arXiv subjects

Maria Kokkou

Publications and source records attributed to Maria Kokkou.

6 recordsLinked to original sources

arXiv2026

Distance-based certification for leader election in meshed graphs and local recognition of their subclasses

In this paper, we present a 2-local proof labeling scheme with labels in $\{ 0,1,2\}$ for leader election in anonymous meshed graphs. Meshed graphs form a general class of graphs defined by a distance condition. They comprise several important classes of graphs, which have long been the subject of intensive studies in metric graph theory, geometric group theory, and discrete mathematics: median graphs, bridged graphs, chordal graphs, Helly graphs, dual polar graphs, modular, weakly modular graphs, and basis graphs of matroids. We also provide 3-local proof labeling schemes to recognize these subclasses of meshed graphs using labels of size $O(\log D)$ (where $D$ is the diameter of the graph). To establish these results, we show that in meshed graphs, we can verify locally that every vertex $v$ is labeled by its distance $d(s,v)$ to an arbitrary root $s$. To design proof labeling schemes to recognize the subclasses of meshed graphs mentioned above, we use this distance verification to ensure that the triangle-square complex of the graph is simply connected and we then rely on existing local-to-global characterizations for the different classes we consider. To get a proof-labeling scheme for leader election with labels of constant size, we then show that we can check locally if every $v$ is labeled by $d(s,v) \pmod{3}$ for some root $s$ that we designate as the leader.

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arXiv2025

Leader Election via Unique Sink Orientation

A Locally Checkable Labeling (LCL) is a distributed constraint satisfaction problem defined on a bounded-degree graph that relates a finite set of input labels to a finite set of output labels through a finite set of locally checkable constraints. In this work we define labels and local constraints that encode solutions to two classical problems: leader election and spanning tree construction. It is known that leader election cannot be expressed as an LCL in arbitrary graphs using constant-size labels. In fact, it is known that there does not exist a finite set of labels and local constraints for leader election even for the class of rings. On the other hand, there exists a finite set of labels and local constraints characterizing leader election on trees. In this work, we prove that there exists a finite set of labels and local constraints for leader election also in the much larger class of dismantlable graphs. Our labels need one bit per edge or equivalently $O(\Delta)$ bits per node (where $\Delta$ is the maximum degree in the graph) and are checkable within the graph induced by the 1-neighborhood of each node. To the best of our knowledge, these are the first local labeling results tailored to dismantlable graphs, potentially highlighting structural properties useful for designing labels and constraints for additional LCL problems. Finally, we present a generic transformation that converts any finite set of labels and local constraints into a silent self-stabilizing algorithm by adding only one extra state, assuming a Gouda fair scheduler. This transformation may be of independent interest.

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arXiv2025

Silent Self-Stabilising Leader Election in Programmable Matter Systems with Holes

Leader election is a fundamental problem in distributed computing, particularly within programmable matter systems, where coordination among simple computational entities is crucial for solving complex tasks. In these systems, particles (i.e., constant-memory computational entities) operate in a regular triangular grid as described in the geometric Amoebot model. While leader election has been extensively studied in non self-stabilising settings, self-stabilising solutions remain more limited. In this work, we study the problem of self-stabilising leader election in connected (but not necessarily simply connected) configurations. We present the first self-stabilising algorithm for connected programmable matter systems that guarantees the election of a unique leader under an unfair scheduler, for oblivious particles (i.e., particles with no persistent memory) that share a common sense of direction. Our approach leverages particle movement, a capability not previously exploited in the self-stabilising context. We show that movement in conjunction with particles sharing a sense of orientation and operating in a grid can overcome classical impossibility results for constant-memory systems established by Dolev, Gouda and Schneider (1999).

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arXiv2025

Finite Pinwheel Scheduling: the k-Visits Problem

Pinwheel Scheduling is a fundamental scheduling problem, in which each task $i$ is associated with a positive integer $d_i$, and the objective is to schedule one task per time slot, ensuring each task perpetually appears at least once in every $d_i$ time slots. Although conjectured to be PSPACE-complete, the complexity of Pinwheel Scheduling remains open. We introduce k-Visits, a finite version of Pinwheel Scheduling, where given n deadlines, the goal is to schedule each task exactly k times. While we observe that the 1-Visit problem is trivial, we prove that 2-Visits is strongly NP-complete through a surprising reduction from Numerical 3-Dimensional Matching (N3DM). As intermediate steps in the reduction, we define NP-complete variants of N3DM which may be of independent interest. We further extend our strong NP-hardness result to a generalization of k-Visits $k\geq 2$ in which the deadline of each task may vary throughout the schedule, as well as to a similar generalization of Pinwheel Scheduling, thus making progress towards settling the complexity of Pinwheel Scheduling. Additionally, we prove that 2-Visits can be solved in linear time if all deadlines are distinct, rendering it one of the rare natural problems which exhibit the interesting dichotomy of being in P if their input is a set and NP-complete if the input is a multiset. We achieve this through a Turing reduction from 2-Visits to a variation of N3DM, which we call Position Matching. Based on this reduction, we also show an FPT algorithm for 2-Visits parameterized by a value related to how close the input deadlines are to each other, as well as a linear-time algorithm for instances with up to two distinct deadlines.

cs.DS↗

arXiv2024

Deterministic Self-Stabilising Leader Election for Programmable Matter with Constant Memory

The problem of electing a unique leader is central to all distributed systems, including programmable matter systems where particles have constant size memory. In this paper, we present a silent self-stabilising, deterministic, stationary, election algorithm for particles having constant memory, assuming that the system is simply connected. Our algorithm is elegant and simple, and requires constant memory per particle. We prove that our algorithm always stabilises to a configuration with a unique leader, under a daemon satisfying some fairness guarantees (Gouda fairness [Gouda 2001]). We use the special geometric properties of programmable matter in 2D triangular grids to obtain the first self-stabilising algorithm for such systems. This result is surprising since it is known that silent self-stabilising algorithms for election in general distributed networks require $\Omega(\log{n})$ bits of memory per node, even for ring topologies [Dolev et al. 1999].

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arXiv2024

Deterministic Leader Election for Stationary Programmable Matter with Common Direction

Leader Election is an important primitive for programmable matter, since it is often an intermediate step for the solution of more complex problems. Although the leader election problem itself is well studied even in the specific context of programmable matter systems, research on fault tolerant approaches is more limited. We consider the problem in the previously studied Amoebot model on a triangular grid, when the configuration is connected but contains nodes the particles cannot move to (e.g., obstacles). We assume that particles agree on a common direction (i.e., the horizontal axis) but do not have chirality (i.e., they do not agree on the other two directions of the triangular grid). We begin by showing that an election algorithm with explicit termination is not possible in this case, but we provide an implicitly terminating algorithm that elects a unique leader without requiring any movement. These results are in contrast to those in the more common model with chirality but no agreement on directions, where explicit termination is always possible but the number of elected leaders depends on the symmetry of the initial configuration. Solving the problem under the assumption of one common direction allows for a unique leader to be elected in a stationary and deterministic way, which until now was only possible for simply connected configurations under a sequential scheduler.

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