Brief Announcement: New Clocks, Fast Line Formation and Self-Replication Population Protocols

In this paper we consider a known variant of the standard population protocol model in which agents can be connected by edges, referred to as the network constructor model. During an interaction between two agents the relevant connecting edge can be formed, maintained or eliminated by the transition function. The state space of agents is fixed (constant size) and the size n of the population is not known, i.e., not hard-coded in the transition function. Since pairs of agents are chosen uniformly at random the status of each edge is updated every Θ(n2) interactions in expectation which coincides with Θ(n) parallel time. This phenomenon provides a natural lower bound on the time complexity for any non-trivial network construction designed for this variant. This is in contrast with the standard population protocol model in which efficient protocols operate in O(poly log n) parallel time. The main focus in this paper is on efficient manipulation of linear structures including formation, self-replication and distribution (including pipelining) of complex information in the adopted model. We propose and analyse a novel edge based phase clock counting parallel time Θ(n log n) in the network constructor model, showing also that its leader based counterpart provides the same time guaranties in the standard population protocol model. Note that all currently known phase clocks can count parallel time not exceeding O(poly log n). The new clock enables a nearly optimal O(n log n) parallel time spanning line construction (a key component of universal network construction), which improves dramatically on the best currently known O(n2) parallel time protocol, solving the main open problem in the considered model [9]. We propose a new probabilistic bubble-sort algorithm in which random comparisons and transfers are allowed only between the adjacent positions in the sequence. Utilising a novel potential function reasoning we show that rather surprisingly this probabilistic sorting (via conditional pipelining) procedure requires O(n2) comparisons in expectation and whp, and is on par with its deterministic counterpart. We propose the first population protocol allowing self-replication of a strand of an arbitrary length k (carrying a k-bit message of size independent of the state space) in parallel time O(n(k + log n)). The pipelining mechanism and the time complexity analysis of the strand self-replication protocol mimic those used in the probabilistic bubble-sort. The new protocol permits also simultaneous self-replication, where l copies of the strand can be created in time O(n(k + log n) log l). Finally, we discuss application of the strand self-replication protocol to pattern matching. Our protocols are always correct and provide time guaranties with high probability defined as 1 - n-η, for a constant η > 0

Upper and lower bounds for dynamic data structures on strings

We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an $O(m^{1/2-\varepsilon})$ time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conjecture is false. We also provide nearly matching upper bounds for most of the problems we consider.Comment: Accepted at STACS'1

Fast and Compact Regular Expression Matching

We study 4 problems in string matching, namely, regular expression matching, approximate regular expression matching, string edit distance, and subsequence indexing, on a standard word RAM model of computation that allows logarithmic-sized words to be manipulated in constant time. We show how to improve the space and/or remove a dependency on the alphabet size for each problem using either an improved tabulation technique of an existing algorithm or by combining known algorithms in a new way

Improved Approximate String Matching and Regular Expression Matching on Ziv-Lempel Compressed Texts

We study the approximate string matching and regular expression matching problem for the case when the text to be searched is compressed with the Ziv-Lempel adaptive dictionary compression schemes. We present a time-space trade-off that leads to algorithms improving the previously known complexities for both problems. In particular, we significantly improve the space bounds, which in practical applications are likely to be a bottleneck