FlashcastHaewon Jeong, Si-Hyeon Lee, and Sae-Young Chung

Department of EE, KAIST, Daejeon, KoreaEmail: addie@kaist.ac.kr, sihyeon@kaist.ac.kr, sychung@ee.kaist.ac.kr

Abstract—In this paper, message dissemination withnode mobility is studied where each node in the networkhaving mobility wants to send its message to all the othernodes. The channel capacity between two nodes is assumedto be large enough for exchanging all the messages theyhave when they get close enough. We show that what typeof network graph enables each node to accumulate allmessages in the network, and investigate the disseminationtime T for all nodes to get all messages. For a generaldirected graph model, the upper and lower bounds on Tare given as Θ(n2) and Θ(1), respectively. For some specialcases, we present tighter bounds. For general undirectedgraph, T is upper bounded by Θ(n). For grid graph, T isan order of Θ(

√n).

I. INTRODUCTION

In this paper, message dissemination with node mo-bility is studied where each node in the network havingmobility wants to send its message to all the other nodes.One example can be forwarding traffic information toother cars on the road. For multiple-unicasts scenario,the capacity scaling with node mobility was studied in[1]. Recently, message dissemination with node mobilitywas studied in [2].

By exploiting mobility, we will use the concept ‘flash-cast’. It is exchanging information within very short timeduring which a strong channel between two nodes isformed. By mobility, two nodes can be very close toform a strong channel so that they can exchange datawithout interference. A strong channel formed for a verysmall amount of time is analogous to a flash light ofcamera which is very strong for a short moment, so wenamed the concept as ‘flash-cast’.

The rest of the paper is organized as follows. InSection II, we model the concept with a general directedgraph. Section III contains our main results: first, weshow which type of graph makes it possible to dissem-inate each node’s message over the graph, and then wepresent the upper and lower bounds of disseminationtime required for all nodes in the network to gather allmessages over the network.

II. MODEL

A network graph is denoted as a directed graphG(V,E) with |V | = n vertices and edges E ⊆ V × V .Note that we will distinguish the term node and vertexhere: a node is an information carrier moving on thegraph while vertex is a fixed point of the graph. Sincethis is a network model with mobility, nodes move from

a vertex to another through edges. We assume that thechannel capacity between two nodes is large enough forexchanging all the messages they have when they getclose enough. In the graph, we regard two nodes areclose enough when they are on the same vertex. Thuswhen some nodes are on the same vertex, they exchangeall messages they have with each other instantly. In otherwords, each node on the vertex transmits all messages ithas and receives messages from all the other nodes onthe vertex. A node is assumed to have an infinite bufferto store information, so it can store as much informationas it wants.

We assume that each node is initially on one of thevertices and has a distinct message. Then every nodemoves to one of its adjacent vertices in one time interval,and a node transmits and receives data only when itmeets other nodes at some vertex. In this paper, weinvestigate a network with heavy traffic, where a largenumber of nodes is assumed so that there exists at leastone node moving on each edge at any time.

A path from u to v is defined as any finite sequenceof vertices in V such that there is a sequence of directededges from u to v in G and the length of a path is thenumber of edges contained in the path. We define Iv(t)as the set of messages that nodes on vertex v have attime t, A(k)

v as the set of vertices that has a path to vwith path length k, and (u, v) as an edge from a vertex uto a vertex v. The dissemination time T means the timerequired for all nodes to disseminate its message overthe graph. A directed graph is called strongly connectedif there is a path from each vertex in the graph to everyother vertex.

III. RESULTS

In our model, Iv(t) is given as follows:

Iv(t) =∪

u∈A(1)v

Iu(t− 1) =∪

u∈A(t)v

Iu(0), (1)

which implies that Iv(t) is the union of initial messageof vertices which has a distance t from v. By theequation (1), we can see that T should satisfy thefollowing condition.

∀v, Iv(T ) =∪

u∈A(T )v

Iu(0) =∪u∈V

Iu(0) (2)

∀v,A(T )v = V (3)

978-1-4673-4728-0/12/$31.00 ©2012 IEEE APCC 2012448

Since Iu(0) is distinct for all u ∈ V , the condition (3)is equivalent to the condition (2).

The following theorem gives the condition which thegraph G should satisfy for finite dissemination time.

Theorem 1: Dissemination time T is finite if and onlyif graph G is strongly connected and contains a set ofcycles whose lengths are relatively prime.

Proof: First, let’t prove that graph G is stronglyconnected and contains a set of cycles whose lengthsare relatively prime if T is finite. By (3), T satisfies∀v ∈ V,A

(T )v = V , which implies that for any vertex

v, there exists a path to v from any vertex u withpath length T . Because there is a path for any pairof vertices, graph is strongly connected. Since for allt > T , I

(t)v =

∪u∈A

(t−T )v

Iu(T ) = V , the adjacencymatrix P associated with G satisfies P (T+i) > 0 for alli = 0, 1, 2, · · · , which means P is regular. By [3], if Pis a a regular matrix then G must contain a set of simplecycles whose lengths are relatively prime.

Next, let’s prove the opposite direction. Let’s assumethat graph G is strongly connected and let c1, c2, · · · , clbe the sequence of cycle lengths in G in an increasingorder which are relatively prime. Since G is stronglyconnected, there exists a path between any pair of vertexu and v. Let muv be a length of the shortest path be-tween vertices u and v which touches cycles with lengthsin c1, c2, · · · , cl. Then there also exists a path betweenu and v with length muv +

∑lk=1 akck. By Schur’s

theorem in [4], for any B ≥ (c1−1)(cl−1), there existpositive integers ak’s satisfying B =

∑lk=1 akck. Let

M = max{muv : ∀u, ∀v} and T = M+(c1−1)(cl−1).Then there is always a path between any two verticeswith length T . Hence, ∀v ∈ V,A

(T )v = V .

If there are both (u, v) and (v, u) between two verticesu and v, and edges (u, v) and (v, u) share the sameroad, then nodes moving toward u and nodes movingtoward v will meet somewhere between u and v. Hencestrong channel between nodes can be formed not onlyon vertices but also on edges and transmissions on theseedges should be considered. We can consider a graphwhich allows edge transmission between vertices byadding self-loop edges at u and v.

Next, we present the upper and lower bounds on Tin the following theorem.

Theorem 2: If dissemination time T is finite, T isupper and lower bounded by n2 − 2n + 3 and 1,respectively.

Proof: The upper bound follows from the result in[3]. The lower bound 1 is trivial.

Furthermore, both the upper and lower bounds can beachieved. The upper bound is tight when a graph withn vertices is consisted of two cycles, one with length nand the other with length n − 1. The lower bound canbe achieved when every vertex has edges to all verticesincluding itself.

From Theorem 2, we can conclude that time requiredfor all nodes to gather all messages in a graph withn vertices is bounded by Θ(n2) and Θ(1). This is,however, a loose bound in general. In the followingsubsections, we present tighter bounds for some specialcases.

A. Undirected Graph

An undirected graph can be represented by a directedgraph if every undirected edge (u, v) is represented bytwo directed edges (u, v) and (v, u). If we assume thatedge transmission happens in undirected graph, thenit can be easily proved that Iv(t) can be written asIv(t) =

∪n=tn=0

∪u∈A

(n)v

Iu(0), which implies that theset of messages that nodes at vertex v have at time t issimply the union of all messages at vertices which havedistance less than t from v. Hence T will be the longestdistance in the graph.

The same result comes out from the discussion inTheorem 1. In Theorem 1, we concluded that T can bewritten as M +(c1 − 1)(cl − 1). In an undirected graphcase, c1 is 1 by self-loop edge and cl is 2 because oneundirected edge becomes a cycle with length 2. Hence,(c1 − 1)(cl − 1) is 0 as c1 is 1, and M would be thelongest distance. T will become the longest distance asa result.

The longest distance is at most n − 1, because thegraph is connected. In an undirected graph case, T isupper bounded by Θ(n). T in an undirected graph willbe somewhere between Θ(n) and Θ(1).

B. Grid graph

A grid graph is a good abstraction of a road network.Let’s consider a r by r grid graph. A grid graph is onespecial type of undirected graphs. Hence T is simplythe longest distance in the graph. The longest distancein the grid is a distance between the leftmost vertex atthe top and the rightmost vertex at the bottom, whichis 2(r − 1). If we represent it again with the numberof vertices n, then T will be 2

√n − 2. So for the grid

network with n vertices, T is order of Θ(√n).

REFERENCES

[1] M. Grossglauser and D. N. C. Tse, “Mobility increases thecapacity of ad hoc wireless networks,” IEEE/ACM Trans. Netw.,vol. 10, pp. 477–486, Aug. 2002.

[2] Y. Chen, S. Shakkottai, and J. G. Andrews, “On therole of mobility for multi-message gossip,” IEEE Trans.Inf. Theory, in revision for publication. [Online]. Available:http://arxiv.org/abs/1204.3114.

[3] P. Perkins, “A theorem on regular matrices,” Pacific Journal ofMathematics, vol. 11, pp. 1529–1533, 1961.

[4] A. Brauer, “On a problem of partitions,” Amer. J. Math, pp. 299–312, 1942.

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