Joel Rybicki

Species–fragmented area relationship

The species–area relationship (SAR) gives a quantitative description of the increasing number of species in a community with increasing area of habitat. In conservation, SARs have been used to predict the number of extinctions when the area of habitat is reduced. Such predictions are most needed for landscapes rather than for individual habitat fragments, but SAR-based predictions of extinctions for landscapes with highly fragmented habitat are likely to be biased because SAR assumes contiguous habitat. In reality, habitat loss is typically accompanied by habitat fragmentation. To quantify the effect of fragmentation in addition to the effect of habitat loss on the number of species, we extend the power-law SAR to the species–fragmented area relationship. This model unites the single-species metapopulation theory with the multispecies SAR for communities. We demonstrate with a realistic simulation model and with empirical data for forest-inhabiting subtropical birds that the species–fragmented area relationship gives a far superior prediction than SAR of the number of species in fragmented landscapes. The results demonstrate that for communities of species that are not well adapted to live in fragmented landscapes, the conventional SAR underestimates the number of extinctions for landscapes in which little habitat remains and it is highly fragmented.

A local 2-approximation algorithm for the vertex cover problem

We present a distributed 2-approximation algorithm for the minimum vertex cover problem. The algorithm is deterministic, and it runs in (Δ + 1)2 synchronous communication rounds, where Δ is the maximum degree of the graph. For Δ = 3, we give a 2-approximation algorithm also for the weighted version of the problem.

A lower bound for the distributed Lovász local lemma

We show that any randomised Monte Carlo distributed algorithm for the Lovász local lemma requires Omega(log log n) communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of d = O(1), where d is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lovász local lemma with a running time of O(log n) rounds in bounded-degree graphs, and the best lower bound before our work was Omega(log* n) rounds [Chung et al. 2014].

Large-scale Habitat Corridors for Biodiversity Conservation: A Forest Corridor in Madagascar

In biodiversity conservation, habitat corridors are assumed to increase landscape-level connectivity and to enhance the viability of otherwise isolated populations. While the role of corridors is supported by empirical evidence, studies have typically been conducted at small spatial scales. Here, we assess the quality and the functionality of a large 95-km long forest corridor connecting two large national parks (416 and 311 km2) in the southeastern escarpment of Madagascar. We analyze the occurrence of 300 species in 5 taxonomic groups in the parks and in the corridor, and combine high-resolution forest cover data with a simulation model to examine various scenarios of corridor destruction. At present, the corridor contains essentially the same communities as the national parks, reflecting its breadth which on average matches that of the parks. In the simulation model, we consider three types of dispersers: passive dispersers, which settle randomly around the source population; active dispersers, which settle only in favorable habitat; and gap-avoiding active dispersers, which avoid dispersing across non-habitat. Our results suggest that long-distance passive dispersers are most sensitive to ongoing degradation of the corridor, because increasing numbers of propagules are lost outside the forest habitat. For a wide range of dispersal parameters, the national parks are large enough to sustain stable populations until the corridor becomes severely broken, which will happen around 2065 if the current rate of forest loss continues. A significant decrease in gene flow along the corridor is expected after 2040, and this will exacerbate the adverse consequences of isolation. Our results demonstrate that simulation studies assessing the role of habitat corridors should pay close attention to the mode of dispersal and the effects of regional stochasticity.

Synchronous Counting and Computational Algorithm Design

Consider a complete communication network on n nodes, each of which is a state machine. In synchronous 2-counting, the nodes receive a common clock pulse and they have to agree on which pulses are “odd” and which are “even”. We require that the solution is self-stabilising (reaching the correct operation from any initial state) and it tolerates f Byzantine failures (nodes that send arbitrary misinformation). Prior algorithms are expensive to implement in hardware: they require a source of random bits or a large number of states. This work consists of two parts. In the first part, we use computational techniques (often known as synthesis) to construct very compact deterministic algorithms for the first non-trivial case of f = 1. While no algorithm exists for n < 4, we show that as few as 3 states per node are sufficient for all values n ≥ 4. Moreover, the problem cannot be solved with only 2 states per node for n = 4, but there is a 2-state solution for all values n ≥ 6. In the second part, we develop and compare two different approaches for synthesising synchronous counting algorithms. Both approaches are based on casting the synthesis problem as a propositional satisfiability (SAT) problem and employing modern SATsolvers. The difference lies in how to solve the SAT problem: either in a direct fashion, or incrementally within a counter-example guided abstraction refinement loop. Empirical results suggest that the former technique is more efficient if we want to synthesise time-optimal algorithms, while the latter technique discovers non-optimal algorithms more quickly. 1 ar X iv :1 30 4. 57 19 v2 [ cs .D C ] 5 J an 2 01 5

Towards Optimal Synchronous Counting

Consider a complete communication network of n nodes, in which the nodes receive a common clock pulse. We study the synchronous c-counting problem: given any starting state and up to f faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes count modulo c in agreement. Thus, we are considering algorithms that are self-stabilising despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have linear stabilisation time in f, (3) use a small number of states, and (4) achieve almost-optimal resilience. Prior algorithms either resort to randomisation, use a large number of states, or have poor resilience. In particular, we achieve an exponential improvement in the state complexity of deterministic algorithms, while still achieving linear stabilisation time and almost-linear resilience.

Exact Bounds for Distributed Graph Colouring

We prove exact bounds on the time complexity of distributed graph colouring. If we are given a directed path that is properly coloured with n colours, by prior work it is known that we can find a proper 3-colouring in

Near-Optimal Self-Stabilising Counting and Firing Squads

\frac{1}{2} \log^{*}n \pm O1

Self-Stabilising Byzantine Clock Synchronisation is Almost as Easy as Consensus

communication rounds. We close the gap between upper and lower bounds: we show that for infinitely many n the time complexity is precisely

Habitat fragmentation and species diversity in competitive species communities

\frac{1}{2} \log^{*} n