In this paper, we address the problem of uniformly deploying mobile robots in complete bipartite graphs. Specifically, when n robots are positioned arbitrarily at distinct nodes in a complete bipartite graph K_{n,n}, which consists of two n-node sets V_L and V_R, the uniform deployment problem requires the robots to achieve one of the following configurations: (a) each node in V_L is occupied by exactly one robot, with no robots in V_R, or (b) each node in V_R is occupied by exactly one robot, with no robots in V_L. In either configuration, the distance between any two robots is 2, ensuring that the robots are uniformly deployed. In this paper, we explore the relationship between the visibility range of robots and the solvability of the uniform deployment problem. First, we characterize solvable and unsolvable initial configurations under the assumption that robots have an infinite visibility range. Next, we demonstrate that visibility range 1 (meaning robots can only observe nodes at a distance of 1 and the robots positioned on them) is insufficient, proving the impossibility of solving the problem under this constraint. Conversely, we show that visibility range Θ(log n) is sufficient by presenting an algorithm that solves the uniform deployment problem in O(1) rounds, starting from any solvable initial configuration. Finally, we briefly introduce an example showing that robots with a constant visibility range (which is 3 in this example) cannot solve the problem in a native way.

Recently, Böckenhauer, Frei, Unger, and Wehner (SIROCCO 2023) introduced a novel variant of the graph exploration problem in which a single memoryless agent must visit all nodes of an unknown, undirected, and connected graph before returning to its starting node. Unlike the standard model for mobile agents, edges are not labeled with port numbers. Instead, the agent can color its current node and observe the color of each neighboring node. To move, it specifies a target color and then moves to an adversarially chosen neighbor of that color. Böckenhauer~et al.~analyzed the minimum number of colors required for successful exploration and proposed an elegant algorithm that enables the agent to explore an arbitrary graph using only eight colors. In this paper, we present a novel graph exploration algorithm that requires only six colors. Furthermore, we prove that five colors are sufficient if we consider only a restricted class of graphs, which we call the $φ$-free graphs, a class that includes every graph with maximum degree at most three and every cactus.

Abstract In this paper, we consider the partial gathering problem of mobile agents in dynamic tori. This problem requires $k$ agents distributed in the network to reach a configuration such that either at least $g$ agents or no agent exists at each node. Thus far, in dynamic graphs, partial gathering is considered in 1-interval connected rings, where one of the links in the ring may be missing at each time step. In this paper, we consider another dynamic topology. Concretely, we consider partial gathering in $n\times n$ dynamic tori such that each of the row and column rings is represented as a 1-interval connected ring. In such networks, when $k = O(gn)$, focusing on the relationship between the values of $k, n$, and $g$, we characterize the solvability of the problem and analyze the move complexity. First, we show that agents cannot solve the problem when $k = o(gn)$. Second, we show that agents can achieve partial gathering with a total number of $O(gn^{3})$ moves when $2gn+2n-1\le k \le 2gn + 6n +16g -12$. Finally, we show that agents can achieve partial gathering with a total number of $\Theta (gn^{2})$ moves when $k\ge 2gn + 6n +16g -11$.

The demand for executing neural network (NN) learning and inference on edge devices is increasing. Memristor crossbar (MC) devices in neuromorphic computing (NC) provide a promising solution for accelerating NNs. However, immature fabrication technology makes faulty memristors inevitable, and the presence of stuck-at faults (SAFs) in MC devices substantially reduces the inference accuracy of NC System. The existing fault-resilient methods, which depend on mapping and retraining algorithms, cause costly and time-consuming processes with hardware overhead. This paper proposes a reliability-aware design framework for an NC system utilizing MC devices by combining fault-resilient training and IC-specific mapping. First, we apply fault-resilient training to enhance the robustness of the NN model. SAF-injected training is developed in this phase to mitigate the effects of SAFs. Then we map the weights of the trained model to MCs. The proposed framework preserves the inference accuracy without re-training, even though MCs are suffered from SAFs. Our proposed method is assessed using two different NN applications across two distinct datasets. The experimental results show that the proposed framework achieves high inference accuracy for several differently faulty MCs and enhances the reliability of NC system.

The population protocol model is a computational model for passive mobile agents. We address the leader election problem, which determines a unique leader on arbitrary communication graphs starting from any configuration. Unfortunately, self-stabilizing leader election is impossible to be solved without knowing the exact number of agents; thus, we consider loosely-stabilizing leader election, which converges to safe configurations in a relatively short time, and holds the specification (maintains a unique leader) for a relatively long time. When agents have unique identifiers, Sudo et al.(2019) proposed a protocol that, given an upper bound $N$ for the number of agents $n$, converges in $O(mN\log n)$ expected steps, where $m$ is the number of edges. When unique identifiers are not required, they also proposed a protocol that, using random numbers and given $N$, converges in $O(mN^2\log{N})$ expected steps. Both protocols have a holding time of $Ω(e^{2N})$ expected steps and use $O(\log{N})$ bits of memory. They also showed that the lower bound of the convergence time is $Ω(mN)$ expected steps for protocols with a holding time of $Ω(e^N)$ expected steps given $N$. In this paper, we propose protocols that do not require unique identifiers. These protocols achieve convergence times close to the lower bound with increasing memory usage. Specifically, given $N$ and an upper bound $Δ$ for the maximum degree, we propose two protocols whose convergence times are $O(mN\log n)$ and $O(mN\log N)$ both in expectation and with high probability. The former protocol uses random numbers, while the latter does not require them. Both protocols utilize $O(Δ\log N)$ bits of memory and hold the specification for $Ω(e^{2N})$ expected steps.

We investigate crash-tolerant perpetual exploration algorithms by myopic luminous robots on ring networks. Myopic robots mean that they can observe nodes only within a certain fixed distance ϕ, and luminous robots mean that they have light devices that can emit a color from a set of colors. The goal of perpetual exploration is to ensure that robots, starting from specific initial positions and colors, move in such a way that every node is visited by at least one robot infinitely often. As a main contribution, we clarify the tight necessary and sufficient number of robots to realize perpetual exploration when at most f robots crash. In the fully synchronous model, we prove that f+2 robots are necessary and sufficient for any ϕ ≥ 1. In the semi-synchronous and asynchronous models, we prove that 3f+3 (resp., 2f+2) robots are necessary and sufficient if ϕ = 1 (resp., ϕ ≥ 2).

We consider the leader election problem in population protocol models. In pragmatic settings of population protocols, self-stabilization is a highly desired feature owing to its fault resilience and the benefit of initialization freedom. However, the design of self-stabilizing leader election is possible only under a strong assumption (i.e. the knowledge of the \emph{exact} size of a network) and rich computational resources (i.e. the number of states). Loose-stabilization, introduced by Sudo et al [Theoretical Computer Science, 2012], is a promising relaxed concept of self-stabilization to address the aforementioned issue. Loose-stabilization guarantees that starting from any configuration, the network will reach a safe configuration where a single leader exists within a short time, and thereafter it will maintain the single leader for a long time, but not forever. The main contribution of the paper is a time-optimal loosely-stabilizing leader election protocol. While the shortest convergence time achieved so far in loosely-stabilizing leader election is $O(\log^3 n)$ parallel time, the proposed protocol with design parameter $τ\ge 1$ attains $O(τ\log n)$ parallel convergence time and $Ω(n^τ)$ parallel holding time (i.e. the length of the period keeping the unique leader), both in expectation. This protocol is time-optimal in the sense of both the convergence and holding times in expectation because any loosely-stabilizing leader election protocol with the same length of the holding time is known to require $Ω(τ\log n)$ parallel time.

In the rendezvous problem, two computing entities (called agents) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous neighborhood rendezvous problem, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in O(Δ) rounds (Δ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in o(Δ) rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of O(√n) rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, and accessibility to neighborhood IDs. The first algorithm runs within Õ(√(nΔ/δ) + n/δ) rounds for graphs of the minimum degree larger than √n, where n is the number of vertices in the graph, and δ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is O(n), and achieves Õ(n/√δ)-round time complexity without using whiteboards. These algorithms attain o(Δ)-round complexity in the case of δ = ω(√n log n) and δ = ω(n2/3log4/3n) respectively. We also prove that three unconventional assumptions of our algorithm, bounded minimum degree, accessibility to neighborhood IDs, and initial distance one, are all inherently necessary for attaining fast rendezvous. That is, one can obtain the Ω(n)-round lower bound if either one of them is removed.

The coalescence protocol plays an important role in the population protocol model. The conceptual structure of the protocol is for two agents holding two non-zero values a, b respectively to take a transition (a,b) -> (a+b, 0), where + is an arbitrary commutative binary operation. Obviously, it eventually aggregates the sum of all initial values. In this paper, we present a fast coalescence protocol that converges in O(sqrt(n) log^2 n) parallel time with high probability in the model with an initial leader (equivalently, the model with a base station), which achieves an substantial speed-up compared with the naive implementation taking Omega(n) time.