arXiv subjects
Publications and source records attributed to Younan Gao.
We extend Baier's foundationial work on tunnelling Burrows-Wheeler Transforms (BWTs) by showing how something that would be a good tunnel except for a strings that diverge from the tunnel (``leaks'') can be patched and turned into a tunnel. Whereas augmenting a BWT with a normal tunnel takes extra space proportional to the tunnels length, augmenting it with a patched tunnels take that plus space proportional to the number of leaks.
Compactly representing a variation graph is a core problem in computational pangenomics that is usually attacked with techniques that have been originated on texts and adapted to graphs. In this paper we propose a new framework that takes a topology-centric perspective instead. A variation graph is modeled as a directed acyclic graph (DAG) together with a set of distinguished paths, where each path is assigned a distinct color. Our compact representation is centered on pebbling the graph, i.e. placing colored pebbles on edges so that every predefined path can be univocally reconstructed from the pebbled edges. In particular, a saturated pebbling marks each chosen edge with every path (color) traversing it. We first propose a data structure to represent and query a variation graph with storage space depending on the size of the pebbling. The supported queries are: (i) path query, which recovers a path given its color, and (ii) edge query, which reports the colors of paths traversing a given edge. We then prove that the problem of finding a pebbling of minimum size is solvable in polynomial time. On the contrary, we prove that finding a saturated pebbling of minimum size is NP-hard, but can be reduced to the minimum-weight set cover problem, allowing us to leverage integer linear programming (ILP) solvers. We show how to exploit saturated pebblings to achieve faster queries times than minimum size pebbling. Our framework opens a new algorithmic viewpoint on developing more efficient variation graph representations rooted on the study of the topology of those graphs.
We revisit the online construction of \emph{smallest suffixient sets} and the online computation of the \emph{longest repeating suffix} (LRS). We give the first compressed-space online construction of smallest suffixient sets, and present two space-time trade-offs for both problems: $O(r\log n+n)$ bits of working space and $O(\log^2 n/\log \log n)$ worst-case time per character, and $O(r\log n+n \log \log n)$ bits of working space and $O((\log n/\log \log n)^2)$ worst-case time per character. Here, $r$ is the number of runs in the Burrows-Wheeler transform of the reverse of $T[1..n]$. In particular, for highly repetitive texts satisfying $r=O(n/\log n)$, the first trade-off uses $O(n)$ bits of working space, while the second uses $O(n\log\log n)$ bits. We also prove that any deterministic online algorithm for computing LRS requires \(Ω(n)\) bits of peak working space in the worst case, even over a constant-size alphabet. Through reductions from online LRS computation, we extend this lower bound to deterministic online algorithms maintaining either an arbitrary smallest suffixient set augmented with the length of the supermaximal right extension represented by each selected position, or the position-only smallest suffixient set obtained by selecting the rightmost occurrence of every such extension. For constructing smallest suffixient sets, our algorithms are the first online solutions using compressed working space, improving the $O(n)$-word space required by previous online constructions. For compressed-space online LRS computation, compared with the algorithm of Prezza and Rosone~[CiE 2020], our bounds improve their $O(\log^2 n)$ amortized time per character by factors of $Θ(\log\log n)$ and $Θ((\log\log n)^2)$, respectively, while also providing worst-case guarantees.
The Positional Burrows-Wheeler Transform (PBWT) is a fundamental data structure for the efficient representation and analysis of large-scale haplotype panels. For a panel of $h$ sequences $\{S_1, \dots, S_h\}$ over $m$ sites, a key operation is the $ϕ_j(i)$ query, which returns the haplotype index immediately preceding $S_i$ in co-lexicographic order at site $j$. Efficient support for $k$ iterative queries $ϕ^1, \dots, ϕ^k$ is essential for haplotype matching and variation analysis. In this work, we introduce a simple and novel decomposition scheme that decomposes each haplotype row into sub-intervals, called refined segments, within which a haplotype's co-lexicographic predecessor for the sites remains unchanged. We show that refined segments satisfy two key properties: (i) each segment $[b,e]$ associated with $S_i$ overlaps with at most a constant number of segments of $S_{ϕ_e(i)}$, and (ii) the total number of segments is bounded by $O(\tilde{r} + h)$, where $\tilde{r}$ denotes the number of runs in the PBWT. Building on this decomposition, we present two space-time tradeoffs for supporting $k$ iterative $ϕ$ queries: (i) a structure using $O((\tilde{r} + h)\log n)$ bits of space that answers $k$ iterative queries in $O(\log \log_w \min(m,h) + k)$ time, where $n = m \cdot h$, and (ii) a more compact structure using $O(\tilde{r} \log h + h \log n)$ bits of space that supports queries in $O(k \log \log_w h)$ time. Prior to our work, supporting these queries required $O((\tilde{r} + h)\log n)$ bits of space and $O(k \cdot \log \log_w m)$ time. Our second tradeoff is expected to be effective in practice for modern genomic datasets, where the number $h$ of haplotypes is typically much smaller than the number $m$ of sites.
Recently, Cenzato et al.\ proposed a new text index, called the \emph{suffixient array}, which is a subset of the suffix array and supports locating a single pattern occurrence or finding its maximal exact matches (MEMs), assuming random access to the input text $T[1..n]$ is available. They show that, given the suffix array, the longest common prefix array, and the Burrows--Wheeler transform (BWT) of the reverse of $T[1..n]$ over an alphabet $\{1,\ldots,σ\}$, a suffixient array can be constructed in linear time. However, their construction algorithms require multiple scans of these arrays. When restricted to a single pass over the arrays, they present an alternative construction algorithm running in $O(n + \overline{r} \log σ)$ time, where $\overline{r}$ is the number of runs in the BWT of the reversed text. In this paper, we present a new one-pass algorithm that constructs a suffixient array in linear time under the standard RAM model.
The Positional Burrows--Wheeler Transform (PBWT) is a data structure designed for efficiently representing and querying large collections of sequences, such as haplotype panels in genomics. Forward and backward stepping operations -- analogues to LF- and FL-mapping in the traditional BWT -- are fundamental to the PBWT, underpinning many algorithms based on the PBWT for haplotype matching and related analyses. Although the run-length encoded variant of the PBWT (also known as the $μ$-PBWT) achieves $O(\newR)$-word space usage, where $\newR$ is the total number of runs, no data structure supporting both forward and backward stepping in constant time within this space bound was previously known. In this paper, we consider the multi-allelic PBWT that is extended from its original binary form to a general ordered alphabet $\{0, \dots, σ-1\}$. We first establish bounds on the size $\newR$ and then introduce a new $O(\newR)$-word data structure built over a list of haplotypes $\{S_1, \dots, S_\height\}$, each of length $\width$, that supports constant-time forward and backward stepping. We further revisit two key applications -- haplotype retrieval and prefix search -- leveraging our efficient forward stepping technique. Specifically, we design an $O(\newR)$-word space data structure that supports haplotype retrieval in $O(\log \log_{\word} h + \width)$ time. For prefix search, we present an $O(\height + \newR)$-word data structure that answers queries in $O(m' \log\log_{\word} σ+ \occ)$ time, where $m'$ denotes the length of the longest common prefix returned and $\occ$ denotes the number of haplotypes prefixed the longest prefix.
Several mobile agents, modelled as deterministic automata, navigate in an infinite line in synchronous rounds. All agents start in the same round. In each round, an agent can move to one of the two neighboring nodes, or stay idle. Agents have distinct labels which are integers from the set $\{1,\dots, L\}$. They start in teams, and all agents in a team have the same starting node. The adversary decides the compositions of teams, and their starting nodes. Whenever an agent enters a node, it sees the entry port number and the states of all collocated agents; this information forms the input of the agent on the basis of which it transits to the next state and decides the current action. The aim is for all agents to gather at the same node and stop. Gathering is feasible, if this task can be accomplished for any decisions of the adversary, and its time is the worst-case number of rounds from the start till gathering. We consider the feasibility and time complexity of gathering teams of agents, and give a complete solution of this problem. It turns out that both feasibility and complexity of gathering depend on the sizes of teams. We first concentrate on the case when all teams have the same size $x$. For the oriented line, gathering is impossible if $x=1$, and it can be accomplished in time $O(D)$, for $x>1$, where $D$ is the distance between the starting nodes of the most distant teams. This complexity is of course optimal. For the unoriented line, the situation is different. For $x=1$, gathering is also impossible, but for $x=2$, the optimal time of gathering is $Θ(D\log L)$, and for $x\geq 3$, the optimal time of gathering is $Θ(D)$. In the case when there are teams of different sizes, we show that gathering is always possible in time $O(D)$, even for the unoriented line. This complexity is of course optimal.
Two identical anonymous mobile agents have to meet at a node of the infinite oriented grid whose nodes are unlabeled. This problem is known as rendezvous. The agents execute the same deterministic algorithm. Time is divided into rounds, and in each round each agent can either stay idle at the current node or move to an adjacent node. An adversary places the agents at two nodes of the grid at a distance at most $D$, and wakes them up in possibly different rounds. Each agent starts executing the algorithm in its wakeup round. If agents cannot leave any marks on visited nodes then they can never meet, even if they start simultaneously at adjacent nodes and know it. Hence, we assume that each agent marks any unmarked node it visits, and that an agent can distinguish if a node it visits has been previously marked or not. The time of a rendezvous algorithm is the number of rounds between the wakeup of the later agent and rendezvous. We ask the question whether the capability of marking nodes enables the agents to meet, and if so, what is the fastest rendezvous algorithm. We consider this problem under three scenarios. First, agents know $D$ but may start with arbitrary delay. Second, they start simultaneously but do not have any {\em a priori} knowledge. Third, most difficult scenario, we do not make any of the above facilitating assumptions. Agents start with arbitrary delay and they do not have any a priori knowledge. We prove that in the first two scenarios rendezvous can be accomplished in time $O(D)$. This is clearly optimal. For the third scenario, we prove that there does not exist any rendezvous algorithm working in time $o(D^{\sqrt{2}})$, and we show an algorithm working in time $O(D^2)$. The above negative result shows a separation between the optimal complexity in the two easier scenarios and the optimal complexity in the most difficult scenario.
Computing the {\em matching statistics} of a string $P[1..m]$ with respect to a text $T[1..n]$ is a fundamental problem which has application to genome sequence comparison. In this paper, we study the problem of computing the matching statistics upon highly repetitive texts. We design three different data structures that are similar to LZ-compressed indexes. The space costs of all of them can be measured by $γ$, the size of the smallest string attractor [STOC'2018] and $δ$, a better measure of repetitiveness [LATIN'2020].
In the two-dimensional orthogonal colored range counting problem, we preprocess a set, $P$, of $n$ colored points on the plane, such that given an orthogonal query rectangle, the number of distinct colors of the points contained in this rectangle can be computed efficiently. For this problem, we design three new solutions, and the bounds of each can be expressed in some form of time-space tradeoff. By setting appropriate parameter values for these solutions, we can achieve new specific results with (the space are in words and $ε$ is an arbitrary constant in $(0,1)$): ** $O(n\lg^3 n)$ space and $O(\sqrt{n}\lg^{5/2} n \lg \lg n)$ query time; ** $O(n\lg^2 n)$ space and $O(\sqrt{n}\lg^{4+ε} n)$ query time; ** $O(n\frac{\lg^2 n}{\lg \lg n})$ space and $O(\sqrt{n}\lg^{5+ε} n)$ query time; ** $O(n\lg n)$ space and $O(n^{1/2+ε})$ query time. A known conditional lower bound to this problem based on Boolean matrix multiplication gives some evidence on the difficulty of achieving near-linear space solutions with query time better than $\sqrt{n}$ by more than a polylogarithmic factor using purely combinatorial approaches. Thus the time and space bounds in all these results are efficient. Previously, among solutions with similar query times, the most space-efficient solution uses $O(n\lg^4 n)$ space to answer queries in $O(\sqrt{n}\lg^8 n)$ time (SIAM. J. Comp.~2008). Thus the new results listed above all achieve improvements in space efficiency, while all but the last result achieve speed-up in query time as well.
Under the word RAM model, we design three data structures that can be constructed in $O(n\sqrt{\lg n})$ time over $n$ points in an $n \times n$ grid. The first data structure is an $O(n\lg^ε n)$-word structure supporting orthogonal range reporting in $O(\lg\lg n+k)$ time, where $k$ denotes output size and $ε$ is an arbitrarily small constant. The second is an $O(n\lg\lg n)$-word structure supporting orthogonal range successor in $O(\lg\lg n)$ time, while the third is an $O(n\lg^ε n)$-word structure supporting sorted range reporting in $O(\lg\lg n+k)$ time. The query times of these data structures are optimal when the space costs must be within $O(n\ polylog\ n)$ words. Their exact space bounds match those of the best known results achieving the same query times, and the $O(n\sqrt{\lg n})$ construction time beats the previous bounds on preprocessing. Previously, among 2d range search structures, only the orthogonal range counting structure of Chan and Pǎtraşcu (SODA 2010) and the linear space, $O(\lg^ε n)$ query time structure for orthogonal range successor by Belazzougui and Puglisi (SODA 2016) can be built in the same $O(n\sqrt{\lg n})$ time. Hence our work is the first that achieve the same preprocessing time for optimal orthogonal range reporting and range successor. We also apply our results to improve the construction time of text indexes.