arXiv subjects

Lata Narayanan

Publications and source records attributed to Lata Narayanan.

At least 19 recordsLinked to original sources

arXiv2026

Online Treasure Hunt in Vertex-Permuted Dynamic Rings

We study the problem of treasure hunt by a group of $k \geq 1$ agents in vertex-permuted dynamic rings (VP). In this model, the $n$ vertices remain on a ring but are permuted at each time step. We first show that treasure hunt is impossible for any $k \leq n-3$ agents, if there are no restrictions on the sequence of permutations used in the dynamic ring. We then study the $VP(\delta)$ setting, in which for every pair $i, j$ of vertices, the edge $(i, j)$ is guaranteed to appear within $\delta$ steps. We show that the class $VP(\delta)$ is feasible only for $\delta \geq \left\lceil \frac{n-1}{2}\right\rceil$. For the one-agent case, we show a tight bound of $\Theta(\delta n)$ on the worst-case search time as well as competitive ratio of any online algorithm for treasure hunt, provided $\delta \geq 2n$. We then give an optimal algorithm for $k$ agents, thereby showing that $k$ agents can obtain a speedup of $k$ on the worst-case search time. Finally, in the R-VP setting, in which in every step, the vertices are arranged as a ring according to a random permutation, we show that treasure hunt takes expected $\Theta(n)$ steps against an oblivious adversary and $\Theta(n \log n)$ steps against an adaptive adversary.

cs.DC↗

arXiv2026

Online Drone Coverage of Targets on a Line

We study a problem of online targets coverage by a drone or a sensor that is equipped with a camera or an antenna of fixed half-angle of view $α$. The targets to be monitored appear at arbitrary positions on a line barrier in an online manner. When a new target appears, the drone has to move to a location that covers the newly arrived target, as well as already existing targets. The objective is to design a coverage algorithm that optimizes the total length of the drone's trajectory. Our results are reported in terms of an algorithm's competitive ratio, i.e., the worst-case ratio (over all inputs) of its cost to that of an optimal offline algorithm. In terms of upper bounds, we present three online algorithms and prove bounds on their competitive ratios for every $α\in [0, π/2]$. The best of them, called \FA is significantly better than the other two for $π/6 < α< π/3$. In particular, for $α=π/4$, its worst case, \FA has competitive ratio $1.25$, while the other two have competitive ratio $\sqrt{2}$. Finally, we prove a lower bound on the competitive ratio of online algorithms for a drone with half-angle $α\in [0, π/4]$; this bound is a function of $α$ that achieves its maximum value at $α= π/4$ equal to $(1+\sqrt{2})/2 \approx 1.207$.

cs.DS↗

arXiv2025

Variety-Seeking Jump Games on Graphs

We consider a class of jump games in which agents of different types occupy the nodes of a graph aiming to maximize the variety of types in their neighborhood. In particular, each agent derives a utility equal to the number of types different from its own in its neighborhood. We show that the jump game induced by the strategic behavior of the agents (who aim to maximize their utility) may in general have improving response cycles, but is a potential game under any of the following four conditions: there are only two types of agents; or exactly one empty node; or the graph is of degree at most 2; or the graph is 3-regular and there are two empty nodes. Additionally, we show that on trees, cylinder graphs, and tori, there is always an equilibrium. Finally, we show tight bounds on the price of anarchy with respect to two different measures of diversity: the social welfare (the total utility of the agents) and the number of colorful edges (that connect agents of different types).

cs.GT↗

arXiv2025

Diversity-seeking swap games in networks

Schelling games use a game-theoretic approach to study the phenomenon of residential segregation as originally modeled by Schelling. Inspired by the recent increase in the number of people and businesses preferring and promoting diversity, we propose swap games under three diversity-seeking utility functions: the binary utility of an agent is 1 if it has a neighbor of a different type, and 0 otherwise; the difference-seeking utility of an agent is equal to the number of its neighbors of a different type; the variety-seeking utility of an agent is equal to the number of types different from its own in its neighborhood. We consider four global measures of diversity: degree of integration, number of colorful edges, neighborhood variety, and evenness, and prove asymptotically tight or almost tight bounds on the price of anarchy with respect to these measures on both general graphs, as well as on cycles, cylinders, and tori that model residential neighborhoods. We complement our theoretical results with simulations of our swap games starting either from random placements of agents, or from segregated placements. Our simulation results are generally consistent with our theoretical results, showing that segregation is effectively removed when agents are diversity-seeking; however strong diversity, such as measured by neighborhood variety and evenness, is harder to achieve by our swap games.

cs.GT↗

arXiv2024

Renting Servers in the Cloud: The Case of Equal Duration Jobs

Renting servers in the cloud is a generalization of the bin packing problem, motivated by job allocation to servers in cloud computing applications. Jobs arrive in an online manner, and need to be assigned to servers; their duration and size are known at the time of arrival. There is an infinite supply of identical servers, each having one unit of computational capacity per unit of time. A server can be rented at any time and continues to be rented until all jobs assigned to it finish. The cost of an assignment is the sum of durations of rental periods of all servers. The goal is to assign jobs to servers to minimize the overall cost while satisfying server capacity constraints. We focus on analyzing two natural algorithms, NextFit and FirstFit, for the case of jobs of equal duration. It is known that the competitive ratio of NextFit and FirstFit are at most 3 and 4 respectively for this case. We prove a tight bound of 2 on the competitive ratio of NextFit. For FirstFit, we establish a lower bound of 2.519 on the competitive ratio, even when jobs have only two distinct arrival times. For the case when jobs have arrival times 0 and 1 and duration 2, we show a lower bound of 1.89 and an upper bound of 2 on the strict competitive ratio of FirstFit. Finally, using the weight function technique, we obtain stronger results for the case of uniform servers.

cs.DS↗

arXiv2024

Renting Servers for Multi-Parameter Jobs in the Cloud

We study the Renting Servers in the Cloud problem (RSiC) in multiple dimensions. In this problem, a sequence of multi-parameter jobs must be scheduled on servers that can be rented on-demand. Each job has an arrival time, a finishing time, and a multi-dimensional size vector that specifies its resource demands. Each server has a multi-dimensional capacity and jobs can be scheduled on a server as long as in each dimension the sum of sizes of jobs does not exceed the capacity of the server in that dimension. The goal is to minimize the total rental time of servers needed to process the job sequence. AF algorithms do not rent new servers to accommodate a job unless they have to. We introduce a sub-family of AF algorithms called monotone AF algorithms. We show this family have a tight competitive ratio of $Theta(d mu)$, where $d$ is the dimension of the problem and $mu$ is the ratio between the maximum and minimum duration of jobs in the input sequence. We also show that upper bounds for the RSiC problem obey the direct-sum property with respect to dimension $d$, that is we show how to transform $1$-dimensional algorithms for RSiC to work in the $d$-dimensional setting with competitive ratio scaling by a factor of $d$. As a corollary, we obtain an $O(d\sqrt{log mu})$ upper bound for $d$-dimensional clairvoyant RSiC. We also establish a lower bound of $\widetilde{Omega}(d mu)$ for both deterministic and randomized algorithms for $d$-dimensional non-clairvoyant RSiC, under the assumption that $mu \le log d - 2$. Lastly, we propose a natural greedy algorithm called Greedy. Greedy, is a clairvoyant algorithm belongs to the monotone AF family, achieves a competitive ratio of $Theta(d mu)$. Our experimental results indicate that Greedy performs better or matches all other existing algorithms, for almost all the settings of arrival rates and values of mu and $d$ that we implemented.

cs.DS↗

arXiv2024

Diversity-seeking Jump Games in Networks

Recently, strategic games inspired by Schelling's influential model of residential segregation have been studied in the TCS and AI literature. In these games, agents of k different types occupy the nodes of a network topology aiming to maximize their utility, which is a function of the fraction of same-type agents they are adjacent to in the network. As such, the agents exhibit similarity seeking strategic behavior. In this paper, we introduce a class of strategic jump games in which the agents are diversity-seeking: The utility of an agent is defined as the fraction of its neighbors that are of different type than itself. We show that in general it is computationally hard to determine the existence of an equilibrium in such games. However, when the network is a tree, diversity-seeking jump games always admit an equilibrium assignment. For regular graphs and spider graphs with a single empty node, we prove a stronger result: The game is potential, that is, the improving response dynamics always converge to an equilibrium from any initial placement of the agents. We also show (nearly tight) bounds on the price of anarchy and price of stability in terms of the social welfare (the total utility of the agents).

cs.GT↗

arXiv2024

The En Route Truck-Drone Delivery Problem

We study the truck-drone cooperative delivery problem in a setting where a single truck carrying a drone travels at constant speed on a straight-line trajectory/street. Delivery to clients located in the plane and not on the truck's trajectory is performed by the drone, which has limited carrying capacity and flying range, and whose battery can be recharged when on the truck. We show that the problem of maximizing the number of deliveries is strongly NP-hard even in this simple setting. We present a 2-approximation algorithm for the problem, and an optimal algorithm for a non-trivial family of instances.

cs.DS↗

arXiv2021

Group Evacuation on a Line by Agents with Different Communication Abilities

We consider evacuation of a group of $n \geq 2$ autonomous mobile agents (or robots) from an unknown exit on an infinite line. The agents are initially placed at the origin of the line and can move with any speed up to the maximum speed $1$ in any direction they wish and they all can communicate when they are co-located. However, the agents have different wireless communication abilities: while some are fully wireless and can send and receive messages at any distance, a subset of the agents are senders, they can only transmit messages wirelessly, and the rest are receivers, they can only receive messages wirelessly. The agents start at the same time and their communication abilities are known to each other from the start. Starting at the origin of the line, the goal of the agents is to collectively find a target/exit at an unknown location on the line while minimizing the evacuation time, defined as the time when the last agent reaches the target. We investigate the impact of such a mixed communication model on evacuation time on an infinite line for a group of cooperating agents. In particular, we provide evacuation algorithms and analyze the resulting competitive ratio ($CR$) of the evacuation time for such a group of agents. If the group has two agents of two different types, we give an optimal evacuation algorithm with competitive ratio $CR=3+2 \sqrt{2}$. If there is a single sender or fully wireless agent, and multiple receivers we prove that $CR \in [2+\sqrt{5},5]$, and if there are multiple senders and a single receiver or fully wireless agent, we show that $CR \in [3,5.681319]$. Any group consisting of only senders or only receivers requires competitive ratio 9, and any other combination of agents has competitive ratio 3.

cs.DS↗

arXiv2020

Evacuating Robots from a Disk Using Face-to-Face Communication

Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed $1$. The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [CGGKMP14] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most $5.740$ and also proved that any algorithm has evacuation time at least $3+ \fracπ{4} + \sqrt{2} \approx 5.199$. We improve both the upper and lower bound on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most $5.628$ and show that any algorithm has evacuation time at least $3+ \fracπ{6} + \sqrt{3} \approx 5.255$. To achieve the upper bound, we designed an algorithm which proposes a forced meeting between the two robots, even if the exit has not been found by either of them. We also show that such a strategy is provably optimal for a related problem of searching for an exit placed at the vertices of a regular hexagon.

cs.DS↗

arXiv2020

The Bike Sharing Problem

Assume that $m \geq 1$ autonomous mobile agents and $0 \leq b \leq m$ single-agent transportation devices (called {\em bikes}) are initially placed at the left endpoint $0$ of the unit interval $[0,1]$. The agents are identical in capability and can move at speed one. The bikes cannot move on their own, but any agent riding bike $i$ can move at speed $v_i > 1$. An agent may ride at most one bike at a time. The agents can cooperate by sharing the bikes; an agent can ride a bike for a time, then drop it to be used by another agent, and possibly switch to a different bike. We study two problems. In the \BS problem, we require all agents and bikes starting at the left endpoint of the interval to reach the end of the interval as soon as possible. In the \RBS problem, we aim to minimize the arrival time of the agents; the bikes can be used to increase the average speed of the agents, but are not required to reach the end of the interval. Our main result is the construction of a polynomial time algorithm for the \BS problem that creates an arrival-time optimal schedule for travellers and bikes to travel across the interval. For the \RBS problem, we give an algorithm that gives an optimal solution for the case when at most one of the bikes can be abandoned.

cs.DS↗

arXiv2019

Evacuation of equilateral triangles by mobile agents of limited communication range

We consider the problem of evacuating $k \geq 2$ mobile agents from a unit-sided equilateral triangle through an exit located at an unknown location on the perimeter of the triangle. The agents are initially located at the centroid of the triangle and they can communicate with other agents at distance at most $r$ with $0\leq r \leq 1$. An agent can move at speed at most one, and finds the exit only when it reaches the point where the exit is located. The agents can collaborate in the search for the exit. The goal of the {\em evacuation problem} is to minimize the evacuation time, defined as the worst-case time for {\em all} the agents to reach the exit. We propose and analyze several algorithms for the problem of evacuation by $k \geq 2$ agents; our results indicate that the best strategy to be used varies depending on the values of $r$ and $k$. For two agents, we give three algorithms, each of which achieves the best performance for different sub-ranges of $r$ in the range $0 \leq r \leq 1$. Finally, we show that for any $r$, evacuation of $k=6 +2\lceil(\frac{1}{r}-1)\rceil$ agents can be done in time $1+\sqrt{3}/3$, which is optimal in terms of time, and asymptotically optimal in terms of the number of agents.

cs.DC↗

arXiv2019

Distributed Pattern Formation in a Ring

Motivated by concerns about diversity in social networks, we consider the following pattern formation problems in rings. Assume $n$ mobile agents are located at the nodes of an $n$-node ring network. Each agent is assigned a colour from the set $\{c_1, c_2, \ldots, c_q \}$. The ring is divided into $k$ contiguous {\em blocks} or neighbourhoods of length $p$. The agents are required to rearrange themselves in a distributed manner to satisfy given diversity requirements: in each block $j$ and for each colour $c_i$, there must be exactly $n_i(j) >0$ agents of colour $c_i$ in block $j$. Agents are assumed to be able to see agents in adjacent blocks, and move to any position in adjacent blocks in one time step. When the number of colours $q=2$, we give an algorithm that terminates in time $N_1/n^*_1 + k + 4$ where $N_1$ is the total number of agents of colour $c_1$ and $n^*_1$ is the minimum number of agents of colour $c_1$ required in any block. When the diversity requirements are the same in every block, our algorithm requires $3k+4$ steps, and is asymptotically optimal. Our algorithm generalizes for an arbitrary number of colours, and terminates in $O(nk)$ steps. We also show how to extend it to achieve arbitrary specific final patterns, provided there is at least one agent of every colour in every pattern.

cs.DC↗

arXiv2019

Time-Energy Tradeoffs for Evacuation by Two Robots in the Wireless Model

Two robots stand at the origin of the infinite line and are tasked with searching collaboratively for an exit at an unknown location on the line. They can travel at maximum speed $b$ and can change speed or direction at any time. The two robots can communicate with each other at any distance and at any time. The task is completed when the last robot arrives at the exit and evacuates. We study time-energy tradeoffs for the above evacuation problem. The evacuation time is the time it takes the last robot to reach the exit. The energy it takes for a robot to travel a distance $x$ at speed $s$ is measured as $xs^2$. The total and makespan evacuation energies are respectively the sum and maximum of the energy consumption of the two robots while executing the evacuation algorithm. Assuming that the maximum speed is $b$, and the evacuation time is at most $cd$, where $d$ is the distance of the exit from the origin, we study the problem of minimizing the total energy consumption of the robots. We prove that the problem is solvable only for $bc \geq 3$. For the case $bc=3$, we give an optimal algorithm, and give upper bounds on the energy for the case $bc>3$. We also consider the problem of minimizing the evacuation time when the available energy is bounded by $Δ$. Surprisingly, when $Δ$ is a constant, independent of the distance $d$ of the exit from the origin, we prove that evacuation is possible in time $O(d^{3/2}\log d)$, and this is optimal up to a logarithmic factor. When $Δ$ is linear in $d$, we give upper bounds on the evacuation time.

cs.DS↗

arXiv2019

Energy Consumption of Group Search on a Line

Consider two robots that start at the origin of the infinite line in search of an exit at an unknown location on the line. The robots can only communicate if they arrive at the same location at exactly the same time, i.e. they use the so-called face-to-face communication model. The group search time is defined as the worst-case time as a function of $d$, the distance of the exit from the origin, when both robots can reach the exit. It has long been known that for a single robot traveling at unit speed, the search time is at least $9d-o(d)$. It was shown recently that $k\geq2$ robots traveling at unit speed also require at least $9d$ group search time. We investigate energy-time trade-offs in group search by two robots, where the energy loss experienced by a robot traveling a distance $x$ at constant speed $s$ is given by $s^2 x$. Specifically, we consider the problem of minimizing the total energy used by the robots, under the constraints that the search time is at most a multiple $c$ of the distance $d$ and the speed of the robots is bounded by $b$. Motivation for this study is that for the case when robots must complete the search in $9d$ time with maximum speed one, a single robot requires at least $9d$ energy, while for two robots, all previously proposed algorithms consume at least $28d/3$ energy. When the robots have bounded memory, we generalize existing algorithms to obtain a family of optimal (and in some cases nearly optimal) algorithms parametrized by pairs of $b,c$ values that can solve the problem for the entire spectrum of these pairs for which the problem is solvable. We also propose a novel search algorithm, with unbounded memory, that simultaneously achieves search time $9d$ and consumes energy $8.42588d$. Our result shows that two robots can search on the line in optimal time $9d$ while consuming less total energy than a single robot within the same search time.

cs.DM↗

arXiv2019

Exploration of High-Dimensional Grids by Finite State Machines

We consider the problem of finding a treasure at an unknown point of an $n$-dimensional infinite grid, $n\geq 3$, by initially collocated finite state agents (scouts/robots). Recently, the problem has been well characterized for 2 dimensions for deterministic as well as randomized agents, both in synchronous and semi-synchronous models. It has been conjectured that $n+1$ randomized agents are necessary to solve this problem in the $n$-dimensional grid. In this paper we disprove the conjecture in a strong sense: we show that three randomized synchronous agents suffice to explore an $n$-dimensional grid for any $n$. Our algorithm is optimal in terms of the number of the agents. Our key insight is that a constant number of finite state machine agents can, by their positions and movements, implement a stack, which can store the path being explored. We also show how to implement our algorithm using: four randomized semi-synchronous agents; four deterministic synchronous agents; or five deterministic semi-synchronous agents. We give a different algorithm that uses $4$ deterministic semi-synchronous agents for the $3$-dimensional grid. This is provably optimal, and surprisingly, matches the result for $2$ dimensions. For $n\geq 4$, the time complexity of the solutions mentioned above is exponential in distance $D$ of the treasure from the starting point of the agents. We show that in the deterministic case, one additional agent brings the time down to a polynomial. Finally, we focus on algorithms that never venture much beyond the distance $D$. We describe an algorithm that uses $O(\sqrt{n})$ semi-synchronous deterministic agents that never go beyond $2D$, as well as show that any algorithm using $3$ synchronous deterministic agents in $3$ dimensions must travel beyond $Ω(D^{3/2})$ from the origin.

cs.MA↗

arXiv2018

Evacuating Equilateral Triangles and Squares in the Face-to-Face Model

Consider $k$ robots initially located at a point inside a region $T$. Each robot can move anywhere in $T$ independently of other robots with maximum speed one. The goal of the robots is to \emph{evacuate} $T$ through an exit at an unknown location on the boundary of $T$. The objective is to minimize the \emph{evacuation time}, which is defined as the time the \emph{last} robot reaches the exit. We consider the \emph{face-to-face} communication model for the robots: a robot can communicate with another robot only when they meet in $T$. In this paper, we give upper and lower bounds for the face-to-face evacuation time by $k$ robots that are initially located at the centroid of a unit-sided equilateral triangle or square. For the case of a triangle with $k=2$ robots, we give a lower bound of $1+2/\sqrt{3} \approx 2.154$, and an algorithm with upper bound of 2.3367 on the worst-case evacuation time. We show that for any $k$, any algorithm for evacuating $k\geq 2$ robots requires at least $\sqrt{3}$ time. This bound is asymptotically optimal, as we show that even a straightforward strategy of evacuation by $k$ robots gives an upper bound of $\sqrt{3} + 3/k$. For $k=3$ and $4$, we give better algorithms with evacuation times of 2.0887 and 1.9816, respectively. For the case of the square and $k=2$, we give an algorithm with evacuation time of $3.4645$ and show that any algorithm requires time at least $3.118$ to evacuate in the worst-case. Moreover, for $k=3$, and $4$, we give algorithms with evacuation times 3.1786 and 2.6646, respectively. The algorithms given for $k=3$ and $4$ for evacuation in the triangle or the square can be easily generalized for larger values of $k$.

cs.CG↗

arXiv2018

God Save the Queen

Queen Daniela of Sardinia is asleep at the center of a round room at the top of the tower in her castle. She is accompanied by her faithful servant, Eva. Suddenly, they are awakened by cries of "Fire". The room is pitch black and they are disoriented. There is exactly one exit from the room somewhere along its boundary. They must find it as quickly as possible in order to save the life of the queen. It is known that with two people searching while moving at maximum speed 1 anywhere in the room, the room can be evacuated (i.e., with both people exiting) in $1 + \frac{2π}{3} + \sqrt{3} \approx 4.8264$ time units and this is optimal~[Czyzowicz et al., DISC'14], assuming that the first person to find the exit can directly guide the other person to the exit using her voice. Somewhat surprisingly, in this paper we show that if the goal is to save the queen (possibly leaving Eva behind to die in the fire) there is a slightly better strategy. We prove that this "priority" version of evacuation can be solved in time at most $4.81854$. Furthermore, we show that any strategy for saving the queen requires time at least $3 + π/6 + \sqrt{3}/2 \approx 4.3896$ in the worst case. If one or both of the queen's other servants (Biddy and/or Lili) are with her, we show that the time bounds can be improved to $3.8327$ for two servants, and $3.3738$ for three servants. Finally we show lower bounds for these cases of $3.6307$ (two servants) and $3.2017$ (three servants). The case of $n\geq 4$ is the subject of an independent study by Queen Daniela's Royal Scientific Team.

cs.MA↗