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.

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 $k\ge 2gn + 6n +16g -11$.

An \emph{$α$-approximate vertex fault-tolerant distance sensitivity oracle} (\emph{$α$-VSDO}) for a weighted input graph $G=(V, E, w)$ and a source vertex $s \in V$ is the data structure answering an $α$-approximate distance from $s$ to $t$ in $G-x$ for any given query $(x, t) \in V \times V$. It is a data structure version of the so-called single-source replacement path problem (SSRP). In this paper, we present a new \emph{nearly linear-time} algorithm of constructing a $(1 + ε)$-VSDO for any directed input graph with polynomially bounded integer edge weights. More precisely, the presented oracle attains $\tilde{O}(m \log (nW)/ ε+ n \log^2 (nW)/ε^2)$ construction time, $\tilde{O}(n \log (nW) / ε)$ size, and $\tilde{O}(1/ε)$ query time, where $n$ is the number of vertices, $m$ is the number of edges, and $W$ is the maximum edge weight. These bounds are all optimal up to polylogarithmic factors. To the best of our knowledge, this is the first non-trivial algorithm for SSRP/VSDO beating $\tilde{O}(mn)$ computation time for directed graphs with general edge weight functions, and also the first nearly linear-time construction breaking approximation factor 3. Such a construction has been unknown even for undirected and unweighted graphs. In addition, our result implies that the known conditional lower bounds for the exact SSRP computation does not apply to the case of approximation.

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).

In this paper, we propose a randomized Õ(µ(G))-round algorithm for the maximum cardinality matching problem in the CONGEST model, where µ(G) means the maximum size of a matching of the input graph G. The proposed algorithm substantially improves the current best worst-case running time. The key technical ingredient is a new randomized algorithm of finding an augmenting path of length ℓ with high probability within Õ(ℓ) rounds, which positively settles an open problem left in the prior work by Ahmadi and Kuhn [DISC’20].

We consider global problems, i.e. problems that take at least diameter time, even when the bandwidth is not restricted. We show that all problems considered admit efficient solutions in low-treewidth graphs.

Abstract Distributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of $${\tilde{\Omega }}(\sqrt{n} + D)$$ Ω ~ ( n + D ) rounds for several global problems, where n denotes the number of nodes and D the diameter of the input graph. Because such a lower bound is derived from special “hard-core” instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts was initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. In particular, given a graph class $${\mathcal {C}}$$ C , an f-round algorithm for constructing shortcuts of quality q for any instance in $${\mathcal {C}}$$ C results in $${\tilde{O}}(q + f)$$ O ~ ( q + f ) -round algorithms for solving several fundamental graph problems such as minimum spanning tree and minimum cut, for $${\mathcal {C}}$$ C . The main interest on this line is to identify the graph classes allowing the shortcuts that are efficient in the sense of breaking $${\tilde{O}}(\sqrt{n}+D)$$ O ~ ( n + D ) -round general lower bounds. In this study, we consider the relationship between the quality of low-congestion shortcuts and the following four major graph parameters: doubling dimension, chordality, diameter, and clique-width. The key ingredient of the upper-bound side is a novel shortcut construction technique known as short-hop extension, which might be of independent interest.

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.

Pachinko is a Japanese mechanical gambling game similar to pinball. Recently, several mathematical models of Pachinko have been proposed. A number of pins are spiked in a field. A ball drops from the top of the playfield and the ball falls down. In the 50-50 model, if the ball hits a pin, it moves to the left or right passage of the pin with an equal probability. An arrangement of pins generates a distribution of the drop probability for all of the columns. This problem was considered by generating uniform distributions. Previous studies have demonstrated that the (1/2a)-uniform distribution is possible for a∈{0,1,2,3,4} and is conjectured so that it is possible for any positive integer a. This study describes the constructive proof for this conjecture. This study also formalizes a natural decision problem yielded by this model while investigating its computational complexity. More precisely, given any drop-probability distribution A and any partial drop-probability distribution B, this study uses non-deterministic polynomial-time (NP) hardness to determine if there exists a pin arrangement that transforms A into B.

Algorithmic meta-theorems, stating that graph properties expressible in some particular logic can be decided efficiently in graph classes having some specific structural properties, are now standard in sequential graph algorithms. One of the most classic examples is Courcelle's theorem: all properties expressible in Monadic Second-Order logic (MSO) are decidable in linear time in graphs of bounded treewidth. We provide here a distributed version of Courcelle's theorem, in the standard CONGEST model for distributed computing: For any MSO formula $φ$ and any constant $k$, there is a CONGEST algorithm that, given an input communication network $G$ of treewidth at most $k$ and of diameter $D$, decides if $G$ satisfies property $φ$ in $\tilde O(D)$ rounds. Simple examples show that the dependency on $D$ is unavoidable. Also, if we drop the assumption of bounded treewidth, deciding MSO properties such as 3-colorability are known to require $\tildeΩ(n^2)$ rounds in the CONGEST model. Our results extend to optimization problems (e.g., computing a maximum size independent set, or a minimum dominating set) and counting (e.g. triangle counting). As usual, the $\tilde{O}$ notation hides polylogarithmic factors in $n$; here it also hides a constant factor depending on $k$ and on the MSO formula $φ$. We also give a distributed algorithm producing a linear approximation for treewidth: For any $k$, it decides that the treewidth of the input network $G$ is larger than $k$ or computes a tree decomposition of width $O(k)$ and depth $O(\log n)$, in $\tilde O(k^{O(k)} D)$ rounds in CONGEST. Our algorithms make use of the low-congestion shortcuts framework introduced by Ghaffari and Haeupler [SODA 2016], and our main technical tool is an $\tilde O(k^4 D)$ algorithm for computing $(s,t)$-vertex separators of size at most $k+1$ in graphs of treewidth at most $k$.