chord (Otherwise if no chord is used in any -path, for all sinks of degree 2, we can remove this chord and, by induction on , we get the inequality). We choose as the last chord in that -path. Let be the last join vertex of in that -path. Let now be the -acyclic orientation of obtained from by reversing the orientation of all edges of the unique -path and maintaining the orientation of all remaining edges.
Case 1: is a sink in .
It follows that the pair (, ) is optimal for because, otherwise, we will get a better maximum for . Thus .
Case 2: is not a sink in .
It follows that the number of sinks of degree 2 in is . Thus , and then, . ∎

Lemma 3.7.

Let be a simple and 2-connected directed acyclic outerplanar graph rooted at with chords, and , , be the number of sinks in with degree . Then:

Proof.

By induction on .
If then, according to Lemma 3.5, .
If then there exists such that . It follows that there is a sink in with degree . Let be the union of all chords adjacent to in . Note that all chords in are one-way directed to . Note also that the number of chords in is .
Let . Since is still -acyclic, then by induction on , , we have because there are chords in . On the other hand, since all except , and ( becomes a sink of degree 2 in ), we have , and we are done. ∎

To solve BRP and rBRP in outerplanar graphs, we need to give some definitions. A mixed graph is a graph where the edges of are undirected and the (remaining) edges of are directed. A partial -acyclic orientation of an undirected graph , where , is a mixed graph such that: (1) The underlying graph of is ; (2) There exists an orientation of such that the obtained graph is an -acyclic orientation of (3) if , i.e., if some edge is oriented in the direction of then all remaining edges incident to are directed.
Note that any undirected graph and any of its acyclic orientations are particular cases of partial acyclic orientations ( and , respectively).
We call an undirected edge of a partial acyclic orientation a forced edge if it has one unique possible orientation in order to get (at the end) an acyclic orientation. For example, if we choose the root , then all edges incident to are forced because they should be oriented from to its adjacent vertices in order to get at the end an -acyclic orientation. Another example is when we choose a sink of degree 2 in an outer chord, then all edges of this outer chord are forced because they should be oriented in the direction of that sink (except if we have a root of degree 2 in the same outer chord). In opposition, a non forced edge is called a free edge. We have then the following condition for forced edges.

Lemma 3.8.

Let be a partial -acyclic orientation of an undirected simple and 2-connected outerplanar graph with chords and . If

then all edges of are forced.

Proof.

We consider the function . We will prove by induction on , that if then all edges of are forced.
If then is a circuit and the edges become forced if we choose the root and the sink, i.e., if .
Suppose now that and .
Case 1: There exists such that and .
Let be a chord for which is a join vertex.
Subcase 1.1: .
It follows that is a partial -acyclic orientation of with . Thus all remaining edges of are forced.
Subcase 1.2: .
Let be the second join vertex of then . If contributes to then , we are in Subcase 1.1. Otherwise, is not acyclic, a contradiction.
Case 2: For all such that , we have: .
It follows that all vertices contributing to are sinks of degree 2. Since the contribution of each such sink is 1 then their number is , according to Lemma 3.6.
Since all chords containing sinks of degree 2 in are outer chords, then it is not difficult to see that the number of outer chords is because chords should be incident successively. So all edges of the outer circuit are forced. Suppose now, by contradiction, that there exists a free undirected edge for some chord . So is a partial -acyclic orientation of with chords and , a contradiction with Lemma 3.6. ∎

Corollary 3.9.

The subgraph formed by all forced chords is a rooted arborescence.

Proof.

Direct from the fact that there is no contribution induced by forced edges, so all vertices incident to them should have indegree 1 in the (full) acyclic orientation (at the end). ∎

Now, we solve BRP and rBRP in simple and 2-connected outerplanar graphs.

Theorem 3.10.

BCNP and rBCNP can be solved in simple and 2-connected outerplanar graphs in polynomial time.

Proof.

We will give a polynomial time algorithm for solving rBCNP in simple and 2-connected outerplanar graphs.
Input: An undirected simple and 2-connected outerplanar graph with , and .
Output: an -acyclic orientation of such that:

(0) Orient all current forced edges (incident to )
(1) Set , , , , and .
(2) While do:
(2.1) Find such that .
(2.2) Set .
(2.3) Orient all edges of in the direction of , and all current forced edges.
(2.4) Set , , and .
(2.5) Set .
(2.6) End of While.
(3) Set .
(4) Orient all edges of as an -arborescence.
(5) End of algorithm.
Step (2.1) has a running time complexity , where . Since we will run the loop of While at most times, then the running time of the whole algorithm is at most because we need to compute a minimum once in step (3).
Now, we will prove the correctness of the algorithm by induction on .
Note that Lemma 3.8 and Corollary 3.9 imply that the obtained directed graph is an -acyclic orientation of .
Let be the number of iterations of the While loop. It is not difficult to see that:

If then is a circuit and according to Lemma 3.5, .
Suppose now that and where is the smallest possible one. We can suppose that because if then we can conclude.
It is not difficult to see that there is an outer chord such that is not an inner vertex of and is a simple and 2-connected outer planar graph. Let the chord adjacent to in such that belongs to an outer chord in . Suppose that is oriented as in .
Case 1: There is a sink of degree 2 in .
Note that all edges of are oriented in the direction of . Since then and . It follows that is the output of the algorithm for the input and , the weight function restricted to . By induction on , we have: .
Whatever the orientation of the edges of , cannot be improved, so .
Case 2: There is no sink of degree 2 in .
Note that all edges of are oriented in the direction of in . Let the incident edge to , , and such that and otherwise. It is not difficult to see that is the output of our algorithm for the input and . So by induction on , we have . ∎

4. Conclusion

We have given a new and the first multipath algorithm for broadcast routing (in computer networks). We have introduced a mathematical formulation for this problem and called it BRP. Then we have solved BRP in outerplanar graphs. Future investigations can be trying to prove the NP-hardness of BRP in general graphs and solve it in larger classes than outerplanar graphs.

Acknowledgements The author is grateful to the deanship of Scientific Research at Al Imam Mohammad Ibn Saud Islamic University (IMSIU) for supporting financially this research under the grant No 351223.

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