Golnaz Badkobeh, Maxime Crochemore
We extend the left-to-right Lyndon factorisation of a word to the left Lyndon tree construction of a Lyndon word. It yields an algorithm to sort the prefixes of a Lyndon word according to the infinite ordering defined by Dolce et al. (2019). A straightforward variant computes the left Lyndon forest of a word. All algorithms run in linear time on a general alphabet, that is, in the letter-comparison model.
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The article describes the structural and algorithmic relations between Cartesian trees and Lyndon Trees. This leads to a uniform presentation of the Lyndon table of a word corresponding to the Next Nearest Smaller table of a sequence of numbers. It shows how to efficiently compute runs, that is, maximal periodicities occurring in a word.
In-Place BWT and Lyndon Array Construction in Constant Space
本文提出一种常空间原地BWT算法，通过维护后缀字典序排名并利用下一个更小值过程构造Lyndon数组。
We introduce a new class of straight-line programs (SLPs), named the Lyndon SLP, inspired by the Lyndon trees (Barcelo, 1990). Based on this SLP, we propose a self-index data structure of $O(g)$ words of space that can be built from a string $T$ in $O(n \lg n)$ expected time, retrieving the starting positions of all occurrences of a pattern $P$ of length $m$ in $O(m + \lg m \lg n + occ \lg g)$ time, where $n$ is the length of $T$, $g$ is the size of the Lyndon SLP for $T$, and $occ$ is the number of occurrences of $P$ in $T$.
We present an algorithm for computing the Lyndon factorization of a string that is given in grammar compressed form, namely, a Straight Line Program (SLP). The algorithm runs in $O(n^4 + mn^3h)$ time and $O(n^2)$ space, where $m$ is the size of the Lyndon factorization, $n$ is the size of the SLP, and $h$ is the height of the derivation tree of the SLP. Since the length of the decompressed string can be exponentially large w.r.t. $n, m$ and $h$, our result is the first polynomial time solution when the string is given as SLP.
We present an algorithm that allows for find-closest-point and kNN-style traversals of left-balanced k-d trees, without the need for either recursion or software-managed stacks; instead using only current and last previously traversed node to compute which node to traverse next.
We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the "runs" conjecture (Kolpakov \& Kucherov (FOCS '99)), which states that the maximum number of runs $ρ(n)$ in a string of length $n$ is less than $n$. The proof is remarkably simple, considering the numerous endeavors to tackle this problem in the last 15 years, and significantly improves our understanding of how runs can occur in strings. In addition, we obtain an upper bound of $3n$ for the maximum sum of exponents $σ(n)$ of runs in a …
GPU-friendly, Parallel, and (Almost-)In-Place Construction of Left-Balanced k-d Trees
提出一种GPU友好、易并行且几乎原地的算法，用于在k维点上构建左平衡且完整的k-d树，每点仅需一个int临时存储，每次迭代包含一次并行排序和CUDA更新内核。
Consider laying out a fixed-topology tree of N nodes into external memory with block size B so as to minimize the worst-case number of block memory transfers required to traverse a path from the root to a node of depth D. We prove that the optimal number of memory transfers is $$ \cases{ \displaystyle Θ\left( {D \over \lg (1{+}B)} \right) & when $D = O(\lg N)$, \cr \displaystyle Θ\left( {\lg N \over \lg \left(1{+}{B \lg N \over D}\right)} \right) & when $D = Ω(\lg N)$ and $D = O(B \lg N)$, \cr \displaystyle Θ\left( {D \over B} \right) & when $D = Ω(B \lg N)$. } $$
Eliminating Left Recursion without the Epsilon
针对标准左递归消除算法导致规则丢失和指数膨胀的问题，提出改进算法：先消除直接左递归，再用图论方法消除间接左递归，并重写解析树以恢复原始结构，兼顾理论与实用性。
This thesis presents analysis of the properties and run-time of the Rapidly-exploring Random Tree (RRT) algorithm. It is shown that the time for the RRT with stepsize $ε$ to grow close to every point in the $d$-dimensional unit cube is $Θ\left(\frac1{ε^d} \log \left(\frac1ε\right)\right)$. Also, the time it takes for the tree to reach a region of positive probability is $O\left(ε^{-\frac32}\right)$. Finally, a relationship is shown to the Nearest Neighbour Tree (NNT). This relationship shows that the total Euclidean path length after $n$ steps is $O(\sqrt n)$ and the expected height of the tre…
Time Efficient Implementation for Online $k$-server Problem on Trees
针对树上在线 $k$-服务器问题，本文为 Chrobak–Larmore 的最优 $k$-竞争算法给出新实现，将预处理降至 $O(n)$、单次查询降至 $O(k\log k)$，且查询时间与树大小无关。
The suffix trees are fundamental data structures for various kinds of string processing. The suffix tree of a string $T$ of length $n$ has $O(n)$ nodes and edges, and the string label of each edge is encoded by a pair of positions in $T$. Thus, even after the tree is built, the input text $T$ needs to be kept stored and random access to $T$ is still needed. The linear-size suffix tries (LSTs), proposed by Crochemore et al. [Linear-size suffix tries, TCS 638:171-178, 2016], are a `stand-alone' alternative to the suffix trees. Namely, the LST of a string $T$ of length $n$ occupies $O(n)$ total s…
Rabin Games and Colourful Universal Trees
本文提出基于彩色通用树与 Rabin progress measure 的提升算法，将图上 Rabin/Streett 博弈求解改进至 $\tilde{O}(mn(k!)^{1+o(1)})$ 时间和近线性相关空间。
Simple Compact Monotone Tree Drawings
该研究针对树的单调绘制问题，提出简单算法，将有根树绘制压缩至n×n网格，同时给出无根有序、无序树的紧凑单调绘制方案，大幅缩小所需网格尺寸。
Red-black (RB) trees are one of the most efficient variants of balanced binary search trees. However, they have always been blamed for being too complicated, hard to explain, and not suitable for pedagogical purposes. In the pioneering work of Guibas & Sedgewick (1978), both 2-3 and 2-3-4 variants of RB trees had been considered, but further study of the former had been abandoned due to the higher number of rotations in the insert algorithm. Sedgewick (2008) proposed a variant of 2-3 RB trees, viz. left-leaning red-black (LLRB) trees, in which red links are restricted to left children and prop…
We give an online algorithm that with high probability computes a $\left(\frac{e}{e-1} + o(1)\right)Δ$ edge coloring on a graph $G$ with maximum degree $Δ= ω(\log n)$ under online edge arrivals against oblivious adversaries, making first progress on the conjecture of Bar-Noy, Motwani, and Naor in this general setting. Our algorithm is based on reducing to a matching problem on locally treelike graphs, and then applying a tree recurrences based approach for arguing correlation decay.
Buffered Partially-Persistent External-Memory Search Trees
本文提出一种缓冲的部分持久外部存储搜索树，以线性空间和摊销最优 I/O 界限同时支持历史版本查询与当前版本更新，并通过改进存储需求实现了最坏情况下的相同性能。
Linear Time Online Algorithms for Constructing Linear-size Suffix Trie
本文提出两种直接在线构造线性大小后缀字典树（LST）的算法，分别从右到左和从左到右增量构建，在 O(n log σ) 时间和 O(n) 空间内完成，且无需存储输入文本串。
We deal with the problem of maintaining the suffix tree indexing structure for a fully-online collection of multiple strings, where a new character can be prepended to any string in the collection at any time. The only previously known algorithm for the problem, recently proposed by Takagi et al. [Algorithmica 82(5): 1346-1377 (2020)], runs in $O(N \log σ)$ time and $O(N)$ space on the word RAM model, where $N$ denotes the total length of the strings and $σ$ denotes the alphabet size. Their algorithm makes heavy use of the nearest marked ancestor (NMA) data structure on semi-dynamic trees, tha…
We consider online algorithms for the $k$-server problem on trees. There is a $k$-competitive algorithm for this problem, and it is the best competitive ratio. M. Chrobak and L. Larmore provided it. At the same time, the existing implementation has $O(n)$ time complexity for processing a query and $O(n)$ for prepossessing, where $n$ is the number of nodes in a tree. Another implementation of the algorithm has $O(k^2+k\log n)$ time complexity for processing a query and $O(n\log n)$ for prepossessing. We provide a new time-efficient implementation of the algorithm. It has $O(n)$ time complexity …
Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The s…
An ordered labeled tree is a tree in which the nodes are labeled and the left-to-right order among siblings is relevant. The edit distance between two ordered labeled trees is the minimum cost of changing one tree into the other through a sequence of edit steps. In the literature, there are a class of algorithms based on different yet closely related path-decomposition schemes. This article reviews the principles of these algorithms, and studies the concepts related to the algorithmic complexities as a consequence of the decomposition schemes.
In the Directed Steiner Tree (DST) problem, we are given a directed graph $G=(V,E)$ on $n$ vertices with edge-costs $c \in \mathbb{R}_{\geq 0}^E$, a root vertex $r \in V$, and a set $K \subseteq V \setminus \{r\}$ of $k$ terminals. The goal is to find a minimum-cost subgraph of $G$ that contains a path from $r$ to every terminal $t \in K$. DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of $O(k^ε)$ for any constant $ε> 0$, and the best quasi-polynomial-time approximation ratio of $O\left(\frac{\log^2 k}{\log \log k…
We consider online algorithms for the $k$-server problem on trees. Chrobak and Larmore proposed a $k$-competitive algorithm for this problem that has the optimal competitive ratio. However, a naive implementation of their algorithm has $O(n)$ time complexity for processing each query, where $n$ is the number of nodes in the tree. We propose a new time-efficient implementation of this algorithm that has $O(n\log n)$ time complexity for preprocessing and $O\left(k^2 + k\cdot \log n\right)$ time for processing a query. We also propose a quantum algorithm for the case where the nodes of the tree a…
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