Fast Structuring of Radio Networks for Multi-Message Communications

We introduce collision free layerings as a powerful way to structure radio networks. These layerings can replace hard-to-compute BFS-trees in many contexts while having an efficient randomized distributed construction. We demonstrate their versatility by using them to provide near optimal distributed algorithms for several multi-message communication primitives. Designing efficient communication primitives for radio networks has a rich history that began 25 years ago when Bar-Yehuda et al. introduced fast randomized algorithms for broadcasting and for constructing BFS-trees. Their BFS-tree construction time was $O(D \log^2 n)$ rounds, where $D$ is the network diameter and $n$ is the number of nodes. Since then, the complexity of a broadcast has been resolved to be $T_{BC} = \Theta(D \log \frac{n}{D} + \log^2 n)$ rounds. On the other hand, BFS-trees have been used as a crucial building block for many communication primitives and their construction time remained a bottleneck for these primitives. We introduce collision free layerings that can be used in place of BFS-trees and we give a randomized construction of these layerings that runs in nearly broadcast time, that is, w.h.p. in $T_{Lay} = O(D \log \frac{n}{D} + \log^{2+\epsilon} n)$ rounds for any constant $\epsilon>0$. We then use these layerings to obtain: (1) A randomized algorithm for gathering $k$ messages running w.h.p. in $O(T_{Lay} + k)$ rounds. (2) A randomized $k$-message broadcast algorithm running w.h.p. in $O(T_{Lay} + k \log n)$ rounds. These algorithms are optimal up to the small difference in the additive poly-logarithmic term between $T_{BC}$ and $T_{Lay}$. Moreover, they imply the first optimal $O(n \log n)$ round randomized gossip algorithm

Distributed Deterministic Broadcasting in Uniform-Power Ad Hoc Wireless Networks

Development of many futuristic technologies, such as MANET, VANET, iThings, nano-devices, depend on efficient distributed communication protocols in multi-hop ad hoc networks. A vast majority of research in this area focus on design heuristic protocols, and analyze their performance by simulations on networks generated randomly or obtained in practical measurements of some (usually small-size) wireless networks. %some library. Moreover, they often assume access to truly random sources, which is often not reasonable in case of wireless devices. In this work we use a formal framework to study the problem of broadcasting and its time complexity in any two dimensional Euclidean wireless network with uniform transmission powers. For the analysis, we consider two popular models of ad hoc networks based on the Signal-to-Interference-and-Noise Ratio (SINR): one with opportunistic links, and the other with randomly disturbed SINR. In the former model, we show that one of our algorithms accomplishes broadcasting in $O(D\log^2 n)$ rounds, where $n$ is the number of nodes and $D$ is the diameter of the network. If nodes know a priori the granularity $g$ of the network, i.e., the inverse of the maximum transmission range over the minimum distance between any two stations, a modification of this algorithm accomplishes broadcasting in $O(D\log g)$ rounds. Finally, we modify both algorithms to make them efficient in the latter model with randomly disturbed SINR, with only logarithmic growth of performance. Ours are the first provably efficient and well-scalable, under the two models, distributed deterministic solutions for the broadcast task.Comment: arXiv admin note: substantial text overlap with arXiv:1207.673

Searching with increasing speeds

© Springer Nature Switzerland AG 2018. In the classical search problem on the line or in higher dimension one is asked to find the shortest (and often the fastest) route to be adopted by a robot R from the starting point s towards the target point t located at unknown location and distance D. It is usually assumed that robot R moves with a fixed unit speed 1. It is well known that one can adopt a “zig-zag” strategy based on the exponential expansion, which allows to reach the target located on the line in time ≤9D and this bound is tight. The problem was also studied in two dimensions where the competitive factor is known to be O(D). In this paper we study an alteration of the search problem in which robot R starts moving with the initial speed 1. However, during search it can encounter a point or a sequence of points enabling faster and faster movement. The main goal is to adopt the route which allows R to reach the target t as quickly as possible. We study two variants of the considered search problem: (1) with the global knowledge and (2) with the local knowledge. In variant (1) robot R knows a priori the location of all intermediate points as well as their expulsion speeds. In this variant we study the complexity of computing optimal search trajectories. In variant (2) the relevant information about points in P is acquired by R gradually, i.e., while moving along the adopted trajectory. Here the focus is on the competitive factor of the solution, i.e., the ratio between the solutions computed in variants (2) and (1). We also consider two types of search spaces with points distributed on the line and subsequently with points distributed in two-dimensional space

Quantum and approximation algorithms for maximum witnesses of Boolean matrix products

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n\times n Boolean matrices of the form O(n^{2.575}) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the standard computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time \tilde{O}(n^{2+\lambda/2})=\tilde{O}(n^{2.434}), where \lambda satisfies the equation \omega(1, \lambda, 1) = 1 + 1.5 \, \lambda and \omega(1, \lambda, 1) is the exponent of the multiplication of an n \times n^{\lambda}$ matrix by an n^{\lambda} \times n matrix. Next, we consider a relaxed version of the MW problem (in the standard model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most \ell in time \tilde{O}((n/\ell)n^{\omega(1,\log_n \ell,1)}). Then, by reducing the relaxed problem to the so called k-witness problem, we provide an algorithm that reports for each non-zero entry C[i,j] of the product matrix C a witness of rank O(\lceil W_C(i,j)/k\rceil ), where W_C(i,j) is the number of witnesses for C[i,j], with high probability. The algorithm runs in \tilde{O}(n^{\omega}k^{0.4653} +n^2k) time, where \omega=\omega(1,1,1).Comment: 14 pages, 3 figure

Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P)

In this paper, a compressed membership problem for finite automata, both deterministic and non-deterministic, with compressed transition labels is studied. The compression is represented by straight-line programs (SLPs), i.e. context-free grammars generating exactly one string. A novel technique of dealing with SLPs is introduced: the SLPs are recompressed, so that substrings of the input text are encoded in SLPs labelling the transitions of the NFA (DFA) in the same way, as in the SLP representing the input text. To this end, the SLPs are locally decompressed and then recompressed in a uniform way. Furthermore, such recompression induces only small changes in the automaton, in particular, the size of the automaton remains polynomial. Using this technique it is shown that the compressed membership for NFA with compressed labels is in NP, thus confirming the conjecture of Plandowski and Rytter and extending the partial result of Lohrey and Mathissen; as it is already known, that this problem is NP-hard, we settle its exact computational complexity. Moreover, the same technique applied to the compressed membership for DFA with compressed labels yields that this problem is in P; for this problem, only trivial upper-bound PSPACE was known

Pushing the online matrix-vector conjecture off-line and identifying its easy cases

© Springer Nature Switzerland AG 2019. Henzinger et al. posed the so called Online Boolean Matrix-vector Multiplication (OMv) conjecture and showed that it implies tight hardness results for several basic partially dynamic or dynamic problems [STOC’15]. We show that the OMv conjecture is implied by a simple off-line conjecture. If a not uniform (i.e., it might be different for different matrices) polynomial-time preprocessing of the matrix in the OMv conjecture is allowed then we can show such a variant of the OMv conjecture to be equivalent to our off-line conjecture. On the other hand, we show that the OMV conjecture does not hold in the restricted cases when the rows of the matrix or the input vectors are clustered

Information Gathering in Ad-Hoc Radio Networks with Tree Topology

We study the problem of information gathering in ad-hoc radio networks without collision detection, focussing on the case when the network forms a tree, with edges directed towards the root. Initially, each node has a piece of information that we refer to as a rumor. Our goal is to design protocols that deliver all rumors to the root of the tree as quickly as possible. The protocol must complete this task within its allotted time even though the actual tree topology is unknown when the computation starts. In the deterministic case, assuming that the nodes are labeled with small integers, we give an O(n)-time protocol that uses unbounded messages, and an O(n log n)-time protocol using bounded messages, where any message can include only one rumor. We also consider fire-and-forward protocols, in which a node can only transmit its own rumor or the rumor received in the previous step. We give a deterministic fire-and- forward protocol with running time O(n^1.5), and we show that it is asymptotically optimal. We then study randomized algorithms where the nodes are not labelled. In this model, we give an O(n log n)-time protocol and we prove that this bound is asymptotically optimal

Anonymous Graph Exploration with Binoculars

International audienceWe investigate the exploration of networks by a mobile agent. It is long known that, without global information about the graph, it is not possible to make the agent halts after the exploration except if the graph is a tree. We therefore endow the agent with binoculars, a sensing device that can show the local structure of the environment at a constant distance of the agent current location.We show that, with binoculars, it is possible to explore and halt in a large class of non-tree networks. We give a complete characterization of the class of networks that can be explored using binoculars using standard notions of discrete topology. This class is much larger than the class of trees: it contains in particular chordal graphs, plane triangulations and triangulations of the projective plane. Our characterization is constructive, we present an Exploration algorithm that is universal; this algorithm explores any network explorable with binoculars, and never halts in non-explorable networks

Fingerprints in Compressed Strings

The Karp-Rabin fingerprint of a string is a type of hash value that due to its strong properties has been used in many string algorithms. In this paper we show how to construct a data structure for a string S of size N compressed by a context-free grammar of size n that answers fingerprint queries. That is, given indices i and j, the answer to a query is the fingerprint of the substring S[i,j]. We present the first O(n) space data structures that answer fingerprint queries without decompressing any characters. For Straight Line Programs (SLP) we get O(logN) query time, and for Linear SLPs (an SLP derivative that captures LZ78 compression and its variations) we get O(log log N) query time. Hence, our data structures has the same time and space complexity as for random access in SLPs. We utilize the fingerprint data structures to solve the longest common extension problem in query time O(log N log l) and O(log l log log l + log log N) for SLPs and Linear SLPs, respectively. Here, l denotes the length of the LCE

More efficient periodic traversal in anonymous undirected graphs

We consider the problem of periodic graph exploration in which a mobile entity with constant memory, an agent, has to visit all n nodes of an arbitrary undirected graph G in a periodic manner. Graphs are supposed to be anonymous, that is, nodes are unlabeled. However, while visiting a node, the robot has to distinguish between edges incident to it. For each node v the endpoints of the edges incident to v are uniquely identified by different integer labels called port numbers. We are interested in minimisation of the length of the exploration period. This problem is unsolvable if the local port numbers are set arbitrarily. However, surprisingly small periods can be achieved when assigning carefully the local port numbers. Dobrev et al. described an algorithm for assigning port numbers, and an oblivious agent (i.e. agent with no memory) using it, such that the agent explores all graphs of size n within period 10n. Providing the agent with a constant number of memory bits, the optimal length of the period was previously proved to be no more than 3.75n (using a different assignment of the port numbers). In this paper, we improve both these bounds. More precisely, we show a period of length at most 4 1/3 n for oblivious agents, and a period of length at most 3.5n for agents with constant memory. Moreover, we give the first non-trivial lower bound, 2.8n, on the period length for the oblivious case