arXiv subjects

Tanvir Kaur

Publications and source records attributed to Tanvir Kaur.

6 recordsLinked to original sources

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 $2\delta_{BH} + 1$ agents are necessary to solve the problem when agents are initially placed arbitrarily across the nodes of the graph (scattered initial configuration), where $\delta_{BH}$ denotes the degree of the black hole. In this work, we present an algorithm that solves the BHS problem using $2\delta_{BH} + 17$ 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.

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arXiv2025

When Agents are Powerful: Black Hole Search with Verification in Time-Varying Graphs

A black hole is a harmful node in a graph that destroys any agent entering it, making its identification a critical task. In the \emph{Black Hole Search with Verification (BHSV)} problem, a team of agents operates on a graph $G$ with the objective that at least one agent survives and correctly identifies an edge incident to the black hole; if no black hole exists, then all agents must terminate. Prior work has studied BHS in arbitrary dynamic graphs under the restrictive \emph{face-to-face} communication model, where agents can exchange information only when co-located. This constraint significantly increases the number of agents required to solve the problem. In this work, we strengthen the capabilities of agents by equipping them with (i) \emph{1-hop visibility}, (ii) \emph{global communication}, and (iii) both \emph{1-hop visibility} and \emph{global communication}. We show that these enhancements lead to more efficient solutions for the BHSV problem in dynamic graphs.

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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 $O(m\Delta)$ synchronous rounds with a total of $O(\log n)$ memory per agent, where $m$ is the number of edges and $\Delta$ 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.

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arXiv2024

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.

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arXiv2024

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 $\delta_v$ is the degree of the node $v$ in $G$. We show it is impossible to solve for $2\delta_{BH}$ many agents starting from an arbitrary configuration where $\delta_{BH}$ 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.

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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.

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