Search

We revisit the linear search problem where a robot, initially placed at the origin on an infinite line, tries to locate a stationary tar-get placed at an unknown position on the line. Unlike previous studies, in which the robot travels along the line at a constant speed, we con-sider settings where the robot’s speed can depend on the direction of travel along the line, or on the profile of the terrain, e.g. when the line is inclined, and the robot can accelerate. Our objective is to design search algorithms that achieve good competitive ratios for the time spent by the robot to complete its search versus the time spent by an omniscient robot that knows the location of the target. We consider several new robot mobility models in which the speed of the robot depends on the terrain. These include (1) different con-stant speeds for different directions, (2) speed with constant acceleration and/or variability depending on whether a certain segment has already been searched, (3) speed dependent on the incline of the terrain. We pro-vide both upper and lower bounds on the competitive ratios of search algorithms for these models, and in many cases, we derive optimal algo-rithms for the search time.

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak bar-rier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (M

We consider Persistence, a new online problem concerning optimizing weighted observations in a stream of data when the observer has limited buffer capacity. A stream of weighted items arrive one at a time at the entrance of a buffer with two holding locations. A processor (or observer) can process (observe) an item at the buffer location it chooses, deriving this way the weight of the observed item as profit. The main constraint is that the processor can only move synchronously with the item stream. Persistence is the online problem of scheduling the processor movements through the buffer so that its total derived value is maximized under this constraint. We study the performance of the straight-forward heuristic Threshold, i.e., forcing the processor to “follow” an item through the whole buffer only if its value is above a threshold. We analyze both the optimal offline and Threshold algorithms in the cases where the input stream is either a random permutation, or its items are iid valued. We show that in both cases the competitive ratio achieved by the Threshold algorithm is at least 2/3 when the only statistical knowledge of the items is the median of all possible values. We generalize our results by showing that Threshold, equipped with some minimal statistical advice about the input, achieves competitive ratios in the whole spectrum between 2/3 and 1, following the variation of a newly defined density-like measure of the input. This result is a significant improvement over the case of arbitrary input streams, where we show that no online algorithm can achieve a competitive ratio better than 1/2.

Consider a theatre consisting of m rows each containing n seats. Theatregoers enter the theatre along aisles and pick a row which they enter along one of its two entrances so as to occupy a seat. Assume they select their seats uniformly and independently at random among the empty ones. A row of seats is narrow and an occupant who is already occupying a seat is blocking passage to new incoming theatregoers. As a consequence, occupying a specific seat depends on the courtesy of theatregoers and their willingness to get up so as to create free space that will allow passage to others. Thus, courtesy facilitates and may well increase the overall seat occupancy of the theatre. We say a theatregoer is courteous if (s)he will get up to let others pass. Otherwise, the theatregoer is selfish. A set of theatregoers is courteous with probability p (or p-courteous, for short) if each theatregoer in the set is courteous with probability p, randomly and independently. It is assumed that the behaviour of a theatregoer does not change during the occupancy of the row. Thus, p=1 represents the case where all theatregoers are courteous and p=0 when they are all selfish. In this paper, we are interested in the following question: what is the expected number of occupied seats as a function of the total number of seats in a theatre, n, and the probability that a theatregoer is courteous, p? We study and analyze interesting variants of this problem reflecting behaviour of the theatregoers as entirely selfish, and p-courteous for a row of seats with one or two entrances and as a consequence for a theatre with m rows of seats with multiple aisles. We also consider the case where seats in a row are chosen according to the geometric distribution and the Zipf distibrution (as opposed to the uniform distribution) and provide bounds on the occupancy of a row (and thus the theatre) in each case. Finally, we propose several open problems for other seating probability distributions and theatre seati

Background: RNA folding depends on the distribution of kinetic traps in the landscape of all secondary structures. Kinetic traps in the Nussinov energy model are precisely those secondary structures that are saturated, meaning that no base pair can be added without introducing either a pseudoknot or base triple. In previous work, we investigated asymptotic combinatorics of both random saturated structures and of quasi-random saturated structures, where the latter are constructed by a natural stochastic process.Results: We prove that for quasi-random saturated structures with the uniform distribution, the asymptotic expected number of external loops is O(logn) and the asymptotic expected maximum stem length is O(logn), while under the Zipf distribution, the asymptotic expected number of external loops is O(log2n) and the asymptotic expected maximum stem length is O(logn/log logn).Conclusions: Quasi-random saturated structures are generated by a stochastic greedy method, which is simple to implement. Structural features of random saturated structures appear to resemble those of quasi-random saturated structures, and the latter appear to constitute a class for which both the generation of sampled structures as well as a combinatorial investigation of structural features may be simpler to undertake.

A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, we are interested in finding final positions of sensors which establish a barrier coverage of I so that the sum of the distances traveled by all sensors from initial to final positions is minimized. It is shown that the problem is NP complete even to approximate up to constant factor when the sensors may have different sensing ranges. When the sensors have an identical sensing range we give several efficient algorithms to calculate the final destinations so that the sensors either establish a barrier coverage or maximize the coverage of the segment if complete coverage is not feasible while at the same time the sum of the distances traveled by all sensors is minimized. Some open problems are also mentioned.

Delay (or disruption) tolerant sensor networks may be modeled as Markovian evolving graphs [1]. We present experimental evidence showing that considering multiple (possibly not shortest) paths instead of one fixed (greedy) path can decrease the expected time to deliver a packet on such a network by as much as 65 per cent depending on the probability that an edge exists in a given time interval. We provide theoretical justification for this result by studying a special case of the Markovian evolving grid graph. We analyze a natural algorithm for routing on such networks and show that it is possible to improve the expected time of delivery by up to a factor of two depending upon the probability of an edge being up during a time step and the relative positions of the source and destination. Furthermore we show that this is optimal, i.e., no other algorithm can achieve a better expected running time. As an aside, our results give high probability bounds for Knuth's toilet paper problem [11].

A graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G \ v is a 2-tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for recognizing and realizing degree sequences of 2-trees.

Recent results on the problem of mobile agent rendezvous on distributed networks are surveyed with an emphasis on outlining the various approaches taken by researchers in the theoretical computer science community.

We consider the rendezvous problem for identical mobile agents (i.e., running the same deterministic algorithm) with tokens in a synchronous torus with a sense of direction and show that there is a striking computational difference between one and more tokens. More specifically, we show that 1) two agents with a constant number of unmovable tokens, or with one movable token, each cannot rendezvous if they have o(log n) memory, while they can perform rendezvous with detection as long as they have one unmovable token and O(log n) memory; in contrast, 2) when two agents have two movable tokens each then rendezvous (respectively, rendezvous with detection) is possible with constant memory in an arbitrary n × m (respectively, n × n) torus; and finally, 3) two agents with three movable tokens each and constant memory can perform rendezvous with detection in a n × m torus. This is the first publication in the literature that studies tradeoffs between the number of tokens, memory and knowledge the agents need in order to meet in such a network.

We present deterministic algorithms to search for an item s contained in a node of a network, without prior knowledge of its exact location. Each node of the network has a database that will answer queries of the form "how do I get to s?" by responding with the first edge on a shortest path to the node containing s. It may happen that some nodes, called liars, give bad advice. If the number of liars k is bounded, we show different strategies to find the item depending on the topology of the network. In particular we consider the complete graph, ring, torus, hypercube and bounded degree trees.

The problem of searching for a piece of information in a fully interconnected computer network by exploiting advice about its location from the network nodes is discussed. Each node contains a database that knows what kind of documents or information are stored in other nodes. The databases in each node, when queried, provide a pointer that leads to the node that contains the information. It is found that the information is up-to-date with some bounded probability.

We consider the problem of distributed deterministic broadcasting in radio networks whose nodes are located on a line. Nodes send messages in synchronous time-slots. Each node v has a given transmission range. All nodes located within this range can receive messages from v. However, a node situated in the range of two or more nodes that send messages simultaneously, cannot receive these messages and hears only noise. Each node knows only its own position and range, as well as the maximum of all ranges. Broadcasting is adaptive: nodes can decide on the action to take on the basis of previously received messages, silence or noise. We prove lower bounds on broadcasting time in this model and construct broadcasting protocols whose performance nearly matches these bounds for the simplest case when nodes are situated on a line. We also show that if nodes do not even know their own range, every broadcasting protocol must be hopelessly slow. While distributed randomized broadcasting algorithms, and, on the other hand, deterministic off-line broadcasting algorithms assuming full knowledge of the radio network, have been extensively studied in the literature, ours are the first results concerning broadcasting algorithms that are distributed and deterministic at the same time. We show that in this case, information available to nodes influences the efficiency of broadcasting in a significant way.

In recent years there has been significant interest in the study of random k-SAT formulae. For a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct, non-complementary literals from its variables (k-clauses). A random k-SAT formula Fk(n,m) is formed by selecting uniformly and independently m clauses from Bk and taking their conjunction. Motivated by insights from statistical mechanics that suggest a possible relationship between the "order" of phase transitions and computational complexity, Monasson and Zecchina (Phys. Rev. E 56(2) (1997) 1357) proposed the random (2+p)-SAT model: for a given p∈[0,1], a random (2+p)-SAT formula, F2+p(n,m), has m randomly chosen clauses over n variables, where pm clauses are chosen from B3 and (1-p)m from B2. Using the heuristic "replica method" of statistical mechanics, Monasson and Zecchina gave a number of non-rigorous predictions on the behavior of random (2+p)-SAT formulae. In this paper we give the first rigorous results for random (2+p)-SAT, including the following surprising fact: for p≤2/5, with probability 1-o(1), a random (2+p)-SAT formula is satisfiable iff its 2-SAT subformula is satisfiable. That is, for p≤2/5, random (2+p)-SAT behaves like random 2-SAT.

In recent years there has been significant interest in the study of random k-SAT formulae. For a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct, non-complementary literals from its variables (k-clauses). A random k-SAT formula Fk(n,m) is formed by selecting uniformly and independently m clauses from Bk and taking their conjunction. Motivated by insights from statistical mechanics that suggest a possible relationship between the "order" of phase transitions and computational complexity, Monasson and Zecchina (Phys. Rev. E 56(2) (1997) 1357) proposed the random (2+p)-SAT model: for a given p∈[0,1], a random (2+p)-SAT formula, F2+p(n,m), has m randomly chosen clauses over n variables, where pm clauses are chosen from B3 and (1-p)m from B2. Using the heuristic "replica method" of statistical mechanics, Monasson and Zecchina gave a number of non-rigorous predictions on the behavior of random (2+p)-SAT formulae. In this paper we give the first rigorous results for random (2+p)-SAT, including the following surprising fact: for p≤2/5, with probability 1-o(1), a random (2+p)-SAT formula is satisfiable iff its 2-SAT subformula is satisfiable. That is, for p≤2/5, random (2+p)-SAT behaves like random 2-SAT.

In this paper we examine the problem of searching for some information item in the nodes of a fully interconnected computer network, where each node contains information relevant to some topic as well as links to other network nodes that also contain information, not necessarily related to locally kept information. These links are used to facilitate the Internet users and mobile software agents that try to locate specific pieces of information. However, the links do not necessarily point to nodes containing information of interest to the user or relevant to the aims of the mobile agent. Thus an element of uncertainty is introduced. For example, when an Internet user or some search agent lands on a particular network node, they see a set of links that point to information that is, supposedly, relevant to the current search. Therefore, we can assume that a link points to relevant information with some unknown probability p that, in general, is related to the number of nodes in the network (intuitively, as the network grows, this probability tends to zero since adding more nodes to the network renders some extant l

Consider a directed rooted tree T = (V,E) of maximal degree d representing a collection V of web pages connected via a set E of links all reachable from a source home page, represented by the root of T. Each leaf web page carries a weight representative of the frequency with which it is visited. By adding hotlinks, shortcuts from a node to one of its descendents, we are interested in minimizing the expected number of steps needed to visit the leaf pages from the home page. We give an O(N 2) time algorithm for assigning hotlinks so that the expected number of steps to reach the leaves from the root of the tree is at most H(p)/ log(d+1)-(d/(d+1)) log d + d+1/d, where H(p) is the entropy of the probability (frequency) distribution p =< p1,p2, pN > on the N leaves of the given tree, i.e., pi is the weight on the ith leaf. The best known lower bound for this problem is H(p)/ log(d+1). Thus our algorithm approximates the optimal hotlink assignment to within a constant for any fixed d.

Consider a set X of n-points in space, each with positive z-coordinate, and a compact plane convex set S. We define a graph G(X,S), called the ray-shooting graph, on X as follows: The points of X are the vertices and {pi,pj},pi,pj∈X, is an edge if and only if the infinite line joining pi to pj intersects S. In this paper we provide a characterization of shooting graphs; they are exactly the set of comparability graphs of containment orders arising from families of plane homothetic convex sets. We also prove that the crossing number of these ordered sets is at most 2, and that not all comparability graphs are three-dimensional ray-shooting graphs.

In wireless communication, the signal of a typical broadcast station is transmited from a broadcast center p and reaches objects at a distance, say, R from it. In addition there is a radius r, r < R, such that the signal originating from the center of the station is so strong that human habitation within distance r from the center p should be avoided. Thus every station determines a region which is an “annulus of permissible habitation". We consider the following station layout (SL) problem: Cover a given (say, rectangular) planar region which includes a collection of orthogonal buildings with a minimum number of stations so that every point in the region is within the reach of a station, while at the same time no building is within the dangerous range of a station. We give algorithms for computing such station layouts in both the one-and two-dimensional cases.

A function is distributed among nodes of a graph in a " continuous" way, i.e., such that the difference between values stored at adjacent nodes is small. The goal is to find a node of maximum value by probing some nodes under a restricted budget. Every node has an associated cost which has to be paid for probing it and a probe reveals the value of the node. If the total budget is too small to allow probing every node, it is impossible to find the maximum value in the worst case. Hence we seek an Approximate Maxima Finding (AMF) algorithm that offers the best worst-case guarantee g, i.e., for any continuous distribution of values it finds a node whose value differs from the maximum value by at most g. Approximate Maxima Finding in graphs is related to a generalization of the multicenter problem and we get new results for this problem as well. For example, we give a polynomial algorithm to find a minimum cost solution for the multicenter problem on a tree, with arbitrary node costs.

We consider graph applications of the well-known paradigm "killing two birds with one stone". In the plane, this gives rise to a stage graph as follows: vertices are the points, and {u, v} is an edge if and only if the (infinite, straight) line segment joining u to v intersects the stage. Such graphs are shown to be comparability graphs of ordered sets of dimension 2. Similar graphs can be constructed when we have a fixed number k of stages on the plane. In this case, {u, v} is an edge if and only if the (straight) line segment uv intersects one of the k stages. In this paper, we study stage representations of stage graphs and give upper and lower bounds on the number of stages needed to represent a graph.

We consider labelings (i.e. assignments of labels to the links that give the network a globally consistent orientation) on anonymous Cayley networks NG constructed from a set G of generators of a group G. Such networks can be endowed with a natural labeling LG to form the oriented Cayley network, denoted by NG[LG]. We show that in general oriented Cayley networks are more powerful than unoriented Cayley networks, in the sense that the former can compute more Boolean functions than the latter. We also give a characterization of those Abelian groups G which have a canonical set of generators G such that the network NG computes more Boolean functions than the network NG[LG].