Title: Cost vs. Information Tradeoffs for Treasure Hunt in the Plane

Authors: A. Pelc R. Yadav

Abstract:

A mobile agent has to find an inert treasure hidden in the plane. Both the agent and the treasure are modeled as points. This is a variant of the task known as treasure hunt. The treasure is at a distance at most $D$ from the initial position of the agent, and the agent finds the treasure when it gets at distance $r$ from it, called the {\em vision radius}. However, the agent does not know the location of the treasure and does not know the parameters $D$ and $r$. The cost of finding the treasure is the length of the trajectory of the agent. We investigate the tradeoffs between the amount of information held {\em a priori} by the agent and the cost of treasure hunt. Following the well-established paradigm of {\em algorithms with advice}, this information is given to the agent in advance as a binary string, by an oracle cooperating with the agent and knowing the location of the treasure and the initial position of the agent. The size of advice given to the agent is the length of this binary string. For any size $z$ of advice and any $D$ and $r$, let $OPT(z,D,r)$ be the optimal cost of finding the treasure for parameters $z$, $D$ and $r$, if the agent has only an advice string of length $z$ as input. We design treasure hunt algorithms working with advice of size $z$ at cost $O(OPT(z,D,r))$ whenever $r\leq 1$ or $r\geq 0.9D$. For intermediate values of $r$, i.e., $1 < r < 0.9D$, we design an almost optimal scheme of algorithms: for any constant $\alpha > 0$, the treasure can be found at cost $O(OPT(z,D,r)^{1+\alpha})$.

Title: Information Complexity of Treasure Hunt in Geometric Terrains

Authors: A. Pelc R. Yadav

Abstract:

Treasure hunt is the task of finding an inert target by a mobile agent in an unknown environment. We consider treasure hunt in geometric terrains with obstacles. Both the terrain and the obstacles are modeled as polygons and both the agent and the treasure are modeled as points. The agent navigates in the terrain, avoiding obstacles, and finds the treasure when there is a segment of length at most 1 between them, unobstructed by the boundary of the terrain or by the obstacles. The cost of finding the treasure is the length of the trajectory of the agent. We investigate the amount of information that the agent needs {\em a priori} in order to find the treasure at cost $O(L)$, where $L$ is the length of the shortest path in the terrain from the initial position of the agent to the treasure, avoiding obstacles. Following the well-established paradigm of {\em algorithms with advice}, this information is given to the agent in advance as a binary string, by an oracle cooperating with the agent and knowing the whole environment: in our case, the terrain, the position of the treasure and the initial position of the agent. Information complexity of treasure hunt is the minimum length of the advice string (up to multiplicative constants) that enables the agent to find the treasure at cost $O(L)$. We first consider treasure hunt in {\em regular} terrains which are defined as convex polygons with convex $c$-fat obstacles, for some constant $c>1$. A polygon is $c$-fat if the ratio of the radius of the smallest disc containing it to the radius of the largest disc contained in it is at most $c$. For the class of regular terrains, we establish the exact information complexity of treasure hunt. We then show that information complexity of treasure hunt for the class of arbitrary terrains (even for non-convex polygons without obstacles, and even for those with only horizontal or vertical sides) is exponentially larger than for regular terrains.

Title: Using Time to Break Symmetry: Universal Deterministic Anonymous Rendezvous

Authors: A. Pelc R. Yadav

Abstract:

Two anonymous mobile agents navigate synchronously in an anonymous graph and have to meet at a node, using a deterministic algorithm. This is a symmetry breaking task called rendezvous, equivalent to the fundamental task of leader election between the agents. When is this feasible in a completely anonymous environment? It is known that agents can always meet if their initial positions are nonsymmetric, and that if they are symmetric and agents start simultaneously then rendezvous is impossible. What happens for symmetric initial positions with non-simultaneous start? Can symmetry between the agents be broken by the delay between their starting times? In order to answer these questions, we consider {\em space-time initial configurations} (abbreviated by STIC). A STIC is formalized as $[(u,v),\delta]$, where $u$ and $v$ are initial nodes of the agents in some graph and $\delta$ is a non-negative integer that represents the difference between their starting times. A STIC is {\em feasible} if there exists a deterministic algorithm, even dedicated to this particular STIC, which accomplishes rendezvous for it. Our main result is a characterization of all feasible STICs and the design of a universal deterministic algorithm that accomplishes rendezvous for all of them without {\em any } a priori knowledge of the agents. Thus, as far as feasibility is concerned, we completely solve the problem of symmetry breaking between two anonymous agents in anonymous graphs. Moreover, we show that such a universal algorithm cannot work for all feasible STICs in time polynomial in the initial distance between the agents.

Title: Latecomers Help to Meet: Deterministic Anonymous Gathering in the Plane

Authors: A. Pelc R. Yadav

Abstract:

A team of anonymous mobile agents represented by points freely moving in the plane have to gather at a single point and stop. Agents start at different points of the plane and at possibly different times chosen by the adversary. They are equipped with compasses, a common unit of distance and clocks. They execute the same deterministic algorithm. When moving, agents travel at the same speed normalized to 1. When agents are at distance at most $\epsilon$, for some positive constant $\epsilon$ unknown to them, they see each other and can exchange all information known to date. Due to the anonymity of the agents and the symmetry of the plane, gathering is impossible, e.g., if agents start simultaneously at distances larger than $\epsilon$. However, if some agents start with a delay with respect to others, gathering may become possible. In which situations such latecomers can enable gathering? To answer this question we consider initial configurations formalized as sets of pairs $\{(p_1,t_1), (p_2,t_2),\dots , (p_n,t_n)\}$, for $n\geq 2$ where $p_i$ is the starting point of the $i$-th agent and $t_i$ is its starting time. An initial configuration is {\em gatherable} if agents starting at it can be gathered by some algorithm, even dedicated to this particular configuration. Our first result is a characterization of all gatherable initial configurations. It is then natural to ask if there is a universal deterministic algorithm that can gather all gatherable configurations of a given size. It turns out that the answer to this question is negative. Indeed, we show that all gatherable configurations can be partitioned into two sets: {\em bad} configurations and {\em good} configurations. We show that bad gatherable configurations (even of size 2) cannot be gathered by a common gathering algorithm. On the other hand, we prove that there is a universal algorithm that gathers all good configurations of a given size. Then we ask the question of whether the exact knowledge of the number of agents is necessary to gather all good configurations. It turns out that the answer is no, and we prove a necessary and sufficient condition on the knowledge concerning the number of agents that an algorithm gathering all good configurations must have.

Title: Deterministic Treasure Hunt in the Plane with Angular Hints

Authors: S. Bouchard, Y. Dieudonné, A. Pelc, F. Petit

Abstract:

A mobile agent equipped with a compass and a measure of length has to find an inert treasure in the Euclidean plane. Both the agent and the treasure are modeled as points. In the beginning, the agent is at a distance at most $D>0$ from the treasure, but knows neither the distance nor any bound on it. Finding the treasure means getting at distance at most 1 from it. The agent makes a series of moves. Each of them consists in moving straight in a chosen direction at a chosen distance. In the beginning and after each move the agent gets a hint consisting of a positive angle smaller than $2\pi$ whose vertex is at the current position of the agent and within which the treasure is contained. We investigate the problem of how these hints permit the agent to lower the cost of finding the treasure, using a deterministic algorithm, where the cost is the worst-case total length of the agent's trajectory. It is well known that without any hint the optimal (worst case) cost is $\Theta(D^2)$. We show that if all angles given as hints are at most $\pi$, then the cost can be lowered to $O(D)$, which is optimal. If all angles are at most $\beta$, where $\beta<2\pi$ is a constant unknown to the agent, then the cost is at most $O(D^{2-\epsilon})$, for some $\epsilon>0$. For both these positive results we present deterministic algorithms achieving the above costs. Finally, if angles given as hints can be arbitrary, smaller than $2\pi$, then we show that cost $\Theta(D^2)$ cannot be beaten.

Title: Impact of Knowledge on Election Time in Anonymous Networks

Authors: Y. Dieudonné, A. Pelc

Abstract:

Leader election is one of the basic problems in distributed computing. For anonymous networks, the task of leader election is formulated as follows: every node v of the network must output a simple path, which is coded as a sequence of port numbers, such that all these paths end at a common node, the leader. In this paper, we study deterministic leader election in arbitrary anonymous networks. It is well known that leader election is impossible in some networks, regardless of the allocated amount of time, even if nodes know the map of the network. However, even in networks in which it is possible to elect a leader knowing the map, the task may be still impossible without any knowledge, regardless of the allocated time. On the other hand, for any network in which leader election is possible knowing the map, there is a minimum time, called the election index, in which this can be done. Informally, the election index of a network is the minimum depth at which views of all nodes are distinct. Our aim is to establish tradeoffs between the allocated time T and the amount of information that has to be given a priori to the nodes to enable leader election in time T in all networks for which leader election in this time is at all possible. Following the framework of algorithms with advice, this information is provided to all nodes at the start by an oracle knowing the entire network. The length of this string (its number of bits) is called the size of advice. For a given time T allocated to leader election, we give upper and lower bounds on the minimum size of advice sufficient to perform leader election in time T. We focus on the two sides of the time spectrum and give tight (or almost tight) bounds on the minimum size of advice for these extremes. We also show that constant advice is not sufficient for leader election in all graphs, regardless of the allocated time.

Title: Exploration of Faulty Hamiltonian Graphs

Authors: D. Caissy, A. Pelc

Abstract:

We consider the problem of exploration of networks, some of whose edges are faulty. A mobile agent, situated at a starting node and unaware of which edges are faulty, has to explore the connected fault-free component of this node by visiting all of its nodes. The cost of the exploration is the number of edge traversals. For a given network and given starting node, the overhead of an exploration algorithm is the worst-case ratio (taken over all fault configurations) of its cost to the cost of an optimal algorithm which knows where faults are situated. An exploration algorithm, for a given network and given starting node, is called perfectly competitive if its overhead is the smallest among all exploration algorithms not knowing the location of faults. We design a perfectly competitive exploration algorithm for any ring, and show that, for networks modeled by hamiltonian graphs, the overhead of any DFS exploration is at most 10/9 times larger than that of a perfectly competitive algorithm. Moreover, for hamiltonian graphs of size at least 24, this overhead is less than 6% larger than that of a perfectly competitive algorithm.

Title: Deterministic Graph Exploration with Advice

Authors: B. Gorain, A. Pelc

Abstract:

We consider the task of graph exploration. An n-node graph has unlabeled nodes, and all ports at any node of degree d are arbitrarily numbered 0,…,d-1. A mobile agent has to visit all nodes and stop. The exploration time is the number of edge traversals. We consider the problem of how much knowledge the agent has to have a priori, in order to explore the graph in a given time, using a deterministic algorithm. This a priori information (advice) is provided to the agent by an oracle, in the form of a binary string, whose length is called the size of advice. We consider two types of oracles. The instance oracle knows the entire instance of the exploration problem, i.e., the port-numbered map of the graph and the starting node of the agent in this map. The map oracle knows the port-numbered map of the graph but does not know the starting node of the agent. We first consider exploration in polynomial time, and determine the exact minimum size of advice to achieve it. This size is logloglogn-\Theta(1), for both types of oracles. When advice is large, there are two natural time thresholds: \Theta(n2) for a map oracle, and \Theta(n) for an instance oracle, that can be achieved with sufficiently large advice. We show that, with a map oracle, time \Theta(n2) cannot be improved in general, regardless of the size of advice. We also show that the smallest size of advice to achieve this time is larger than n^\delta, for any \delta<1/3. For an instance oracle, advice of size O(nlogn) is enough to achieve time O(n). We show that, with any advice of size o(nlogn), the time of exploration must be at least n^\epsilon, for any \epsilon<2, and with any advice of size O(n), the time must be \Omega(n2). We also investigate minimum advice sufficient for fast exploration of hamiltonian graphs.

Title: Asynchronous Broadcasting with Bivalent Beeps

Authors: K. Hounkanli, A. Pelc

Abstract:

In broadcasting, one node of a network has a message that must be learned by all other nodes. We study deterministic algorithms for this fundamental communication task in a very weak model of wireless communication. The only signals sent by nodes are beeps. Moreover, they are delivered to neighbors of the beeping node in an asynchronous way: the time between sending and reception is finite but unpredictable. We first observe that under this scenario, no communication is possible, if beeps are all of the same strength. Hence we study broadcasting in the bivalent beeping model, where every beep can be either soft or loud. At the receiving end, if exactly one soft beep is received by a node in a round, it is heard as soft. Any other combination of beeps received in a round is heard as a loud beep. The cost of a broadcasting algorithm is the total number of beeps sent by all nodes. We consider four levels of knowledge that nodes may have about the network: anonymity (no knowledge whatsoever), ad-hoc (all nodes have distinct labels and every node knows only its own label), neighborhood awareness (every node knows its label and labels of all neighbors), and full knowledge (every node knows the entire labeled map of the network and the identity of the source). We first show that in the anonymous case, broadcasting is impossible even for very simple networks. For each of the other three knowledge levels we provide upper and lower bounds on the minimum cost of a broadcasting algorithm. Our results show separations between all these scenarios. Perhaps surprisingly, the jump in broadcasting cost between the ad-hoc and neighborhood awareness levels is much larger than between the neighborhood awareness and full knowledge levels, although in the two former levels knowledge of nodes is local, and in the latter it is global.

Title: Deterministic Meeting of Sniffing Agents in the Plane

Authors: S. Elouasbi, A. Pelc

Abstract:

Two mobile agents, starting at arbitrary, possibly different times from arbitrary locations in the plane, have to meet. Agents are modeled as discs of diameter 1, and meeting occurs when these discs touch. Agents have different labels which are integers from the set {0,…,L-1} . Each agent knows L and knows its own label, but not the label of the other agent. Agents are equipped with compasses and have synchronized clocks. They make a series of moves. Each move specifies the direction and the duration of moving. This includes a null move which consists in staying inert for some time, or forever. In a non-null move agents travel at the same constant speed, normalized to 1. Agents have sensors enabling them to estimate the distance from the other agent, but not the direction towards it. We consider two models of estimation. In both models an agent reads its sensor at the moment of its appearance in the plane and then at the end of each move. This reading (together with the previous ones) determines the decision concerning the next move. In both models the reading of the sensor tells the agent if the other agent is already present. Moreover, in the monotone model, each agent can find out, for any two readings in moments t1 and t2, whether the distance from the other agent at time t1was smaller, equal or larger than at time t2. In the weaker binary model, each agent can find out, at any reading, whether it is at distance less than \rho or at distance at least \rho from the other agent, for some real \rho>1 unknown to them. Such distance estimation mechanism can be implemented, e.g., using chemical sensors. Each agent emits some chemical substance (scent), and the sensor of the other agent detects it, i.e., sniffs. The intensity of the scent decreases with the distance. In the monotone model it is assumed that the sensor is ideally accurate and can measure any change of intensity. In the binary model it is only assumed that the sensor can detect the scent below some distance (without being able to measure intensity) above which the scent is too weak to be detected. We show the impact of the two ways of sensing on the time of meeting, measured from the start of the later agent. For the monotone model we show an algorithm achieving meeting in time O(D), where D is the initial distance between the agents. This complexity is optimal. For the binary model we show that, if agents start at distance smaller than \rho (i.e., when they sense each other initially) then meeting can be guaranteed within time O(\rho logL), and that this time cannot be improved in general. Finally we observe that, if agents start at distance \alpha \rho, for some constant \alpha>1 in the binary model, then sniffing does not help, i.e., the worst-case optimal meeting time is of the same order of magnitude as without any sniffing ability.