Analysing local algorithms in location-aware quasi-unit-disk graphs

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.Peer reviewe

Survey of local algorithms

A local algorithm is a distributed algorithm that runs in constant time, independently of the size of the network. Being highly scalable and fault-tolerant, such algorithms are ideal in the operation of large-scale distributed systems. Furthermore, even though the model of local algorithms is very limited, in recent years we have seen many positive results for non-trivial problems. This work surveys the state-of-the-art in the field, covering impossibility results, deterministic local algorithms, randomised local algorithms, and local algorithms for geometric graphs.Peer reviewe

Optimisation problems in wireless sensor networks : Local algorithms and local graphs

This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating–dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating–dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs – these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating–dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

Local algorithms for edge colorings in UDGs

AbstractIn this paper, we consider two problems: the edge coloring and the strong edge coloring problems on unit disk graphs (UDGs). Both problems have important applications in wireless sensor networks as they can be used to model link scheduling problems in such networks. It is well known that both problems are NP-complete, and approximation algorithms for them have been extensively studied under the centralized model of computation. Centralized algorithms, however, are not suitable for ad hoc wireless sensor networks whose devices typically have limited resources, and lack the centralized coordination.We develop local distributed approximation algorithms for the edge coloring and the strong edge coloring problems on unit disk graphs. For the edge coloring problem, our local distributed algorithm has approximation ratio 2 and locality 50. For the strong edge coloring problem on UDGs, we present two local distributed algorithms with different tradeoffs between their approximation ratio and locality. The first algorithm has ratio 128 and locality 22, whereas the second algorithm has ratio 10 and locality 180

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Finding structure in data streams : correlations, independent sets, and matchings

The streaming model supposes that, rather than being available all at once, the data is received in a piecemeal fashion. In a world of massive data sets, streaming algorithms give a complementary approach to distributed algorithms: with the data all being available in one place but at different times, rather than at the same time in different places. We examine three different single-pass streaming problems where existing results show limited feasibility. We consider realistic relaxations or restrictions of these problems which allow for more efficient algorithms. In the correlation outliers problem, we wish to identify pairs of unusually correlated signals from a streamed matrix of observations. We show that a simple application of existing technique is space-optimal but has slow query time when the outlier threshold is small. We demonstrate how we can achieve faster query times at the cost of storing a larger data summary. In the maximum independent set problem, we wish to find an edge-less induced subgraph of maximum size. For arbitrary graphs, given as a stream of edges, it is known that no space-efficient algorithm exists. We consider a variant streaming model, where the graph is received vertex by vertex. While we show this model still does not admit efficient algorithms for general graphs, we demonstrate efficient approximation algorithms for various special graph classes. In the maximum matching problem, we wish to find a disjoint subset of edges of largest possible size. The greedy algorithm gives us an easy 2-approximation for streams of edges, but the problem becomes infeasible to solve if we allow unlimited edge deletions. We consider a model where, instead, a limited number of deletions are allowed. We describe several new approximation algorithms with complexity parameterised by the number of deletions. We also present new techniques which may lead to the development of corresponding tight lower bounds

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum