arXiv subjects

Kaushik Mondal

Publications and source records attributed to Kaushik Mondal.

At least 19 recordsLinked to original sources

arXiv2026

Cops and Lethal Robber on Rings: A Distributed Perspective

The Cops and Robber game is extensively studied in the sequential setting where the main goal is to capture the robber. Capturing the robber means at least one cop, and the robber will be at the same vertex together at some time. There are several variants, including variants where the goal of the cops is to surround the robber. Surrounding the robber means there is at least one cop in each of the neighboring vertices of the robber's position. In this paper, we introduce it in the distributed setting while empowering the robber by saying it can even kill cops. Specifically, in our model, the robber moves in odd rounds and has unbounded speed, cops move in even rounds, and if one or more cops move into a vertex where the robber is currently residing, all these cops get killed. We call this {\it lethal robber}. This also connects our work to the Intruder Capture and Black Hole Search problems by introducing an entity which is dynamic as well as lethal, a setting that, to the best of our knowledge, has not been studied. In this work, we introduce the lethal robber, define the {\it cops and lethal robber} problem in the distributed setting and study it on a static ring of size $n$. We prove $n$ cops are not enough, even if all start from the same vertex, and provide an algorithm starting from an arbitrary initial configuration that requires $n+\lfloor\log n \rfloor+4$ cops in the worst case.

cs.DC↗

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.

cs.DC↗

arXiv2026

Black Hole Search: Dynamics, Distribution, and Emergence

A black hole is a malicious node in a graph that destroys resources entering into it without leaving any trace. The problem of Black Hole Search (BHS) using mobile agents requires that at least one agent survives and terminates after locating the black hole. Recently, this problem has been studied on 1-bounded 1-interval connected dynamic graphs \cite{BHS_gen}, where there is a footprint graph, and at most one edge can disappear from the footprint in a round, provided that the graph remains connected. In this setting, the authors in \cite{BHS_gen} proposed an algorithm that solves the BHS problem when all agents start from a single node (rooted initial configuration). They also proved that at least $2δ_{BH} + 1$ agents are necessary to solve the problem when agents are initially placed arbitrarily across the nodes of the graph (scattered initial configuration), where $δ_{BH}$ denotes the degree of the black hole. In this work, we present an algorithm that solves the BHS problem using $2δ_{BH} + 17$ initially scattered agents. Our result matches asymptotically with the rooted algorithm of \cite{BHS_gen} under the same model assumptions. Further, we study the Eventual Black Hole Search (\textsc{Ebhs}) problem, in which the black hole may appear at any node and at any time during the execution of the algorithm, destroying all agents located on that node at the time of its appearance. However, the black hole cannot emerge at the home base in round~0, where the home base is the node at which all agents are initially co-located. Once the black hole appears, it remains active at that node for the rest of the execution. This problem has been studied on static rings~\cite{Bonnet25}; here we extend it to arbitrary static graphs and provide a solution using four agents. Moreover, it does not require any knowledge of global parameters or additional model assumptions.

cs.DC↗

arXiv2026

Exploration on Highly Dynamic Graphs

We study the exploration problem by mobile agents in two prominent models of dynamic graphs: $1$-Interval Connectivity and Connectivity Time. The $1$-Interval Connectivity model was introduced by Kuhn et al.~[STOC 2010], and the Connectivity Time model was proposed by Michail et al.~[JPDC 2014]. Recently, Saxena et al.~[TCS 2025] investigated the exploration problem under both models. In this work, we first strengthen the existing impossibility results for the $1$-Interval Connectivity model. We then show that, in Connectivity Time dynamic graphs, exploration is impossible with $\frac{(n-1)(n-2)}{2}$ mobile agents, even when the agents have full knowledge of all system parameters, global communication, full visibility, and infinite memory. This significantly improves the previously known bound of $n$. Moreover, we prove that to solve exploration with $\frac{(n-1)(n-2)}{2}+1$ agents, $1$-hop visibility is necessary. Finally, we present an exploration algorithm that uses $\frac{(n-1)(n-2)}{2}+1$ agents, assuming global communication, $1$-hop visibility, and $O(\log n)$ memory per agent.

cs.DC↗

arXiv2025

Black Hole Search in Dynamic Graphs

A black hole is considered to be a dangerous node present in a graph that disposes of any resources that enter that node. Therefore, it is essential to find such a node in the graph. Let a group of agents be present on a graph $G$. The Black Hole Search (BHS) problem aims for at least one agent to survive and terminate after {finding} the black hole. This problem is already studied for specific dynamic graph classes such as rings, cactuses, and tori {where finding the black hole means at least one agent needs to survive and terminate after knowing at least one edge associated with the black hole. In this work, we investigate the problem of BHS for general graphs.} In the dynamic graph, adversary may remove edges at each round keeping the graph connected. We consider two cases: (a) at any round at most one edge can be removed (b) at any round at most $f$ edges can be removed. For both scenarios, we study the problem when the agents start from a rooted initial configuration. We consider each agent has $O(\log n)$ memory and each node has $O(\log n)$ storage. For case (a), we present an algorithm with $9$ agents that solves the problem of BHS in $O(|E|^2)$ time where $|E|$ is the number of edges and $δ_v$ is the degree of the node $v$ in $G$. We show it is impossible to solve for $2δ_{BH}$ many agents starting from an arbitrary configuration where $δ_{BH}$ is the degree of the black hole in $G$. We also provide another improved algorithm that uses $6$ agents from a rooted initial configuration to solve the problem of BHS. For case (b), we provide an algorithm using $6f$ agents to solve the problem of BHS, albeit taking exponential time. We also provide an impossibility result for $2f+1$ agents starting from a rooted initial configuration. This result holds even if unlimited storage is available on each node and the agents have infinite memory.

cs.DC↗

arXiv2025

Balanced Dispersion on Time-Varying Dynamic Graphs

We aim to connect two problems, namely, dispersion and load balancing. Both problems have already been studied over static as well as dynamic graphs. Though dispersion and load balancing share some common features, the tools used in solving load balancing differ significantly from those used in solving dispersion. One of the reasons is that the load balancing problem is introduced and studied heavily over graphs where nodes are the processors and work under the message passing model, whereas dispersion is a task for mobile agents to achieve on graphs. To bring the (load) balancing aspect in the dispersion problem, we say, mobile agents move to balance themselves as equally as possible across the nodes of the graph, instead of stationary nodes sharing loads in the load balancing problem. We call it the \emph{$k$-balanced dispersion} problem and study it on dynamic graphs. This is equivalent to the load balancing problem considering movable loads in form of the agents. Earlier, on static graphs, the \emph{$k$-dispersion} problem [TAMC 2019] aimed for the same by putting an upper bound on the number of agents on each node in the final configuration; however, the absence of a lower bound on the number of agents in their problem definition hampers the load-balancing aspect, as some nodes may end up with no agents in the final configuration. We take care of this part in our \emph{$k$-balanced dispersion} problem definition and thus produce a stronger connection between the two domains.

cs.DC↗

arXiv2025

Path Connected Dynamic Graphs with a Study of Dispersion and Exploration

In dynamic graphs, edges may be added or deleted in each synchronous round. Various connectivity models exist based on constraints on these changes. One well-known model is the $T$-Interval Connectivity model, where the graph remains connected in every round, and the parameter $T$ reflects the duration of structural stability. Another model is Connectivity Time, where the union of edges across any $T$ consecutive rounds forms a connected graph. This is a weaker model, as the graph may be disconnected in individual rounds. In this work, we introduce a new connectivity model called $T$-Path Connectivity. Unlike $T$-Interval Connectivity, the graph may not be connected in each round, but for every pair of nodes $u,v$, there must exist a path connecting them in at least one round within any $T$ consecutive rounds. This model is strictly weaker than $T$-Interval Connectivity but stronger than the Connectivity Time model. We study the dispersion problem in the $T$-Path Connectivity model. While dispersion has been explored in the 1-Interval Connectivity model, we show that the existing algorithm with termination does not work in our model. We then identify the minimal necessary assumptions required to solve dispersion in the $T$-Path Connectivity model and provide an algorithm that solves it optimally under those conditions. Moreover, we prove that dispersion is unsolvable in the Connectivity Time model, even under several strong assumptions. We further initiate the study of the exploration problem under all three connectivity models. We present multiple impossibility results and, in most cases, establish tight bounds on the number of agents and time required. Our results demonstrate that, in both dispersion and exploration, the Connectivity Time model is strictly the weakest among the three.

cs.DC↗

arXiv2024

Efficient D-2-D via Leader Election: Arbitrary Initial Configuration and No Global Knowledge

Distance-2-Dispersion (D-2-D) problem aims to disperse $k$ mobile agents starting from an arbitrary initial configuration on an anonymous port-labeled graph $G$ with $n$ nodes such that no two agents occupy adjacent nodes in the final configuration, though multiple agents may occupy a single node if there is no other empty node whose all adjacent nodes are also empty. In the existing literature, this problem is solved starting from a rooted configuration for $k$ $(\geq 1)$ agents in $O(mΔ)$ synchronous rounds with a total of $O(\log n)$ memory per agent, where $m$ is the number of edges and $Δ$ is the maximum degree of the graph. In this work we study the D-2-D problem using $n$ mobile agents starting from an arbitrary initial configuration. Solving D-2-D with $n$ agents is equivalent to finding a maximal independent set of the graph as size of any maximal independent set must be less than $n$. We solve this problem and terminate in $O(max\{n\log^2 n, m\})$ rounds using $O(\log n)$ memory per agent. The agents do not have any prior knowledge of any graph parameters. During the run of our algorithm, we also solve the leader election problem that elects an agent as a leader in $O(max\{n\log^2 n, m\})$ rounds with $O(\log n)$ bits of memory at each agent without requiring any prior global knowledge.

cs.DC↗

arXiv2024

Optimal Dispersion of Silent Robots in a Ring

Given a set of co-located mobile robots in an unknown anonymous graph, the robots must relocate themselves in distinct graph nodes to solve the dispersion problem. In this paper, we consider the dispersion problem for silent robots \cite{gorain2024collaborative}, i.e., no direct, explicit communication between any two robots placed in the nodes of an oriented $n$ node ring network. The robots operate in synchronous rounds. The dispersion problem for silent mobile robots has been studied in arbitrary graphs where the robots start from a single source. In this paper, we focus on the dispersion problem for silent mobile robots where robots can start from multiple sources. The robots have unique labels from a range $[0,\;L]$ for some positive integer $L$. Any two co-located robots do not have the information about the label of the other robot. The robots have weak multiplicity detection capability, which means they can determine if it is alone on a node. The robots are assumed to be able to identify an increase or decrease in the number of robots present on a node in a particular round. However, the robots can not get the exact number of increase or decrease in the number of robots. We have proposed a deterministic distributed algorithm that solves the dispersion of $k$ robots in an oriented ring in $O(\log L+k)$ synchronous rounds with $O(\log L)$ bits of memory for each robot. A lower bound $Ω(\log L+k)$ on time for the dispersion of $k$ robots on a ring network is presented to establish the optimality of the proposed algorithm.

cs.DC↗

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 $n$.

cs.DC↗

arXiv2023

Distance-2-Dispersion: Dispersion with Further Constraints

The aim of the dispersion problem is to place a set of $k(\leq n)$ mobile robots in the nodes of an unknown graph consisting of $n$ nodes such that in the final configuration each node contains at most one robot, starting from any arbitrary initial configuration of the robots on the graph. In this work we propose a variant of the dispersion problem where we start with any number of robots, and put an additional constraint that no two adjacent nodes contain robots in the final configuration. We name this problem as Distance-2-Dispersion (D-2-D). However, even if the number of robots $k$ is less than $n$, it may not possible for each robot to find a distinct node to reside, maintaining our added constraint. Specifically, if a maximal independent set is already formed by the nodes which contain a robot each, then other robots, if any, who are searching for a node to seat, will not find one. Hence we allow multiple robots to seat on some nodes only if there is no place to seat. If $k\geq n$, it is guaranteed that the nodes with robots form a maximal independent set of the underlying network. The graph $G=(V, E)$ has $n$ nodes and $m$ edges, where nodes are anonymous. It is a port labelled graph, i.e., each node $u$ assigns a distinct port number to each of its incident edges from a range $[0,δ-1]$ where $δ$ is the degree of the node $u$. The robots have unique ids in the range $[1, L]$, where $L \ge k$. Co-located robots can communicate among themselves. We provide an algorithm that solves D-2-D starting from a rooted configuration (i.e., initially all the robots are co-located) and terminate after $2Δ(8m-3n+3)$ synchronous rounds using $O(log Δ)$ memory per robot without using any global knowledge of the graph parameters $m$, $n$ and $Δ$, the maximum degree of the graph. We also provide $Ω(mΔ)$ lower bound on the number of rounds for the D-2-D problem.

cs.DC↗

arXiv2022

Collaborative Dispersion by Silent Robots

In the dispersion problem, a set of $k$ co-located mobile robots must relocate themselves in distinct nodes of an unknown network. The network is modeled as an anonymous graph $G=(V,E)$, where the nodes of the graph are not labeled. The edges incident to a node $v$ with degree $d$ are labeled with port numbers in the range $0,1, \cdots, d-1$ at $v$. The robots have unique ids in the range $[0,L]$, where $L \ge k$, and are initially placed at a source node $s$. Each robot knows only its own id but does not know the ids of the other robots or the values of $L,k$. The task of dispersion was traditionally achieved with the assumption of two types of communication abilities: (a) when some robots are at the same node, they can communicate by exchanging messages between them (b) any two robots in the network can exchange messages between them. In this paper, we ask whether this ability of communication among co-located robots is necessary to achieve dispersion. We show that even if the ability of communication is not available, the task of dispersion by a set of mobile robots can be achieved in a much weaker model where a robot at a node $v$ has the access of following very restricted information at the beginning of any round: (1) am I alone at $v$? (2) the number of robots at $v$ increased or decreased compare to the previous round? We propose a deterministic algorithm that achieves dispersion on any given graph $G=(V,E)$ in time $O\left( k\log L+k^2 \log Δ\right)$, where $Δ$ is the maximum degree of a node in $G$. Each robot uses $O(\log L+ \log Δ)$ additional memory. We also prove that the task of dispersion cannot be achieved by a set of mobile robots with $o(\log L + \log Δ)$ additional memory.

cs.DS↗

arXiv2021

Byzantine Dispersion on Graphs

This paper considers the problem of Byzantine dispersion and extends previous work along several parameters. The problem of Byzantine dispersion asks: given $n$ robots, up to $f$ of which are Byzantine, initially placed arbitrarily on an $n$ node anonymous graph, design a terminating algorithm to be run by the robots such that they eventually reach a configuration where each node has at most one non-Byzantine robot on it. Previous work solved this problem for rings and tolerated up to $n-1$ Byzantine robots. In this paper, we investigate the problem on more general graphs. We first develop an algorithm that tolerates up to $n-1$ Byzantine robots and works for a more general class of graphs. We then develop an algorithm that works for any graph but tolerates a lesser number of Byzantine robots. We subsequently turn our focus to the strength of the Byzantine robots. Previous work considers only ``weak" Byzantine robots that cannot fake their IDs. We develop an algorithm that solves the problem when Byzantine robots are not weak and can fake IDs. Finally, we study the situation where the number of the robots is not $n$ but some $k$. We show that in such a scenario, the number of Byzantine robots that can be tolerated is severely restricted. Specifically, we show that it is impossible to deterministically solve Byzantine dispersion when $\lceil k/n \rceil > \lceil (k-f)/n \rceil$.

cs.DC↗

arXiv2021

Pebble Guided Near Optimal Treasure Hunt in Anonymous Graphs

We study the problem of treasure hunt in a graph by a mobile agent. The nodes in the graph are anonymous and the edges at any node $v$ of degree $deg(v)$ are labeled arbitrarily as $0,1,\ldots, deg(v)-1$. A mobile agent, starting from a node, must find a stationary object, called {\it treasure} that is located on an unknown node at a distance $D$ from its initial position. The agent finds the treasure when it reaches the node where the treasure is present. The {\it time} of treasure hunt is defined as the number of edges the agent visits before it finds the treasure. The agent does not have any prior knowledge about the graph or the position of the treasure. An Oracle, that knows the graph, the initial position of the agent, and the position of the treasure, places some pebbles on the nodes, at most one per node, of the graph to guide the agent towards the treasure. We target to answer the question: what is the fastest possible treasure hunt algorithm regardless of the number of pebbles are placed? We show an algorithm that uses $O(D \log Δ)$ pebbles to find the treasure in a graph $G$ in time $O(D \log Δ+ \log^3 Δ)$, where $Δ$ is the maximum degree of a node in $G$ and $D$ is the distance from the initial position of the agent to the treasure. We show an almost matching lower bound of $Ω(D \log Δ)$ on time of the treasure hunt using any number of pebbles.

cs.DS↗

arXiv2020

NP-Completeness Results for Graph Burning on Geometric Graphs

Graph burning runs on discrete time steps. The aim is to burn all the vertices in a given graph in the least number of time steps. This number is known to be the burning number of the graph. The spread of social influence, an alarm, or a social contagion can be modeled using graph burning. The less the burning number, the faster the spread. Optimal burning of general graphs is NP-Hard. There is a 3-approximation algorithm to burn general graphs where as better approximation factors are there for many sub classes. Here we study burning of grids; provide a lower bound for burning arbitrary grids and a 2-approximation algorithm for burning square grids. On the other hand, burning path forests, spider graphs, and trees with maximum degree three is already known to be NP-Complete. In this article we show burning problem to be NP-Complete on connected interval graphs, permutation graphs and several other geometric graph classes as corollaries.

cs.DS↗

arXiv2020

Efficient Dispersion on an Anonymous Ring in the Presence of Weak Byzantine Robots

The problem of dispersion of mobile robots on a graph asks that $n$ robots initially placed arbitrarily on the nodes of an $n$-node anonymous graph, autonomously move to reach a final configuration where exactly each node has at most one robot on it. This problem is of significant interest due to its relationship to other fundamental robot coordination problems, such as exploration, scattering, load balancing, relocation of self-driving electric cars to recharge stations, etc. The robots have unique IDs, typically in the range $[1,poly(n)]$ and limited memory, whereas the graph is anonymous, i.e., the nodes do not have identifiers. The objective is to simultaneously minimize two performance metrics: (i) time to achieve dispersion and (ii) memory requirement at each robot. This problem has been relatively well-studied when robots are non-faulty. In this paper, we introduce the notion of Byzantine faults to this problem, i.e., we formalize the problem of dispersion in the presence of up to $f$ Byzantine robots. We then study the problem on a ring while simultaneously optimizing the time complexity of algorithms and the memory requirement per robot. Specifically, we design deterministic algorithms that attempt to match the time lower bound ($Ω(n)$ rounds) and memory lower bound ($Ω(\log n)$ bits per robot). Our main result is a deterministic algorithm that is both time and memory optimal, i.e., $O(n)$ rounds and $O(\log n)$ bits of memory required per robot, subject to certain constraints. We subsequently provide results that require less assumptions but are either only time or memory optimal but not both. We also provide a primitive, utilized often, that takes robots initially gathered at a node of the ring and disperses them in a time and memory optimal manner without additional assumptions required.

cs.DC↗

arXiv2020

Push-Down Trees: Optimal Self-Adjusting Complete Trees

This paper studies a fundamental algorithmic problem related to the design of demand-aware networks: networks whose topologies adjust toward the traffic patterns they serve, in an online manner. The goal is to strike a tradeoff between the benefits of such adjustments (shorter routes) and their costs (reconfigurations). In particular, we consider the problem of designing a self-adjusting tree network which serves single-source, multi-destination communication. The problem has interesting connections to self-adjusting datastructures. We present two constant-competitive online algorithms for this problem, one randomized and one deterministic. Our approach is based on a natural notion of Most Recently Used (MRU) tree, maintaining a working set. We prove that the working set is a cost lower bound for any online algorithm, and then present a randomized algorithm RANDOM-PUSH which approximates such an MRU tree at low cost, by pushing less recently used communication partners down the tree, along a random walk. Our deterministic algorithm MOVE-HALF does not directly maintain an MRU tree, but its cost is still proportional to the cost of an MRU tree, and also matches the working set lower bound.

cs.DS↗

arXiv2017

Demand-Aware Network Designs of Bounded Degree

Traditionally, networks such as datacenter interconnects are designed to optimize worst-case performance under arbitrary traffic patterns. Such network designs can however be far from optimal when considering the actual workloads and traffic patterns which they serve. This insight led to the development of demand-aware datacenter interconnects which can be reconfigured depending on the workload. Motivated by these trends, this paper initiates the algorithmic study of demand-aware networks (DANs) designs, and in particular the design of bounded-degree networks. The inputs to the network design problem are a discrete communication request distribution, D, defined over communicating pairs from the node set V , and a bound, d, on the maximum degree. In turn, our objective is to design an (undirected) demand-aware network N = (V,E) of bounded-degree d, which provides short routing paths between frequently communicating nodes distributed across N. In particular, the designed network should minimize the expected path length on N (with respect to D), which is a basic measure of the efficiency of the network. We show that this fundamental network design problem exhibits interesting connections to several classic combinatorial problems and to information theory. We derive a general lower bound based on the entropy of the communication pattern D, and present asymptotically optimal network-aware design algorithms for important distribution families, such as sparse distributions and distributions of locally bounded doubling dimensions.

cs.DC↗