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.

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.

We consider the problem of finding a treasure at an unknown point of an n-dimensional infinite grid, n >= 3, by initially collocated finite automaton 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 [S. Brandt et al., 2018; Y. Emek et al., 2015]. It has been conjectured that n+1 randomized agents are necessary to solve this problem in the n-dimensional grid [L. Cohen et al., 2017]. 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 automaton 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 >= 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, if it exists, must travel beyond Omega(D^{3/2}) from the origin.

Consider k robots initially located at the centroid of an equilateral triangle T of sides of length one. The goal of the robots is to evacuate T through an exit at an unknown location on the boundary of T. Each robot can move anywhere in T independently of other robots with maximum speed one. The objective is to minimize the evacuation time, which is defined as the time required for all k robots to reach the exit. We consider the 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. We show that for any k, any algorithm for evacuating k >= 1 robots from T requires at least sqrt(3) time. This bound is asymptotically optimal, as we show that a straightforward strategy of evacuation by k robots gives an upper bound of sqrt(3) + 3/k. For k = 3, 4, 5, 6, we show significant improvements on the obvious upper bound by giving algorithms with evacuation times of 2.0887, 1.9816, 1.876, and 1.827, respectively. For k = 2 robots, we give a lower bound of 1 + 2/sqrt(3) ~= 2.154, and an algorithm with upper bound of 2.3367 on the evacuation time.

We consider a problem of routing on directed paths and trees to a single destination, with rate-limited, adversarial traffic. In particular, we focus on local buffer management algorithms that ensure no packet loss, while minimizing the size of the required buffers. While a centralized algorithm for the problem that uses constant-sized buffers has been recently shown [21], there is no known local algorithm that achieves a sub-linear buffer size. In this paper we show tight bounds for the maximum buffer size needed by l-local algorithms for information gathering on directed paths and trees, where an algorithm is called l-local if the decision made by each node v depends only on the sizes of the buffers at most l hops away from v.

We consider the problem of fault-tolerant parallel search on an infinite line by n robots. Starting from the origin, the robots are required to find a target at an unknown location. The robots can move with maximum speed 1 and can communicate in wireless mode among themselves. However, among the n robots, there are f robots that exhibit byzantine faults. A faulty robot can fail to report the target even after reaching it, or it can make malicious claims about having found the target when in fact it has not. Given the presence of such faulty robots, the search for the target can only be concluded when the non-faulty robots have sufficient verification that the target has been found. We aim to design algorithms that minimize the value of S_d (n, f), the time to find a target at a distance d from the origin by n robots among which f are faulty. We give several different algorithms whose running time depends on the ratio f/n, the density of faulty robots, and also prove lower bounds. Our algorithms are optimal for some densities of faulty robots.

We consider the problem of searching on a line using n mobile robots, of which at most f are faulty, and the remaining are reliable. The robots start at the same location and move in parallel along the line with the same speed. There is a target placed on the line at a location unknown to the robots. Reliable robots can find the target when they reach its location, but faulty robots cannot detect the target. Our goal is to design a parallel algorithm minimizing the competitive ratio, represented by the worst case ratio between the time of arrival of the first reliable robot at the target, and the distance from the source to the target.

Broadcasting has been used in mobile ad hoc networks to send a piece of data to every node throughout the network. Using a simple flooding scheme for such purpose causes redundant rebroadcasting at some nodes. Most approaches developed in the literature assume nodes are aware of the topological changes of the network via beacon messages, which cause an overhead. To address redundant broadcasting at some nodes as well as the inefficiency of the use of beacon messages, in this work we propose a beacon-less broadcasting algorithm for MANETs, where we prove every node in the network receives the message. We also give analysis of our algorithm presenting lower and upper bounds on the number of forwarding nodes.

We consider the k-barrier coverage problem, that is, the problem of deploying sensors on a border or perimeter to ensure that any intruder would be detected by at least k sensors. With random deployment of sensors, there is always a chance of gaps in coverage, thereby necessitating multiple rounds of deployment. In this paper, we study multi-round wireless sensor deployment on a border modeled as a line segment. We present two different classes of deployment strategies: complete and partial. In complete strategies, in every round, sensors are deployed over the entire border segment, while in partial strategies, sensors are deployed over only some part(s) of the border. First, we analyze the probability of k-coverage for any complete strategy as a function of parameters such as length of barrier to be covered, the width of the intruder, the sensing range of sensors, as well as the density of deployed sensors. Second, we propose two specific deployment strategies - Fixed-Density Complete and Fixed-Density Partial - and analyze the expected number of deployment rounds and expected total number of deployed sensors for each strategy. Next, we present a model for cost analysis of multi-round sensor deployment and calculate, for each deployment strategy, the expected total cost as a function of problem parameters and density of sensor deployment. Finally we find the optimal density of sensors in each round that minimizes the total expected cost of deployment for each deployment strategy. We validate our analysis by extensive simulation results.

We study the barrier coverage problem using relocatable sensor nodes. We assume each sensor can sense an intruder or event inside its sensing range. Sensors are initially located at arbitrary positions on the barrier and can move along the barrier. The goal is to find final positions for sensors so that the entire barrier is covered. In recent years, the problem has been studied extensively in the centralized setting. In this paper, we study the problem in the distributed setting. We assume each sensor repeatedly executes a Look-Compute-Move cycle: based on what it sees in its vicinity, it makes a decision on where to move, and moves to its next position. We make two strong but realistic restrictions on the capabilities of sensors: they have a constant visibility range and can move only a constant distance in every cycle. In this model, we give the first two distributed algorithms that achieve barrier coverage for a line segment barrier when there are enough nodes in the network to cover the entire barrier. Our algorithms are synchronous, and local in the sense that sensors make their decisions independently based only on what they see within their constant visibility range. One of our algorithms is oblivious whereas the other uses two bits of memory at each sensor to store the type of move made in the previous step. We show that our oblivious algorithm terminates within Θ(n2) steps with the barrier fully covered, while the constant-memory algorithm is shown to take Θ(n) steps to terminate in the worst case. Since any algorithm that can only move a constant distance in one step requires Ω(n) steps on some inputs, our second algorithm is asymptotically optimal. Finally, both our algorithms are self-stabilizing, and can be easily extended to the case of non-homogeneous sensors, and for the case when the barrier is a circle.

Assume that n sensors with identical range r = f(n)⁄2n, for some f(n) ≥ 1 for all n, are thrown randomly and independently with the uniform distribution in the unit interval [0, 1]. They are required to move to new positions so as to cover the entire unit interval in the sense that every point in the interval is within the range of a sensor. We obtain tradeoffs between the expected sum and maximum of displacements of the sensors and their range required to accomplish this task. In particular, when f(n) -- 1 the expected total displacement is shown to be Θ(√n). For senors with larger ranges we present two algorithms that prove the upper bound for the sum drops sharply as f(n) increases. The first of these holds for f(n) ≥ 6 and shows the total movement of the sensors is O(√ ln n/f(n)) while the second holds for 12 ≤ f(n) ≤ ln n -- 2 ln ln n and gives an upper bound of O(lnn⁄ f(n)ef(n)/2). Note that the second algorithm improves upon the first for f(n) > ln ln n -- ln ln ln n. Further we show a lower bound, for any 1 < f(n) < √n of Ω(εf(n)ε--(1+ε)f(n)), ε > 0.

We study the load balancing aspect of routing algorithms in wireless ad hoc networks. We define a statistical measure called local coefficient of variance (lcv) to study the smoothness of the load distribution in the network. The importance of keeping lcv as low as possible in designing load balanced routing algorithms is demonstrated. We analyze how number of nodes, transmission range, network area and different routing algorithms can affect this metric. We introduce a class of algorithms called elliptic routing that reduce the maximum load of nodes in the network by avoiding the highly loaded network center at the same time as keeping the lcv of the load distribution low. Experimental results show that our algorithms outperform other existing algorithms in reducing the maximum load of the network. We also give a technique to reduce the lcv of the load distribution, and hence decrease the maximum load of the nodes in the network further. This technique can be combined with any location-based routing algorithm. We evaluate the performance gain obtained by this technique via simulations.

We study the problem of achieving maximum barrier coverage by sensors on a barrier modeled by a line segment, by moving the minimum possible number of sensors, initially placed at arbitrary positions on the line containing the barrier. We consider several cases based on whether or not complete coverage is possible, and whether non-contiguous coverage is allowed in the case when complete coverage is impossible. When the sensors have unequal transmission ranges, we show that the problem of finding a minimum-sized subset of sensors to move in order to achieve maximum contiguous or non-contiguous coverage on a finite line segment barrier is NP-complete. In contrast, if the sensors all have the same range, we give efficient algorithms to achieve maximum contiguous as well as non-contiguous coverage. For some cases, we reduce the problem to finding a maximum-hop path of a certain minimum (maximum) weight on a related graph, and solve it using dynamic programming.

A Connected Dominating Set (CDS) can be used as a routing backbone in ad hoc networks and a data gathering/dissemination infrastructure in sensor networks. This virtual backbone can efficiently narrow down the search space for a route to the nodes in the CDS and thus be used by any routing protocol. Ideally, the backbone should constitute the smallest percentage of nodes in the network. However, finding a Minimum CDS (MCDS) is an NP-hard problem. In this paper, we propose an efficient distributed algorithm to construct a CDS in general graphs. The time and message complexity of our algorithm is linear in the number of nodes and degree of the network. Extensive simulations on Unit Disk Graphs (UDGs) show that this algorithm outperforms the distributed algorithms proposed in terms of the size of the CDS. We also present a local implementation of our algorithm in location-aware UDGs. Our algorithm provides the flexibility to arbitrarily adjust the tradeoff between the degree of locality and the size of the generated CDS.

A crucial problem in ad hoc networks is finding an efficient and correct route between a source and a destination; however for many networks, a more important problem is providing an energy efficient route because of, for example, the limited battery life of the wireless nodes. Most previous routing protocols make the routing decision without taking into account the energy budget of the nodes. In addition, when using a fixed transmission power, nodes may waste power by transmitting with more power than is needed for correct reception. In position- based routing algorithms, the nodes use the geographical position of the nodes to make the routing decisions. In this paper we present several localized power-aware 3D position-based routing algorithms that increase the life-time of the network by maximizing the life time of the nodes. These new algorithms use the idea of replacing the constant transmission power of the node with an adjusted transmission power during two stages - first a lower power while discovering the neighboring nodes, and, if needed, a second higher transmission power during the routing process. We evaluate our algorithms and compare their power savings with the current power-aware routing algorithms. The simulation results show a significant improvement in the energy saving (up to 50%).

In position-based routing algorithms for ad-hoc networks, the nodes use the geographical information to make the routing decisions. Recent research in this field primarily addresses such routing algorithms in two dimensional space (2D). However, in real applications, nodes may be distributed in 3D space. In this paper we extend previous randomized routing algorithms from 2D space to 3D space, and we propose two new position-based routing algorithms that combine randomized AB3D routing algorithms with a deterministic CFace (coordinate face) algorithm. The first algorithm AB3D-CFace(I)-AB3D starts with AB3D routing algorithm until a local minimum is reached. The algorithm then switches to CFace routing using one projected coordinate. IfCFace(I) enters a loop, the algorithm switches back to AB3D. The second algorithm AB3D-CFace(3) starts with AB3D, until a local minimum is reached. The algorithm then permanently switches to CFace routing using three projected coordinates, in order. We evaluate our mechanisms and compare them with the current routing algorithms. The simulation results show the significant improvement in delivery rate over pure AB3D randomized routing (97% compared to 70%) and reduction in path dilation (up to 50%) over pure CFace algorithm

Abstract We address the problem of discovering routes in strongly connected planar geometric networks with directed links. Motivated by the necessity for establishing communication in wireless ad hoc networks in which the only information available to a vertex is its immediate neighborhood, we are considering routing algorithms that use the neighborhood information of a vertex for routing with constant memory only. We solve the problem for three types of directed planar geometric networks: Eulerian (in which every vertex has the same number of incoming and outgoing edges), Outerplanar in which a single face contains all vertices of the network, and Strongly Face Connected, a new class of geometric networks that we define in the article, consisting of several faces, each face being a strongly connected outerplanar graph. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(1), 7–15 2006

We consider the problem of routing with guaranteed delivery in ad-hoc wireless networks using the positions of the mobile hosts. Such networks can be modeled as geometric graphs. FACE ROUTING [Bose, P et al. (1999), Karp, B et al. (2000)] is a position-based routing algorithm for planar geometric graphs that guarantees delivery of messages without flooding control packets throughout the network. For general ad hoc networks, FACE ROUTING can use a planar sub-graph of the original graph; many local and distributed algorithms have been proposed to extract such a planar sub-graph. However, these planarization algorithms may fail in some situations, such as when the transmission ranges are not the same, for example, due to the presence of obstacles, which in turn may cause a routing failure. In this paper, we describe a generalization of FACE ROUTING that can guarantee delivery in planar graphs with disjoint crossing edges added. Our algorithm needs O(/spl lscr/) memory, where /spl lscr/ is the maximum number of edges in any face in a graph obtained by removing one edge in each pair of crossing edges.

Summary form only given. The problem of traversal of planar subdivisions or other graph-like structures without using mark bits is central to many real-world applications. The first such algorithms were able to traverse triangulated subdivisions. Later these algorithms were extended to traverse vertices of an arrangement or a convex polytope. The research progress culminated in an algorithm that can traverse any planar subdivision. We extend the notion of planar subdivision to quasiplanar subdivision in which we allow many edges to cross each other. We describe an algorithm to traverse any quasiplanar subdivision that satisfies a simple requirement. The worst case running time of our algorithm is O(|E| log |E|), which matches the running time of the traversal algorithm for planar subdivisions.

Abstract Several papers considered the problem of routing inad hocwireless networks using the positions of the mobile hosts. Perimeter routing 1 , 2 gives an algorithm that guarantees delivery of messages in such networks without the use of flooding of control packets. However, this protocol is likely to fail if the transmission ranges of the mobile hosts vary because of natural or man‐made obstacles. It may fail because either some connections are not considered, which effectively results in a disconnection of the network, or because some crossing connections are used, which could misdirect the message. In this paper, we describe a robust routing protocol, a variant of perimeter routing, which tolerates up to 40% of variation in the transmission ranges of the mobile hosts. More precisely, our protocol guarantees message delivery in a connected ad hoc wireless network without the use of message flooding whenever the ratio of the maximum transmission range to the minimum transmission range is at most √2. Copyright © 2003 John Wiley & Sons, Ltd.

Let n be an integer and F={f/sub 1/:1/spl les/i/spl les/t for some integer t} be a finite set of linear functions. We define a linear congruential graph G(F, n) as a graph on the vertex set V={0, 1, ..., n-1}, in which any x/spl isin/V is adjacent to f/sub i/(x) mod n, 1/spl les/i/spl les/t. For a linear function g, and a subset V/sub 1/ of V we define a linear congruential graph G(F, n, g,V/sub 1/) as a graph on vertex set V, in which any x/spl isin/V is adjacent to f/sub i/(x) mod n, 1/spl les/i/spl les/t, and any x/spl isin/V/sub 1/ is also adjacent to g(x) mod n. These graphs generalize several well known families of graphs, e.g. the de Bruijn graphs. We give a family of linear functions, called DCC linear functions, that generate regular, highly connected graphs which are of substantially larger order than de Bruijn graphs of the same degree and diameter. Some theoretical and empirical properties of these graphs are given and their structural properties are studied.

Abstract For a given connected graph G of order n, a routing R is a set of n(n ‐ 1) simple paths R(x, y) specified for every ordered pair (x, y) of vertices in G. A routing R is consistent if for every vertex z on R(x, y) the paths R(x, z) and R(z, y) are induced by R(x, y). A network is defined by an ordered pair (G, R), where G is a connected graph and R is a routing of G. The vertex‐forwarding index ζ(G, R) of a network (G, R) is the maximum number of paths of R passing through any vertex of G. The minimum of ζ(G, R) over all possible routings of a connected graph G is denoted by ζ(G). Similarly, the notion of the edge‐forwarding index of a network can be defined. In this paper, we prove that, in general, the vertex‐forwarding index cannot be obtained by taking the minimum of ζ(G, R) over all possible consistent routings of G. However, this can be done for the class of Cayley graphs. We prove for a specific class of routings that the computation of ζ(G) is NP‐complete for graphs of diameter at least 4 and is polynomial for graphs of diameter 2. Similar problems are studied for the edge‐forwarding index. © 1994 John Wiley & Sons, Inc.

The problem of finding a total ordering of a finite set satisfying a given set of in-between restrictions is considered. It is shown that the problem is $NP$-complete.