Suffix trees and suffix arrays are two fundamental data structures in the field of string algorithms. For a string (a.k.a. text or sequence) of length n over an alphabet of size sigma, these structures typically require O(n log n) bits of space. The FM-index provides a compressed representation of the suffix array in approximate to n log sigma bits, allowing for efficient queries on both the suffix array and its inverse array in near logarithmic time. In certain applications, such as approximate pattern matching (i.e., with wildcards, mismatches, edits), there is a need to access the suffix array of a text, as well as the suffix array of text's reverse. Motivated by this, we explore the possibility of encoding the suffix array of the reversed text in a compact form, assuming the availability of the FM-index for the original text. Our first solution is an O(n)-bit (relative) encoding of the suffix array of the reversed text, with the time for decoding an entry being only O(log* n) times that of decoding an entry in the text's suffix array using FM-index. We then demonstrate how to reduce the space to O(n/kappa) bits for a parameter kappa, while multiplicative factor in time becomes approximately O(kappa log* n + kappa(3)). We can also support inverse suffix array and longest common extension queries on the reversed text. These results are achieved through some careful and non-trivial application of various succinct data structure techniques.
We prove that, paying a polynomial increase in size only, every unrestricted two-way nondeterministic finite automaton (2NFA) can be complemented by a 1-limited automaton (1-LA), a nondeterministic extension of 2NFAS still characterizing regular languages. The resulting machine is actually a restricted form of 1-LAS - known as 2NFAS with common guess - and is self-verifying. A corollary of our construction is that a single exponential is necessary and sufficient for complementing 1-LAS.
We study the rent-or-buy variant of the online Steiner forest problem on node- and edge-weighted graphs. For n-node graphs with at most (n) over bar nodes of non-zero weight, and at most (k) over tilde different arriving terminal pairs, we obtain the following: - A deterministic, O(log n log (n) over bar)-competitive algorithm against adaptive adversaries. This improves on the previous best, O(log(4) n)-competitive algorithm obtained by the black-box reduction from [5] combined with the previously best deterministic algorithms for the simpler "buy-only" setting. - A deterministic, O((n) over bar log (k) over tilde)-competitive algorithm against adaptive adversaries. This generalizes the O(log (k) over tilde)-competitive algorithm for the purely edge-weighted setting from [24]. - A randomized, O(log (k) over tilde log (n) over bar)-competitive algorithm against oblivious adversaries. All previous approaches were based on the randomized, black-box reduction from [3] that achieves a O(log (k) over tildek log n)-competitive ratio when combined with an algorithm for the "buy-only" setting. Our key technical ingredient is a novel charging scheme to an instance of online prize-collecting set cover. This allows us to extend the witness-technique of [24] to the node-weighted setting and obtain refined guarantees with respect to (n) over bar, already in the much simpler "buy-only" setting.
With substantial progress in automated reasoning, algebraic approaches emerged to automatically analyse program loops in an exact manner. In this invited talk, we discuss recent results in characterizing the functional behaviour of loops with polynomial arithmetic and probabilistic updates. This problem remains unsolved even when we restrict consideration to loops that are non-nested, without conditionals, and/or without exit conditions [5, 11]. We are motivated by applications of computer-aided verification, in particular to assess the safety, security, and sensitivity of computer systems [8, 3, 2, 9, 1, 6]. We are interested in modeling, deciding, and solving loop analysis. The key to our work are moment-computable loops [7, 10] which allow us to set limits on what is decidable and solvable in loop analysis. Our approach combines algebra, statistics, and automated reasoning to mechanize loop analysis. Various techniques, such as martingale theory and quantifier elimination, can be seen as examples of moment-computable loop analysis. This talk is structured within three inter-connected parts. We first bring moment-based loop analysis into the landscape of loop invariant synthesis and extend moment-computable loops with termination guarantees. We next automate the reasoning about (probabilistic) loops by summarizing loop semantics as (probabilistic) algebraic recurrences, whose closed-form solutions capture (higher-order) moments, and hence invariants, among loop variables. These recurrences together with loop tests yield moment-based (super)martingales necessary to prove loop termination and compute probability bounds on termination. We finally describe moment-computable loops whose invariant synthesis decidable or as hard as open problems, such as the Skolem problem [4, 12].
