arXiv subjects

Anisur Rahaman Molla

Publications and source records attributed to Anisur Rahaman Molla.

At least 19 recordsLinked to original sources

arXiv2026

Agent-Based Triangle Counting: Unlocking Truss Decomposition, Triangle Centrality, and Local Clustering Coefficient

In this paper, we study the problem of \emph{triangle counting} in an arbitrary anonymous graph $G$ with $n$ nodes and $m$ edges using the \emph{mobile-agent model}. Our triangle-counting method serves as a building block for solving related problems such as truss decomposition, triangle centrality, and local clustering coefficient computation. The agents operate synchronously, have distinct identifiers and limited memory, and communicate only when co-located. Starting from an arbitrary placement of $n$ agents, we first obtain a dispersed configuration, elect a leader, construct a spanning tree, and determine the maximum degree $Δ$ and maximum agent identifier $λ$. A BFS tree is constructed separately, which is needed for repeated global communication. Using this setup, the agents enumerate triangles and compute node- and edge-level triangle information, which is subsequently used for truss and centrality computations. We also complement the theoretical analysis with simulation-based evaluations on representative graph instances. Overall, our results establish a mobile-agent-based framework for these fundamental graph analytics problems in anonymous networks.

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arXiv2026

Semi-Synchronous Exploration in Dynamic Graphs

We study the fundamental problem of graph exploration in dynamic graphs using mobile agents. We consider $1$-interval connected dynamic graphs, where the topology may change arbitrarily from round to round as long as the graph remains connected, and edges are assigned with the dynamic port labeling at each round. The execution follows a semi-synchronous scheduler, under which an adversary may deactivate an arbitrary subset of agents in each round. For a graph with $n$ nodes and $k$ agents, we show that exploration is impossible if the adversary can deactivate at least $ \left\lceil \frac{k}{n-2} \right\rceil - 1$ agents per round, even when agents are equipped with unbounded memory, have global communication and full visibility. This yields an upper bound, implying that exploration is solvable only when the adversary deactivates at most $\left\lceil \frac{k}{n-2} \right\rceil - 2$ agents per round. We further establish that achieving exploration at this threshold requires agents to have both $1$-hop visibility and $1$-hop communication. Finally, we present the exploration algorithm using $k$ agents when the adversary deactivates at most $ \left\lceil \frac{k}{n-2} \right\rceil - 2$ agents, assuming agents are equipped with $1$-hop visibility and global communication, and matches the adversarial deactivation bound implied by the impossibility results.

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arXiv2026

Toward Optimality: A Tighter Analysis of Message Complexity for Leader Election in Diameter-Two Networks

We study the message complexity of leader election in synchronous networks of diameter two. Our main contribution is a refined analysis of the randomized algorithm proposed by Chatterjee et al. [DC, 2020]. In their work, the authors established a lower bound of $Ω(n)$ messages ($n$ is the number of nodes in the network) and presented a randomized algorithm that elects a leader in ${O}(1)$ rounds using $O(n \log^3 n)$ messages with high probability. In this paper, we improve their $\polylog n$ gap in the message bound by providing a tighter analysis of their algorithm, reducing the message complexity to $O(n\log n)$, while preserving the $O(1)$-round complexity and high-probability correctness guarantee.

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arXiv2025

Improved Linear-Time Construction of Minimal Dominating Set via Mobile Agents

Mobile agents have emerged as a powerful framework for solving fundamental graph problems in distributed settings in recent times. These agents, modelled as autonomous physical or software entities, possess local computation power, finite memory and have the ability to traverse a graph, offering efficient solutions to a range of classical problems. In this work, we focus on the problem of computing a \emph{minimal dominating set} (mDS) in anonymous graphs using mobile agents. Building on the recently proposed optimal dispersion algorithm on the synchronous mobile agent model, we design two new algorithms that achieve a \emph{linear-time} solution for this problem in the synchronous setting. Specifically, given a connected $n$-node graph with $n$ agents initially placed in either rooted or arbitrary configurations, we show that an mDS can be computed in $O(n)$ rounds using only $O(\log n)$ bits of memory per agent, without using any prior knowledge of any global parameters. This improves upon the best-known complexity results in the literature over the same model. In addition, as natural by-products of our methodology, our algorithms also construct a spanning tree and elect a unique leader in $O(n)$ rounds, which are also important results of independent interest in the mobile-agent framework.

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arXiv2025

Agentic Distributed Computing

The most celebrated and extensively studied model of distributed computing is the {\em message-passing model,} in which each vertex/node of the (distributed network) graph corresponds to a static computational device that communicates with other devices through passing messages. In this paper, we consider the {\em agentic model} of distributed computing which extends the message-passing model in a new direction. In the agentic model, computational devices are modeled as relocatable or mobile computational devices (called agents in this paper), i.e., each vertex/node of the graph serves as a container for the devices, and hence communicating with another device requires relocating to the same node. We study two fundamental graph level tasks, leader election, and minimum spanning tree, in the agentic model, which will enhance our understanding of distributed computation across paradigms. The objective is to minimize both time and memory complexities. Following the literature, we consider the synchronous setting in which each agent performs its operations synchronously with others, and hence the time complexity can be measured in rounds. In this paper, we present two deterministic algorithms for leader election: one for the case of $k<n$ and another for the case of $k=n$, minimizing both time and memory complexities, where $k$ and $n$, respectively, are the number of agents and number of nodes of the graph. Using these leader election results, we develop deterministic algorithms for agents to construct a minimum spanning tree of the graph, minimizing both time and memory complexities. To the best of our knowledge, this is the first study of distributed graph level tasks in the agentic model with $k\leq n$. Previous studies only considered the case of $k=n$.

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arXiv2025

Optimal Dispersion Under Asynchrony

We study the dispersion problem in anonymous port-labeled graphs: $k \leq n$ mobile agents, each with a unique ID and initially located arbitrarily on the nodes of an $n$-node graph with maximum degree $Δ$, must autonomously relocate so that no node hosts more than one agent. Dispersion serves as a fundamental task in distributed computing of mobile agents, and its complexity stems from key challenges in local coordination under anonymity and limited memory. The goal is to minimize both the time to achieve dispersion and the memory required per agent. It is known that any algorithm requires $Ω(k)$ time in the worst case, and $Ω(\log k)$ bits of memory per agent. A recent result [SPAA'25] gives an optimal $O(k)$-time algorithm in the synchronous setting and an $O(k \log k)$-time algorithm in the asynchronous setting, both using $O(\log(k+Δ))$ bits. In this paper, we close the complexity gap in the asynchronous setting by presenting the first dispersion algorithm that runs in optimal $O(k)$ time using $O(\log(k+Δ))$ bits of memory per agent. Our solution is based on a novel technique we develop in this paper that constructs a port-one tree in anonymous graphs, which may be of independent interest.

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arXiv2025

Computing Tree Structures in Anonymous Graphs via Mobile Agents

Minimum Spanning Tree (MST) and Breadth-First Search (BFS) tree constructions are classical problems in distributed computing, traditionally studied in the message-passing model, where static nodes communicate via messages. This paper investigates MST and BFS tree construction in an agent-based network, where mobile agents explore a graph and compute. Each node hosts one agent, and communication occurs when agents meet at a node. We consider $n$ agents initially dispersed (one per node) in an anonymous, arbitrary $n$-node, $m$-edge graph $G$. The goal is to construct the BFS and MST trees from this configuration such that each tree edge is known to at least one of its endpoints, while minimizing time and memory per agent. We work in a synchronous model and assume agents have no prior knowledge of any graph parameters such as $n$, $m$, $D$, $Δ$ (graph diameter and maximum degree). Prior work solves BFS in $O(DΔ)$ rounds with $O(\log n)$ bits per agent, assuming the root is known. We give a deterministic algorithm that constructs the BFS tree in $O(\min(DΔ, m\log n) + n\log n + Δ\log^2 n)$ rounds using $O(\log n)$ bits per agent without root knowledge. To determine the root, we solve leader election and MST construction. We elect a leader and construct the MST in $O(n\log n + Δ\log^2 n)$ rounds, with $O(\log n)$ bits per agent. Prior MST algorithms require $O(m + n\log n)$ rounds and $\max(Δ, \log n) \log n$ bits. Our results significantly improve memory efficiency and time, achieving nearly linear-time leader election and MST. Agents are assumed to know $λ$, the maximum identifier, bounded by a polynomial in $n$.

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arXiv2025

Distributed Butterfly Analysis using Mobile Agents

Butterflies, or 4-cycles in bipartite graphs, are crucial for identifying cohesive structures and dense subgraphs. While agent-based data mining is gaining prominence, its application to bipartite networks remains relatively unexplored. We propose distributed, agent-based algorithms for \emph{Butterfly Counting} in a bipartite graph $G((A,B),E)$. Agents first determine their respective partitions and collaboratively construct a spanning tree, electing a leader within $O(n \log λ)$ rounds using only $O(\log λ)$ bits per agent. A novel meeting mechanism between adjacent agents improves efficiency and eliminates the need for prior knowledge of the graph, requiring only the highest agent ID $λ$ among the $n$ agents. Notably, our techniques naturally extend to general graphs, where leader election and spanning tree construction maintain the same round and memory complexities. Building on these foundations, agents count butterflies per node in $O(Δ)$ rounds and compute the total butterfly count of $G$ in $O(Δ+\min\{|A|,|B|\})$ rounds.

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arXiv2025

Dispersion is (Almost) Optimal under (A)synchrony

The dispersion problem has received much attention recently in the distributed computing literature. In this problem, $k\leq n$ agents placed initially arbitrarily on the nodes of an $n$-node, $m$-edge anonymous graph of maximum degree $Δ$ have to reposition autonomously to reach a configuration in which each agent is on a distinct node of the graph. Dispersion is interesting as well as important due to its connections to many fundamental coordination problems by mobile agents on graphs, such as exploration, scattering, load balancing, relocation of self-driven electric cars (robots) to recharge stations (nodes), etc. The objective has been to provide a solution that optimizes simultaneously time and memory complexities. There exist graphs for which the lower bound on time complexity is $Ω(k)$. Memory complexity is $Ω(\log k)$ per agent independent of graph topology. The state-of-the-art algorithms have (i) time complexity $O(k\log^2k)$ and memory complexity $O(\log(k+Δ))$ under the synchronous setting [DISC'24] and (ii) time complexity $O(\min\{m,kΔ\})$ and memory complexity $O(\log(k+Δ))$ under the asynchronous setting [OPODIS'21]. In this paper, we improve substantially on this state-of-the-art. Under the synchronous setting as in [DISC'24], we present the first optimal $O(k)$ time algorithm keeping memory complexity $O(\log (k+Δ))$. Under the asynchronous setting as in [OPODIS'21], we present the first algorithm with time complexity $O(k\log k)$ keeping memory complexity $O(\log (k+Δ))$, which is time-optimal within an $O(\log k)$ factor despite asynchrony. Both results were obtained through novel techniques to quickly find empty nodes to settle agents, which may be of independent interest.

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arXiv2024

Optimal Fault-Tolerant Dispersion on Oriented Grids

Dispersion of mobile robots over the nodes of an anonymous graph is an important problem and turns out to be a crucial subroutine for designing efficient algorithms for many fundamental graph problems via mobile robots. In this problem, starting from an arbitrary initial distribution of $n$ robots across the $n$ nodes, the goal is to achieve a final configuration where each node holds at most one robot. This paper investigates the dispersion problem on an oriented grid, considering the possibility of robot failures (crashes) at any time during the algorithm's execution. We present a crash-tolerant dispersion algorithm that solves the dispersion problem on an anonymous oriented grid in $O(\sqrt{n})$ time and using $O(\log n)$ bits of memory per robot. The algorithm is optimal in terms of both time and memory per robot. We further extend this algorithm to deal with weak Byzantine robots. The weak Byzantine fault dispersion algorithm takes optimal $O(\sqrt{n})$ rounds but requires $O(n\log n)$ bits of memory per robot.

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arXiv2024

Agent-based Leader Election, MST, and Beyond

Leader election is one of the fundamental and well-studied problems in distributed computing. In this paper, we initiate the study of leader election using mobile agents. Suppose $n$ agents are positioned initially arbitrarily on the nodes of an arbitrary, anonymous, $n$-node, $m$-edge graph $G$. The agents relocate themselves autonomously on the nodes of $G$ and elect an agent as a leader such that the leader agent knows it is a leader and the other agents know they are not leaders. The objective is to minimize time and memory requirements. Following the literature, we consider the synchronous setting in which each agent performs its operations synchronously with others and hence the time complexity can be measured in rounds. The quest in this paper is to provide solutions without agents knowing any graph parameter, such as $n$, a priori. We first establish that, without agents knowing any graph parameter a priori, there exists a deterministic algorithm to elect an agent as a leader in $O(m)$ rounds with $O(n\log n)$ bits at each agent. Using this leader election result, we develop a deterministic algorithm for agents to construct a minimum spanning tree of $G$ in $O(m+n\log n)$ rounds using $O(n \log n)$ bits memory at each agent, without agents knowing any graph parameter a priori. Finally, using the same leader election result, we provide improved time/memory results for other fundamental distributed graph problems, namely, gathering, maximal independent set, and minimal dominating sets, removing the assumptions on agents knowing graph parameters a priori.

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arXiv2024

Optimizing Robot Dispersion on Grids: with and without Fault Tolerance

The introduction and study of dispersing mobile robots across the nodes of an anonymous graph have recently gained traction and have been explored within various graph classes and settings. While optimal dispersion solution was established for {\em oriented} grids [Kshemkalyani et al., WALCOM 2020], a significant unresolved question pertains to whether achieving optimal dispersion is feasible on an {\em unoriented} grid. This paper investigates the dispersion problem on unoriented grids, considering both non-faulty and faulty robots. The challenge posed by unoriented grids lies in the absence of a clear sense of direction for a single robot moving between nodes, as opposed to the straightforward navigation of oriented grids. We present three deterministic algorithms tailored to our robot model. The first and second algorithms deal with the dispersion of faulty and non-faulty robots, ensuring both time and memory optimization in oriented and unoriented grids, respectively. Faulty robots that are prone to crashing at any time, causing permanent failure. In both settings, we achieve dispersion in $O(\sqrt{n})$ rounds while requiring $O(\log n)$ bits of memory per robot. The third algorithm tackles faulty robots prone to crash faults in an unoriented grid. In this scenario, our algorithm operates within $O(\sqrt{n} \log n)$ time and uses $O(\sqrt{n} \log n)$ bits of memory per robot. The robots need to know the value of $n$ for termination.

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arXiv2023

Run for Cover: Dominating Set via Mobile Agents

Research involving computing with mobile agents is a fast-growing field, given the advancement of technology in automated systems, e.g., robots, drones, self-driving cars, etc. Therefore, it is pressing to focus on solving classical network problems using mobile agents. In this paper, we study one such problem -- finding small dominating sets of a graph $G$ using mobile agents. Dominating set is interesting in the field of mobile agents as it opens up a way for solving various robotic problems, e.g., guarding, covering, facility location, transport routing, etc. In this paper, we first present two algorithms for computing a {\em minimal dominating set}: (i) an $O(m)$ time algorithm if the robots start from a single node (i.e., gathered initially), (ii) an $O(\ellΔ\log(λ)+n\ell+m)$ time algorithm, if the robots start from multiple nodes (i.e., positioned arbitrarily), where $m$ is the number of edges and $Δ$ is the maximum degree of $G$, $\ell$ is the number of clusters of the robot initially and $λ$ is the maximum ID-length of the robots. Then we present a $\ln (Δ)$ approximation algorithm for the {\em minimum} dominating set which takes $O(nΔ\log (λ))$ rounds.

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arXiv2023

Sublinear Message Bounds of Authenticated Implicit Byzantine Agreement

This paper studies the message complexity of authenticated Byzantine agreement (BA) in synchronous, fully-connected distributed networks under an honest majority. We focus on the so-called {\em implicit} Byzantine agreement problem where each node starts with an input value and at the end a non-empty subset of the honest nodes should agree on a common input value by satisfying the BA properties (i.e., there can be undecided nodes). We show that a sublinear (in $n$, number of nodes) message complexity BA protocol under honest majority is possible in the standard PKI model when the nodes have access to an unbiased global coin and hash function. In particular, we present a randomized Byzantine agreement algorithm which, with high probability achieves implicit agreement, uses $\tilde{O}(\sqrt{n})$ messages, and runs in $\tilde{O}(1)$ rounds while tolerating $(1/2 - ε)n$ Byzantine nodes for any fixed $ε> 0$, the notation $\Tilde{O}$ hides a $O(\polylog{n})$ factor. The algorithm requires standard cryptographic setup PKI and hash function with a static Byzantine adversary. The algorithm works in the CONGEST model and each node does not need to know the identity of its neighbors, i.e., works in the $KT_0$ model. The message complexity (and also the time complexity) of our algorithm is optimal up to a $\polylog n$ factor, as we show a $Ω(\sqrt{n})$ lower bound on the message complexity.

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arXiv2023

Fast Deterministic Gathering with Detection on Arbitrary Graphs: The Power of Many Robots

Over the years, much research involving mobile computational entities has been performed. From modeling actual microscopic (and smaller) robots, to modeling software processes on a network, many important problems have been studied in this context. Gathering is one such fundamental problem in this area. The problem of gathering $k$ robots, initially arbitrarily placed on the nodes of an $n$-node graph, asks that these robots coordinate and communicate in a local manner, as opposed to global, to move around the graph, find each other, and settle down on a single node as fast as possible. A more difficult problem to solve is gathering with detection, where once the robots gather, they must subsequently realize that gathering has occurred and then terminate. In this paper, we propose a deterministic approach to solve gathering with detection for any arbitrary connected graph that is faster than existing deterministic solutions for even just gathering (without the requirement of detection) for arbitrary graphs. In contrast to earlier work on gathering, it leverages the fact that there are more robots present in the system to achieve gathering with detection faster than those previous papers that focused on just gathering. The state of the art solution for deterministic gathering~[Ta-Shma and Zwick, TALG, 2014] takes $\Tilde{O}$$(n^5 \log \ell)$ rounds, where $\ell$ is the smallest label among robots and $\Tilde{O}$ hides a polylog factor. We design a deterministic algorithm for gathering with detection with the following trade-offs depending on how many robots are present: (i) when $k \geq \lfloor n/2 \rfloor + 1$, the algorithm takes $O(n^3)$ rounds, (ii) when $k \geq \lfloor n/3 \rfloor + 1$, the algorithm takes $O(n^4 \log n)$ rounds, and (iii) otherwise, the algorithm takes $\Tilde{O}$$(n^5)$ rounds. The algorithm is not required to know $k$, but only $n$.

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arXiv2023

Improved Deterministic Leader Election in Diameter-Two Networks

In this paper, we investigate the leader election problem in diameter-two networks. Recently, Chatterjee et al. [DC 2020] studied the leader election in diameter-two networks. They presented a $O(\log n)$-round deterministic {implicit} leader election algorithm which incurs optimal $O(n\log n)$ messages, but a drawback of their algorithm is that it requires knowledge of $n$. An important question -- whether it is possible to remove the assumption on the knowledge of $n$ was left open in their paper. Another interesting open question raised in their paper is whether {\em explicit} leader election can be solved in $\tilde{O}(n)$ messages deterministically. In this paper, we give an affirmative answer to them. Further, we solve the {\em broadcast problem}, another fundamental problem in distributed computing, deterministically in diameter-two networks with $\tilde{O}(n)$ messages and $\tilde{O}(1)$ rounds without the knowledge of $n$. In fact, we address all the open questions raised by Chatterjee et al. for the deterministic leader election problem in diameter-two networks. To the best of our knowledge, this is the first $\tilde{O}(n)$ deterministic result for the explicit leader election in the diameter-two networks, that too without the knowledge of $n$.

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arXiv2022

Fault-Tolerant Dispersion of Mobile Robots

We consider the mobile robot dispersion problem in the presence of faulty robots (crash-fault). Mobile robot dispersion consists of $k\leq n$ robots in an $n$-node anonymous graph. The goal is to ensure that regardless of the initial placement of the robots over the nodes, the final configuration consists of having at most one robot at each node. In a crash-fault setting, up to $f \leq k$ robots may fail by crashing arbitrarily and subsequently lose all the information stored at the robots, rendering them unable to communicate. In this paper, we solve the dispersion problem in a crash-fault setting by considering two different initial configurations: i) the rooted configuration, and ii) the arbitrary configuration. In the rooted case, all robots are placed together at a single node at the start. The arbitrary configuration is a general configuration (a.k.a. arbitrary configuration in the literature) where the robots are placed in some $l<k$ clusters arbitrarily across the graph. For the first case, we develop an algorithm solving dispersion in the presence of faulty robots in $O(k^2)$ rounds, which improves over the previous $O(f\cdot\text{min}(m,kΔ))$-round result by \cite{PS021}. For the arbitrary configuration, we present an algorithm solving dispersion in $O((f+l)\cdot\text{min}(m, k Δ, k^2))$ rounds, when the number of edges $m$ and the maximum degree $Δ$ of the graph is known to the robots.

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arXiv2022

Fault-Tolerant Graph Realizations in the Congested Clique

In this paper, we study the graph realization problem in the Congested Clique model of distributed computing under crash faults. We consider {\em degree-sequence realization}, in which each node $v$ is associated with a degree value $d(v)$, and the resulting degree sequence is realizable if it is possible to construct an overlay network with the given degrees. Our main result is a $O(f)$-round deterministic algorithm for the degree-sequence realization problem in a $n$-node Congested Clique, of which $f$ nodes could be faulty ($f<n$). The algorithm uses $O(n^2)$ messages. We complement the result with lower bounds to show that the algorithm is tight w.r.t the number of rounds and the messages simultaneously. We also extend our result to the Node Capacitated Clique (NCC) model, where each node is restricted to sending and receiving at-most $O(\log n)$ messages per round. In the NCC model, our algorithm solves degree-sequence realization in $O(nf/\log n)$ rounds and $O(n^2)$ messages. For both settings, our algorithms work without the knowledge of $f$, the number of faults. To the best of our knowledge, these are the first results for the graph realization problem in the crash-fault distributed network.

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