Trees are useful entities allowing to model data structures and hierarchical relationships in networked decision systems ubiquitously. An ordered tree is a rooted tree where the order of the subtrees (children) of a node is significant. In combinatorial optimization, generating ordered trees is relevant to evaluate candidate combinatorial objects. In this paper, we present an algebraic scheme to generate ordered trees with $n$ vertices with utmost efficiency; whereby our approach uses $\mathcal{O}(n)$ space and $\mathcal{O}(1)$ time in average per tree. Our computational studies have shown the feasibility and efficiency to generate ordered trees in constant time in average, in about one tenth of a millisecond per ordered tree. Due to the 1-1 bijective nature to other combinatorial classes, our approach is favorable to study the generation of binary trees with $n$ external nodes, trees with $n$ nodes, legal sequences of $n$ pairs of parentheses, triangulated $n$-gons, gambler's sequences and lattice paths. We believe our scheme may find its use in devising algorithms for planning and combinatorial optimization involving Catalan numbers.
The $B^ε$-tree [Brodal and Fagerberg 2003] is a simple I/O-efficient external-memory-model data structure that supports updates orders of magnitude faster than B-tree with a query performance comparable to the B-tree: for any positive constant $ε<1$ insertions and deletions take $O(\frac{1}{B^{1-ε}}\log_{B}N)$ time (rather than $O(\log_BN)$ time for the classic B-tree), queries take $O(\log_BN)$ time and range queries returning $k$ items take $O(\log_BN+\frac{k}{B})$ time. Although the $B^ε$-tree has an optimal update/query tradeoff, the runtimes are amortized. Another structure, the write-opt…
We present a decremental data structure for maintaining the SPQR-tree of a planar graph subject to edge contractions and deletions. The update time, amortized over $Ω(n)$ operations, is $O(\log^2 n)$. Via SPQR-trees, we give a decremental data structure for maintaining $3$-vertex connectivity in planar graphs. It answers queries in $O(1)$ time and processes edge deletions and contractions in $O(\log^2 n)$ amortized time. This is an exponential improvement over the previous best bound of $O(\sqrt{n}\,)$ that has stood for over 20 years. In addition, the previous data structures only supported e…
Chazelle [JACM00] introduced the soft heap as a building block for efficient minimum spanning tree algorithms, and recently Kaplan et al. [SOSA2019] showed how soft heaps can be applied to achieve simpler algorithms for various selection problems. A soft heap trades-off accuracy for efficiency, by allowing $εN$ of the items in a heap to be corrupted after a total of $N$ insertions, where a corrupted item is an item with artificially increased key and $0 < ε\leq 1/2$ is a fixed error parameter. Chazelle's soft heaps are based on binomial trees and support insertions in amortized $O(\lg(1/ε))$ t…
We show how a collection of Euler-tour trees for a forest on $n$ vertices can be stored in $2 n + o (n)$ bits such that simple queries take constant time, more complex queries take logarithmic time and updates take polylogarithmic amortized time.
We use a greedy strategy to list the spanning trees of the fan graph, $F_n$, such that successive trees differ by pivoting a single edge around a vertex. It is the first greedy algorithm for exhaustively generating spanning trees using such a minimal change operation. The resulting listing is then studied to find a recursive algorithm that produces the same listing in $O(1)$-amortized time using $O(n)$ space. Additionally, we present $O(n)$-time algorithms for ranking and unranking the spanning trees for our listing; an improvement over the generic $O(n^3)$-time algorithm for ranking and unran…
We start by summarizing the recently proposed implementation of the first non-blocking concurrent interpolation search tree (C-IST) data structure. We then analyze the individual operations of the C-IST, and show that they are correct and linearizable. We furthermore show that lookup (and several other non-destructive operations) are wait-free, and that the insert and delete operations are lock-free. We continue by showing that the C-IST has the following properties. For arbitrary key distributions, this data structure ensures worst-case $O(\log n + p)$ amortized time for search, insertion and…
The pairing heap is a simple "self-adjusting" implementation of a heap (priority queue). Inserting an item into a pairing heap or decreasing the key of an item takes O(1) time worst-case, as does melding two heaps. But deleting an item of minimum key can take time linear in the heap size in the worst case. The paper that introduced the pairing heap proved an O(log n) amortized time bound for each heap operation, where n is the number of items in the heap or heaps involved in the operation, by charging all but O(log n) of the time for each deletion to non-deletion operations, O(log n) to each. …
We study the problem of generating, ranking and unranking of unlabeled ordered trees whose nodes have maximum degree of $Δ$. This class of trees represents a generalization of chemical trees. A chemical tree is an unlabeled tree in which no node has degree greater than 4. By allowing up to $Δ$ children for each node of chemical tree instead of 4, we will have a generalization of chemical trees. Here, we introduce a new encoding over an alphabet of size 4 for representing unlabeled ordered trees with maximum degree of $Δ$. We use this encoding for generating these trees in A-order with constant…
University of Milan（米兰大学）；Institut Polytechnique de Paris（巴黎理工学院）；Télécom Paris（巴黎电信学院）
We develop the first fully dynamic algorithm that maintains a decision tree over an arbitrary sequence of insertions and deletions of labeled examples. Given $ε> 0$ our algorithm guarantees that, at every point in time, every node of the decision tree uses a split with Gini gain within an additive $ε$ of the optimum. For real-valued features the algorithm has an amortized running time per insertion/deletion of $O\big(\frac{d \log^3 n}{ε^2}\big)$, which improves to $O\big(\frac{d \log^2 n}ε\big)$ for binary or categorical features, while it uses space $O(n d)$, where $n$ is the maximum number o…
In this paper we study a family of algorithms, introduced by Chan [SODA 1999] and called LR-algorithms, for drawing ordered rooted binary trees. In particular, we are interested in constructing LR-drawings (that are drawings obtained via LR-algorithms) with small width. Chan showed three different LR-algorithms that achieve, for an ordered rooted binary tree with $n$ nodes, width $O(n^{0.695})$, width $O(n^{0.5})$, and width $O(n^{0.48})$. We prove that, for every $n$-node ordered rooted binary tree, an LR-drawing with minimum width can be constructed in $O(n^{1.48})$ time. Further, we show an…
The degree distribution of an ordered tree $T$ with $n$ nodes is $\vec{n} = (n_0,\ldots,n_{n-1})$, where $n_i$ is the number of nodes in $T$ with $i$ children. Let $\mathcal{N}(\vec{n})$ be the number of trees with degree distribution $\vec{n}$. We give a data structure that stores an ordered tree $T$ with $n$ nodes and degree distribution $\vec{n}$ using $\log \mathcal{N}(\vec{n})+O(n/\log^t n)$ bits for every constant $t$. The data structure answers tree queries in constant time. This improves the current data structures with lowest space for ordered trees: The structure of Jansson et al.\ […
We consider a sliding window $W$ over a stream of characters from some alphabet of constant size. The user wants to perform deterministic substring matching on the current sliding window content and obtain positions of the matches. We present an indexed version of the sliding window using the suffix tree, the link tree and the lowest common ancestor. The data structure of size $Θ(|W|)$ has optimal time queries $Θ(m+occ)$ and amortized constant time updates, where $m$ is the length of the query string and $occ$ is the number of its occurrences.
BFS versus DFS for fixed-level targets in ordered trees
Dartmouth college（达特茅斯学院）；Faculty of Mathematics and Informatics, Sofia University（索非亚大学数学与信息学院）；Department of Mathematics and Computer Science, Davidson College（戴维森学院数学与计算机科学系）
The ordered set is one of the most important data type in both theoretical algorithm design and analysis and practical programming. In this paper we study the set operations on two ordered sets, including Union, Intersect and Difference, based on four types of balanced Binary Search Trees (BST) including AVL trees, red-black trees, weight balanced trees and treaps. We introduced only one subroutine Join that needs to be implemented differently for each balanced BST, and on top of which we can implement generic, simple and efficient parallel functions for ordered sets. We first prove the work-e…
In the field of algorithmic analysis, one of the more well-known exercises is the subset sum problem. That is, given a set of integers, determine whether one or more integers in the set can sum to a target value. Aside from the brute-force approach of verifying all combinations of integers, several solutions have been found, ranging from clever uses of various data structures to computationally-efficient approximation solutions. In this paper, a unique approach is discussed which builds upon the existing min-heap solution for positive integers, introducing a tree-based data structure influence…
We consider the tree consensus problem, an important problem in bioinformatics. Given a rooted tree $t$ and another tree $T$, one would like to incorporate compatible information from $T$ to $t$. This problem is a subproblem in the tree refinement problem called the RF-Optimal Tree Refinement Problem defined by in Christensen, Molloy, Vachaspati and Warnow [WABI'19] who employ the greedy algorithm by Gawrychowski, Landau, Sung, and Weimann [ICALP'18] that runs in time $O(n^{1.5}\log n)$. We give a faster algorithm for this problem that runs in time $O(n\log n)$. Our key ingredient is a biparti…
We study the growth of a time-ordered rooted tree by probabilistic attachment of new vertices to leaves. We construct a likelihood function of the leaves based on the connectivity of the tree. We take such connectivity to be induced by the merging of directed ordered paths from leaves to the root. Combining the likelihood with an assigned prior distribution leads to a posterior leaf distribution from which we sample attachment points for new vertices. We present computational examples of such Bayesian tree growth. Although the discussion is generic, the initial motivation for the paper is the …
In this paper we introduce the notion of explicit worst-case bounded adaptive algorithms for applications with fixed process-completion requirements. Such applications demand that a process be guaranteed to complete within an established time interval while adaptively reducing computational overhead during that interval, e.g., so as to reduce total energy usage. Our principal contribution is the access-adaptive priority search tree (AAPST), which can provide efficient distribution-sensitive performance comparable to the splay tree, but do so within strict - and O(logn) optimal - worst-case per…
Designing dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms. While a few such algorithms are known for spanning trees, matchings, and single-source shortest paths, very little was known for an important primitive like graph sparsifiers. The challenge is how to approximately preserve so much information about the graph (e.g., all-pairs distances and all cuts) without revealing the algorithms' underlying randomness to the adaptive adversary. In this paper we present the first non-trivial efficient adaptive algorithms for maintaining s…
We consider drawings of graphs in the plane in which vertices are assigned distinct points in the plane and edges are drawn as simple curves connecting the vertices and such that the edges intersect only at their common endpoints. There is an intuitive quality measure for drawings of a graph that measures the height of a drawing $ϕ: G \rightarrow \mathbb{R}^2$ as follows. For a vertical line $\ell$ in $\mathbb{R}^2$, let the height of $\ell$ be the cardinality of the set $\ell \cap ϕ(G)$. The height of a drawing of $G$ is the maximum height over all vertical lines. In this paper, instead of ab…
In-memory data management systems, such as key-value stores, have become an essential infrastructure in today's big-data processing and cloud computing. They rely on efficient index structures to access data. While unordered indexes, such as hash tables, can perform point search with O(1) time, they cannot be used in many scenarios where range queries must be supported. Many ordered indexes, such as B+ tree and skip list, have a O(log N) lookup cost, where N is number of keys in an index. For an ordered index hosting billions of keys, it may take more than 30 key-comparisons in a lookup, which…
This article introduces hinted dictionaries for expressing efficient ordered sets and maps functionally. As opposed to the traditional ordered dictionaries with logarithmic operations, hinted dictionaries can achieve better performance by using cursor-like objects referred to as hints. Hinted dictionaries unify the interfaces of imperative ordered dictionaries (e.g., C++ maps) and functional ones (e.g., Adams' sets). We show that such dictionaries can use sorted arrays, unbalanced trees, and balanced trees as their underlying representations. Throughout the article, we use Scala to present the…
We study the problem of maintaining a breadth-first spanning tree (BFS tree) in partially dynamic distributed networks modeling a sequence of either failures or additions of communication links (but not both). We present deterministic $(1+ε)$-approximation algorithms whose amortized time (over some number of link changes) is sublinear in $D$, the maximum diameter of the network. Our technique also leads to a deterministic $(1+ε)$-approximate incremental algorithm for single-source shortest paths (SSSP) in the sequential (usual RAM) model. Prior to our work, the state of the art was the classic…
Pairwise ordered tree alignment are combinatorial objects that appear in RNA secondary structure comparison. However, the usual representation of tree alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce identical sets of matches between identical pairs of trees. This ambiguity is uninformative, and detrimental to any probabilistic analysis.In this work, we consider tree alignments up to equivalence. Our first result is a precise asymptotic enumeration of tree alignments, obtained from a context-free grammar by mean of basic analytic combinatorics. Our second result f…
The need for scalable concurrent ordered set data structures with linearizable range query support is increasing due to the rise of multicore computers, data processing platforms and in-memory databases. This paper presents a new concurrent ordered set with linearizable range query support. The new data structure is based on the contention adapting search tree and an immutable data structure. Experimental results show that the new data structure is as much as three times faster compared to related data structures. The data structure scales well due to its ability to adapt the sizes of its immu…
