Hierarchical neighbor graphs: A low stretch connected structure for points in Euclidean space
📝 Abstract
We introduce hierarchical neighbor graphs, a new architecture for connecting ad hoc wireless nodes distributed in a plane. The structure has the flavor of hierarchical clustering and requires only local knowledge and minimal computation at each node to be formed and repaired. Hence, it is a suitable interconnection model for an ad hoc wireless sensor network. The structure is able to use energy efficiently by reorganizing dynamically when the battery power of heavily utilized nodes degrades and is able to achieve throughput, energy efficiency and network lifetimes that compare favorably with the leading proposals for data collation in sensor networks such as LEACH (Heinzelman et. al., 2002). Additionally, hierarchical neighbor graphs have low power stretch i.e. the power required to connect nodes through the network is a small factor higher than the power required to connect them directly. Our structure also compares favorably to mathematical structures proposed for connecting points in a plane e.g. nearest-neighbor graphs (Ballister et. al., 2005), $\theta $-graphs (Ruppert and Seidel, 1991), in that it has expected constant degree and does not require any significant computation or global information to be formed.
💡 Analysis
We introduce hierarchical neighbor graphs, a new architecture for connecting ad hoc wireless nodes distributed in a plane. The structure has the flavor of hierarchical clustering and requires only local knowledge and minimal computation at each node to be formed and repaired. Hence, it is a suitable interconnection model for an ad hoc wireless sensor network. The structure is able to use energy efficiently by reorganizing dynamically when the battery power of heavily utilized nodes degrades and is able to achieve throughput, energy efficiency and network lifetimes that compare favorably with the leading proposals for data collation in sensor networks such as LEACH (Heinzelman et. al., 2002). Additionally, hierarchical neighbor graphs have low power stretch i.e. the power required to connect nodes through the network is a small factor higher than the power required to connect them directly. Our structure also compares favorably to mathematical structures proposed for connecting points in a plane e.g. nearest-neighbor graphs (Ballister et. al., 2005), $\theta $-graphs (Ruppert and Seidel, 1991), in that it has expected constant degree and does not require any significant computation or global information to be formed.
📄 Content
Topology control is a fundamental problem in the study of wireless ad hoc and sensor networks. A primary issue in this area is that of constructing a connected network between the nodes while keeping in mind the various constraints wireless devices operate under. Several architectures have been suggested to solve this problem, each striving to achieve multiple objectives vital to obtaining high throughput and low latency while expending as little energy as possible. Bounded degree is one such objective, needed to reduce the overhead of channel state exchange between neighbors when nodes use MIMO antennas [28]. Another important criterion of a good solution is that the number of hops between nodes be small, required because multi-hop wireless networks are error-prone and the probability of packet loss increases with path length. For wireless nodes with sufficient battery power, a connected topology must also ensure that the power stretch-the ratio of power spent by two nodes in connecting through the network to power spent in communicating directly-must be kept low [13,Chapter 9].
In this paper we propose a novel architecture for connectivity in wireless networks: Hierarchical neighbor graphs. Our structure is a randomized one that effectively combines ideas from skip list data structures [17,2] and nearest-neighbor graphs [25,3] to give a hierarchical bounded degree structure for connecting points in a plane that has short paths between nodes both in terms of number of hops and distance. Hierarchical neighbor graphs can be built with local information and without any substantial computation. In order to find connections, nodes do not need to estimate any angles or distances, they only need to know the relative distance of their neighbors, which can be easily determined from signal strength. Unlike other nearest-neighbor flavored structures, hierarchical neighbor graphs are able to incorporate battery power as a parameter and ensure that nodes with low battery power are not expected to transmit to nodes that are located far away. This property, along with ease of deployment and reformation as battery power decreases, makes our structure a good candidate for collating data from a field of wireless sensor nodes. And, in fact, we demonstrate that our structure is more energy efficient and has better throughput than one of the leading proposals in this area, LEACH [8], while having network lifetime comparable to it.
The key idea of our construction is that each node is assigned a level and chooses one neighbor from a level above it and its other neighbors from its own level. This assignment of levels is done through a random process which helps ensure that the degree of each node is bounded in expectation no matter what the positions of the points. For the special case where the locations of the points are generated by a Poisson point process we show that even when all nodes have equal battery power, the probability of the edges formed being long is very small. Encouraged by this we define a edge-length bounded version of our structure in which all edges above a certain length, determined by the battery power of the node, are deleted. We demonstrate that for a Poisson point process of sufficiently high density, this more practical structure is still connected. Paper organization and our contributions. In Section 2 we define hierarchical neighbor graphs. We also describe proactive and reactive routing protocols for this architecture and discuss how it can be easily adapted when nodes join or leave or when battery power changes. Additionally:
• We show analytically that hierarchical neighbor graphs have bounded degree in expectation (Section 3.1).
• When the location of the points are determined by a Poisson point process, the probability of long-range connections being formed is very low. We also define a bounded connection length version of our structure and show through simulation that it is connected for Poisson point processes of sufficiently high density (Section 3.2).
• We show analytically that the number of hops between any two nodes in our structure is low (Section 3.3).
• We show through simulation that the distance stretch of our structure has an exponentially decaying distribution, decaying faster for pairs of points that are further apart. We show an initial theorem in this direction (Section 3.4).
• We simulate the use of our architecture to collate data from a field of wireless sensors and show that it performs better than LEACH [8] in terms throughput and energy efficiency while matching it in terms of network lifetime (Section 4).
The area of topology control for wireless ad hoc and sensor networks has seen a lot of activity over the years. Several surveys and books are available on this topic (see e.g. [19,21,13]). The proposed architectures range from the geometry-based Gabriel Graphs [6] and Relative Neighborhood Graphs [22] and their numerous variants, and direction based proposals li
This content is AI-processed based on ArXiv data.