Footnote 1

Synonyms

Wireless information dissemination in geometric networks

Years and Authors of Summarized Original Work

2001; Dessmark, Pelc

Problem Definition

The Model Overview

Consider a set of stations (nodes) modeled as points in the plane, labeled by natural numbers, and equipped with transmitting and receiving capabilities. Every node u has a range r u depending on the power of its transmitter, and it can reach all nodes at distance at most r u from it. The collection of nodes equipped with ranges determines a directed graph on the set of nodes, called a geometric radio network (GRN), in which a directed edge (uv) exists if node v can be reached from u. In this case u is called a neighbor of v. If the power of all transmitters is the same, then all ranges are equal and the corresponding GRN is symmetric.

Nodes send messages in synchronous rounds. In every round, every node acts either as a transmitter or as a receiver. A node gets a message in a given round, if and only if it acts as a receiver and exactly one of its neighbors transmits in this round. The message received in this case is the one that was transmitted. If at least two neighbors of a receiving node u transmit simultaneously in a given round, none of the messages is received by u in this round. In this case, it is said that a collision occurred at u.

The Problem

Broadcasting is one of the fundamental network communication primitives. One node of the network, called the source, has to transmit a message to all other nodes. Remote nodes are informed via intermediate nodes, along directed paths in the network. One of the basic performance measures of a broadcasting scheme is the total time, i.e., the number of rounds it uses to inform all the nodes of the network.

For a fixed real s ≥ 0, called the knowledge radius, it is assumed that each node knows the part of the network within the circle of radius s centered at it, i.e., it knows the positions, labels, and ranges of all nodes at distance at most s. The following problem is considered:

How does the size of the knowledge radius influence deterministic broadcasting time in GRN?

Terminology and Notation

Fix a finite set R = { r 1, …, r ρ } of positive reals such that \(r_{1} < \cdots < r_{\rho }\). Reals r i are called ranges. A node v is a triple [l, (x, y), r i ], where l is a binary sequence called the label of v; (x, y) are coordinates of a point in the plane, called the position of v; and r i  ∈ R is called the range of v. It is assumed that labels are consecutive integers 1 to n, where n is the number of nodes, but all the results hold if labels are integers in the set {1, …, M}, where M ∈ O(n). Moreover, it is assumed that all nodes know an upper bound \(\Gamma \) on n, where \(\Gamma \) is polynomial in n. One of the nodes is distinguished and called the source. Any set of nodes C with a distinguished source, such that positions and labels of distinct nodes are different, is called a configuration.

With any configuration C, the following directed graph \(\mathcal{G}(C)\) is associated. Nodes of the graph are nodes of the configuration and a directed edge (uv) exists in the graph, if and only if the distance between u and v does not exceed the range of u. (The word “distance” always means the geometric distance in the plane and not the distance in a graph.) In this case u is called a neighbor of v. Graphs of the form \(\mathcal{G}(C)\) for some configuration C are called geometric radio networks (GRN). In what follows, only configurations C such that in \(\mathcal{G}(C)\) there exists a directed path from the source to any other node are considered. If the size of the set R of ranges is ρ, a resulting configuration and the corresponding GRN are called a ρ-configuration and ρ-GRN, respectively. Clearly, all 1-GRN are symmetric graphs. D denotes the eccentricity of the source in a GRN, i.e., the maximum length of all shortest paths in this graph from the source to all other nodes. D is of order of the diameter if the graph is symmetric but may be much smaller in general. \(\Omega (D)\) is an obvious lower bound on broadcasting time.

Given any configuration, fix a nonnegative real s, called the knowledge radius, and assume that every node of C has initial input consisting of all nodes whose positions are at distance at most s from its own. Thus, it is assumed that every node knows a priori labels, positions, and ranges of all nodes within a circle of radius s centered at it. All nodes also know the set R of available ranges.

It is not assumed that nodes know any global parameters of the network, such as its size or diameter. The only global information that nodes have about the network is a polynomial upper bound on its size. Consequently, the broadcast process may be finished but no node needs to be aware of this fact. Hence, the adopted definition of broadcasting time is the same as in [3]. An algorithm accomplishes broadcasting in t rounds, if all nodes know the source message after round t, and no messages are sent after round t.

Only deterministic algorithms are considered. Nodes can transmit messages even before getting the source message, which enables preprocessing in some cases. The algorithms are adaptive, i.e., nodes can schedule their actions based on their local history. A node can obviously gain knowledge from previously obtained messages. There is, however, another potential way of acquiring information during the communication process. The availability of this method depends on what happens during a collision, i.e., when u acts as a receiver and two or more neighbors of u transmit simultaneously. As mentioned above, u does not get any of the messages in this case. However, two scenarios are possible. Node u may either hear nothing (except for the background noise), or it may receive interference noise different from any message received properly but also different from background noise. In the first case, it is said that there is no collision detection, and in the second case – that collision detection is available (cf., e.g., [1]). A discussion justifying both scenarios can be found in [1, 7].

Related Work

Broadcasting in geometric radio networks and some of their variations was considered, e.g., in [6, 8, 9, 11, 12]. In [12] the authors proved that scheduling optimal broadcasting is NP hard even when restricted to such graphs and gave an O(nlogn) algorithm to schedule an optimal broadcast when nodes are situated on a line. In [11] broadcasting was considered in networks with nodes randomly placed on a line. In [9] the authors discussed fault-tolerant broadcasting in radio networks arising from regular locations of nodes on the line and in the plane, with reachability regions being squares and hexagons, rather than circles. Finally, in [6] broadcasting with restricted knowledge was considered but the authors studied only the special case of nodes situated on the line.

Key Results

The results summarized below are based on the paper [5], of which [4] is a preliminary version.

Arbitrary GRN in the Model Without Collision Detection

Clearly all upper bounds and algorithms are valid in the model with collision detection as well.

Large Knowledge Radius

Theorem 1.

The minimum time to perform broadcasting in an arbitrary GRN with source eccentricity D and knowledge radius s > r ρ (or with global knowledge of the network) is \(\Theta (D)\) .

This result yields a centralized O(D) broadcasting algorithm when global knowledge of the GRN is available. This is in sharp contrast with broadcasting in arbitrary graphs, as witnessed by the graph from [10] which has bounded diameter but requires time \(\Omega (\log n)\) for broadcasting.

Knowledge Radius Zero

Next consider the case when knowledge radius s = 0, i.e., when every node knows only its own label, position, and range. In this case, it is possible to broadcast in time O(n) for arbitrary GRN. It should be stressed that this upper bound is valid for arbitrary GRN, not only symmetric, unlike the algorithm from [3] designed for arbitrary symmetric graphs.

Theorem 2.

It is possible to broadcast in arbitrary n-node GRN with knowledge radius zero in time O(n).

The above upper bound for GRN should be contrasted with the lower bound from [2, 3] showing that some graphs require broadcasting time \(\Omega (n\log n)\). Indeed, the graphs constructed in [2, 3] and witnessing to this lower bound are not GRN. Surprisingly, this sharper lower bound does not require very unusual graphs. While counterexamples from [2, 3] are not GRN, it turns out that the reason for a longer broadcasting time is really not the topology of the graph but the difference in knowledge available to nodes. Recall that in GRN with knowledge radius 0, it is assumed that each node knows its own position (apart from its label and range): the upper bound O(n) uses this geometric information extensively.

If this knowledge is not available to nodes (i.e., each node knows only its label and range), then there exists a family of GRN requiring broadcasting time \(\Omega (n\log n)\). Moreover, it is possible to show such GRN resulting from configurations with only 2 distinct ranges. (Obviously for 1 configurations, this lower bound does not hold, as these configurations yield symmetric GRN, and in [3], the authors showed an O(n) algorithm working for arbitrary symmetric graphs).

Theorem 3.

If every node knows only its own label and range (and does not know its position), then there exist n-node GRN requiring broadcasting time \(\Omega (n\log n)\) .

Symmetric GRN

The Model with Collision Detection

In the model with collision detection and knowledge radius zero, optimal broadcast time is established by the following pair of results.

Theorem 4.

In the model with collision detection and knowledge radius zero, it is possible to broadcast in any n-node symmetric GRN of diameter D in time O(D + log n).

The next result is the lower bound \(\Omega (\log n)\) for broadcasting time, holding for some GRN of diameter 2. Together with the obvious bound, \(\Omega (D)\) this matches the upper bound from Theorem 4.

Theorem 5.

For any broadcasting algorithm with collision detection and knowledge radius zero, there exist n-node symmetric GRN of diameter 2 for which this algorithm requires time \(\Omega (\log n)\) .

The Model Without Collision Detection

For the model without collision detection, it is possible to maintain complexity O(D + logn) of broadcasting. However, we need a stronger assumption concerning knowledge radius: it is no longer 0, but positive, although arbitrarily small.

Theorem 6.

In the model without collision detection, it is possible to broadcast in any n-node symmetric GRN of diameter D in time O(D + log n), for any positive knowledge radius.

Applications

The radio network model is applicable to wireless networks using a single frequency. The specific model of geometric radio networks described in section “Problem Definition” is applicable to wireless networks where stations are located in a relatively flat region without large obstacles (natural or human made), e.g., in the sea or a desert, as opposed to a large city or a mountain region. In such a terrain, the signal of a transmitter reaches receivers at the same distance in all directions, i.e., the set of potential receivers of a transmitter is a disc.

Open Problems

  1. 1.

    Is it possible to broadcast in time o(n) in arbitrary n-node GRN with eccentricity D sublinear in n for knowledge radius zero? Note: In view of Theorem 2, it is possible to broadcast in time O(n).

  2. 2.

    Is it possible to broadcast in time O(D + logn) in all symmetric n-node GRN with eccentricity D, without collision detection, when knowledge radius is zero? Note: In view of Theorems 4 and 6, the answer is positive if either collision detection or a positive (even arbitrarily small) knowledge radius is assumed.

Cross-References