Flocking with discrete symmetry: the 2d Active Ising Model

We study in detail the active Ising model, a stochastic lattice gas where collective motion emerges from the spontaneous breaking of a discrete symmetry. On a 2d lattice, active particles undergo a diffusion biased in one of two possible directions (left and right) and align ferromagnetically their direction of motion, hence yielding a minimal flocking model with discrete rotational symmetry. We show that the transition to collective motion amounts in this model to a bona fide liquid-gas phase transition in the canonical ensemble. The phase diagram in the density/velocity parameter plane has a critical point at zero velocity which belongs to the Ising universality class. In the density/temperature "canonical" ensemble, the usual critical point of the equilibrium liquid-gas transition is sent to infinite density because the different symmetries between liquid and gas phases preclude a supercritical region. We build a continuum theory which reproduces qualitatively the behavior of the microscopic model. In particular we predict analytically the shapes of the phase diagrams in the vicinity of the critical points, the binodal and spinodal densities at coexistence, and the speeds and shapes of the phase-separated profiles.Comment: 20 pages, 25 figure

Active Brownian Particles and Run-and-Tumble Particles: a Comparative Study

Active Brownian particles (ABPs) and Run-and-Tumble particles (RTPs) both self-propel at fixed speed $v$ along a body-axis ${\bf u}$ that reorients either through slow angular diffusion (ABPs) or sudden complete randomisation (RTPs). We compare the physics of these two model systems both at microscopic and macroscopic scales. Using exact results for their steady-state distribution in the presence of external potentials, we show that they both admit the same effective equilibrium regime perturbatively that breaks down for stronger external potentials, in a model-dependent way. In the presence of collisional repulsions such particles slow down at high density: their propulsive effort is unchanged, but their average speed along ${\bf u}$ becomes $v(\rho) < v$. A fruitful avenue is then to construct a mean-field description in which particles are ghost-like and have no collisions, but swim at a variable speed $v$ that is an explicit function or functional of the density $\rho$. We give numerical evidence that the recently shown equivalence of the fluctuating hydrodynamics of ABPs and RTPs in this case, which we detail here, extends to microscopic models of ABPs and RTPs interacting with repulsive forces.Comment: 32 pages, 6 figure

Optimal paths on the road network as directed polymers

We analyze the statistics of the shortest and fastest paths on the road network between randomly sampled end points. To a good approximation, these optimal paths are found to be directed in that their lengths (at large scales) are linearly proportional to the absolute distance between them. This motivates comparisons to universal features of directed polymers in random media. There are similarities in scalings of fluctuations in length/time and transverse wanderings, but also important distinctions in the scaling exponents, likely due to long-range correlations in geographic and man-made features. At short scales the optimal paths are not directed due to circuitous excursions governed by a fat-tailed (power-law) probability distribution.Comment: 5 pages, 7 figure

Sparse Suffix and LCP Array: Simple, Direct, Small, and Fast

Sparse suffix sorting is the problem of sorting $b=o(n)$ suffixes of a string of length $n$. Efficient sparse suffix sorting algorithms have existed for more than a decade. Despite the multitude of works and their justified claims for applications in text indexing, the existing algorithms have not been employed by practitioners. Arguably this is because there are no simple, direct, and efficient algorithms for sparse suffix array construction. We provide two new algorithms for constructing the sparse suffix and LCP arrays that are simultaneously simple, direct, small, and fast. In particular, our algorithms are: simple in the sense that they can be implemented using only basic data structures; direct in the sense that the output arrays are not a byproduct of constructing the sparse suffix tree or an LCE data structure; fast in the sense that they run in $\mathcal{O}(n\log b)$ time, in the worst case, or in $\mathcal{O}(n)$ time, when the total number of suffixes with an LCP value greater than $2^{\lfloor \log \frac{n}{b} \rfloor + 1}-1$ is in $\mathcal{O}(b/\log b)$, matching the time of the optimal yet much more complicated algorithms [Gawrychowski and Kociumaka, SODA 2017; Birenzwige et al., SODA 2020]; and small in the sense that they can be implemented using only $8b+o(b)$ machine words. Our algorithms are simplified, yet non-trivial, space-efficient adaptations of the Monte Carlo algorithm by I et al. for constructing the sparse suffix tree in $\mathcal{O}(n\log b)$ time [STACS 2014]. We also provide proof-of-concept experiments to justify our claims on simplicity and efficiency.Comment: 16 pages, 1 figur

Range Shortest Unique Substring queries

Let be a string of length n and be the substring of starting at position i and ending at position j. A substring of is a repeat if it occurs more than once in; otherwise, it is a unique substring of. Repeats and unique substrings are of great interest in computational biology and in information retrieval. Given string as input, the Shortest Unique Substring problem is to find a shortest substring of that does not occur elsewhere in. In this paper, we introduce the range variant of this problem, which we call the Range Shortest Unique Substring problem. The task is to construct a data structure over answering the following type of online queries efficiently. Given a range, return a shortest substring of with exactly one occurrence in. We present an -word data structure with query time, where is the word size. Our construction is based on a non-trivial reduction allowing us to apply a recently introduced optimal geometric data structure [Chan et al. ICALP 2018]

Longest common substring made fully dynamic

Given two strings S and T, each of length at most n, the longest common substring (LCS) problem is to find a longest substring common to S and T. This is a classical problem in computer science with an O(n)-time solution. In the fully dynamic setting, edit operations are allowed in either of the two strings, and the problem is to find an LCS after each edit. We present the first solution to this problem requiring sublinear time in n per edit operation. In particular, we show how to find an LCS after each edit operation in Õ(n2/3) time, after Õ(n)-time and space preprocessing. 1 This line of research has been recently initiated in a somewhat restricted dynamic variant by Amir et al. [SPIRE 2017]. More specifically, they presented an Õ(n)-sized data structure that returns an LCS of the two strings after a single edit operation (that is reverted afterwards) in Õ(1) time. At CPM 2018, three papers (Abedin et al., Funakoshi et al., and Urabe et al.) studied analogously restricted dynamic variants of problems on strings. We show that the techniques we develop can be applied to obtain fully dynamic algorithms for all of these variants. The only previously known sublinear-time dynamic algorithms for problems on strings were for maintaining a dynamic collection of strings for comparison queries and for pattern matching, with the most recent advances made by Gawrychowski et al. [SODA 2018] and by Clifford et al. [STACS 2018]. As an intermediate problem we consider computing the solution for a string with a given set of k edits, which leads us, in particular, to answering internal queries on a string. The input to such a query is specified by a substring (or substrings) of a given string. Data structures for answering internal string queries that were proposed by Kociumaka et al. [SODA 2015] and by Gagie et al. [CCCG 2013] are used, along with new ones, based on ingredients such as the suffix tree, heavy-path decomposition, orthogonal range queries, difference covers, and string periodicity

Constructing Antidictionaries of Long Texts in Output-Sensitive Space

A word x that is absent from a word y is called minimal if all its proper factors occur in y. Given a collection of k words y1, … , yk over an alphabet Σ, we are asked to compute the set M{y1,…,yk}ℓ of minimal absent words of length at most ℓ of the collection {y1, … , yk}. The set M{y1,…,yk}ℓ contains all the words x such that x is absent from all the words of the collection while there exist i,j, such that the maximal proper suffix of x is a factor of yi and the maximal proper prefix of x is a factor of yj. In data compression, this corresponds to computing the antidictionary of k documents. In bioinformatics, it corresponds to computing words that are absent from a genome of k chromosomes. Indeed, the set Myℓ of minimal absent words of a word y is equal to M{y1,…,yk}ℓ for any decomposition of y into a collection of words y1, … , yk such that there is an overlap of length at least ℓ − 1 between any two consecutive words in the collection. This computation generally requires Ω(n) space for n = |y| using any of the plenty available O(n) -time algorithms. This is because an Ω(n)-sized text index is constructed over y which can be impractical for large n. We do the identical computation incrementally using output-sensitive space. This goal is reasonable when ∥M{y1,…,yN}ℓ∥=o(n), for all N ∈ [1,k], where ∥S∥ denotes the sum of the lengths of words in set S. For instance, in the human genome, n ≈ 3 × 109 but ∥M{y1,…,yk}12∥≈106. We consider a constant-sized alphabet for stating our results. We show that allMy1ℓ,…,M{y1,…,yk}ℓ can be computed in O(kn+∑N=1k∥M{y1,…,yN}ℓ∥) total time using O(MaxIn+MaxOut) space, where MaxIn is the length of the longest word in {y1, … , yk} and MaxOut=max{∥M{y1,…,yN}ℓ∥:N∈[1,k]}. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution

Efficient data structures for range shortest unique substring queries†

Let T[1, n] be a string of length n and T[i, j] be the substring of T starting at position i and ending at position j. A substring T[i, j] of T is a repeat if it occurs more than once in T; otherwise, it is a unique substring of T. Repeats and unique substrings are of great interest in computational biology and information retrieval. Given string T as input, the Shortest Unique Substring problem is to find a shortest substring of T that does not occur elsewhere in T. In this paper, we introduce the range variant of this problem, which we call the Range Shortest Unique Substring problem. The task is to construct a data structure over T answering the following type of online queries efficiently. Given a range [α, β], return a shortest substring T[i, j] of T with exactly one occurrence in [α, β]. We present an O(n log n)-word data structure with O(logw n) query time, where w = Ω(log n) is the word size. Our construction is based on a non-trivial reduction allowing for us to apply a recently introduced optimal geometric data structure [Chan et al., ICALP 2018]. Additionally, we present an O(n)-word data structure with O(√ n logɛ n) query time, where ɛ > 0 is an arbitrarily small constant. The latter data structure relies heavily on another geometric data structure [Nekrich and Navarro, SWAT 2012]