Fabian Kuhn

Geometric ad-hoc routing: of theory and practice

All too often a seemingly insurmountable divide between theory and practice can be witnessed. In this paper we try to contribute to narrowing this gap in the field of ad-hoc routing. In particular we consider two aspects: We propose a new geometric routing algorithm which is outstandingly efficient on practical average-case networks, however is also in theory asymptotically worst-case optimal. On the other hand we are able to drop the formerly necessary assumption that the distance between network nodes may not fall below a constant value, an assumption that cannot be maintained for practical networks. Abandoning this assumption we identify from a theoretical point of view two fundamentamentally different classes of cost metrics for routing in ad-hoc networks.

Worst-Case optimal and average-case efficient geometric ad-hoc routing

In this paper we present GOAFR, a new geometric ad-hoc routing algorithm combining greedy and face routing. We evaluate this algorithm by both rigorous analysis and comprehensive simulation. GOAFR is the first ad-hoc algorithm to be both asymptotically optimal and average-case efficient. For our simulations we identify a network density range critical for any routing algorithm. We study a dozen of routing algorithms and show that GOAFR outperforms other prominent algorithms, such as GPSR or AFR.

Distributed computation in dynamic networks

In this paper we investigate distributed computation in dynamic networks in which the network topology changes from round to round. We consider a worst-case model in which the communication links for each round are chosen by an adversary, and nodes do not know who their neighbors for the current round are before they broadcast their messages. The model captures mobile networks and wireless networks, in which mobility and interference render communication unpredictable. In contrast to much of the existing work on dynamic networks, we do not assume that the network eventually stops changing; we require correctness and termination even in networks that change continually. We introduce a stability property called T -interval connectivity (for T >= 1), which stipulates that for every T consecutive rounds there exists a stable connected spanning subgraph. For T = 1 this means that the graph is connected in every round, but changes arbitrarily between rounds. We show that in 1-interval connected graphs it is possible for nodes to determine the size of the network and compute any com- putable function of their initial inputs in O(n2) rounds using messages of size O(log n + d), where d is the size of the input to a single node. Further, if the graph is T-interval connected for T > 1, the computation can be sped up by a factor of T, and any function can be computed in O(n + n2/T) rounds using messages of size O(log n + d). We also give two lower bounds on the token dissemination problem, which requires the nodes to disseminate k pieces of information to all the nodes in the network. The T-interval connected dynamic graph model is a novel model, which we believe opens new avenues for research in the theory of distributed computing in wireless, mobile and dynamic networks.

Asymptotically optimal geometric mobile ad-hoc routing

In this paper we present AFR, a new geometric mobile ad-hoc routing algorithm. The algorithm is completely distributed; nodes need to communicate only with direct neighbors in their transmission range. We show that if a best route has cost c, AFR finds a route and terminates with cost &Ogr;(c2) in the worst case. AFR is the first algorithm with cost bounded by a function of the optimal route. We also give a tight lower bound by showing that any geometric routing algorithm has worst-case cost

The price of being near-sighted

Ogr;(c2). Thus AFR is asymptotically optimal. We give a non-geometric algorithm that also matches the lower bound, but needs some memory at each node. This establishes an intriguing trade-off between geometry and memory.

Initializing newly deployed ad hoc and sensor networks

Achieving a global goal based on local information is challenging, especially in complex and large-scale networks such as the Internet or even the human brain. In this paper, we provide an almost tight classification of the possible trade-off between the amount of local information and the quality of the global solution for general covering and packing problems. Specifically, we give a distributed algorithm using only small messages which obtains an (ρΔ)1/k-approximation for general covering and packing problems in time O(k2), where ρ depends on the LPs coefficients. If message size is unbounded, we present a second algorithm that achieves an O(n1/k) approximation in O(k) rounds. Finally, we prove that these algorithms are close to optimal by giving a lower bound on the approximability of packing problems given that each node has to base its decision on information from its k-neighborhood.

On the complexity of distributed graph coloring

A newly deployed multi-hop radio network is unstructured and lacks a reliable and efficient communication scheme. In this paper, we take a step towards analyzing the problems existing during the initialization phase of ad hoc and sensor networks. Particularly, we model the network as a multi-hop quasi unit disk graph and allow nodes to wake up asynchronously at any time. Further, nodes do not feature a reliable collision detection mechanism, and they have only limited knowledge about the network topology. We show that even for this restricted model, a good clustering can be computed efficiently. Our algorithm efficiently computes an asymptotically optimal clustering. Based on this algorithm, we describe a protocol for quickly establishing synchronized sleep and listen schedule between nodes within a cluster. Additionally, we provide simulation results in a variety of settings.

Unit disk graph approximation

Coloring the nodes of a graph with a small number of colors is one of the most fundamental problems in theoretical computer science. In this paper, we study graph coloring in a distributed setting. Processors of a distributed system are nodes of an undirected graph <i>G</i>. There is an edge between two nodes whenever the corresponding processors can directly communicate with each other. We assume that distributed coloring algorithms start with an initial <i>m</i>-coloring of <i>G</i>. In the paper, we prove new strong lower bounds for two special kinds of coloring algorithms. For algorithms which run for a single communication round---i.e., every node of the network can only send its initial color to all its neighbors---, we show that the number of colors of the computed coloring has to be at least Ω(Δ²/log² Δ+ log log <i>m</i>). If such one-round algorithms are iteratively applied to reduce the number of colors step-by-step, we prove a time lower bound of Ω(Δ/log² Δ+ log*<i>m</i>) to obtain an <i>O</i>(Δ)-coloring. The best previous lower bounds for the two types of algorithms are Ω(log log <i>m</i>) and Ω(log*<i>m</i>), respectively.

Fast deterministic distributed maximal independent set computation on growth-bounded graphs

Finding a good embedding of a unit disk graph given by its connectivity information is a problem of practical importance in a variety of fields. In wireless ad hoc and sensor networks, such an embedding can be used to obtain virtual coordinates. In this paper, we prove a non-approximability result for the problem of embedding a given unit disk graph. Particularly, we show that if non-neighboring nodes are not allowed to be closer to each other than distance 1, then two neighbors can be as far apart as √3/2 - ε, where ε goes to 0 as n goes to infinity, unless P=NP. We further show that finding a realization of a d-quasi unit disk graph with d ≥ 1/√2 is NP-hard.

Dynamic networks: models and algorithms

The distributed complexity of computing a maximal independent set in a graph is of both practical and theoretical importance. While there exists an elegant O(log n) time randomized algorithm for general graphs [20], no deterministic polylogarithmic algorithm is known. In this paper, we study the problem in graphs with bounded growth, an important family of graphs which includes the well-known unit disk graph and many variants thereof. Particularly, we propose a deterministic algorithm that computes a maximal independent set in time O(log Δ· log*n) in graphs with bounded growth, where n and Δ denote the number of nodes and the maximal degree in G, respectively.