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Register a new userOn the Count of Trees
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Added by
Everardo Barcenas
Publication date
2010
fields
Informatics Engineering
and research's language is
English
Authors
Everardo Barcenas
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Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on frameworks for reasoning about path expressions where node cardinality constraints occur along a path in a tree. We present a logic capable of expressing deep counting along paths which may include arbitrary recursive forward and backward navigation. The counting extensions can be seen as a generalization of graded modalities that count immediate successor nodes. While the combination of graded modalities, nominals, and inverse modalities yields undecidable logics over graphs, we show that these features can be combined in a decidable tree logic whose main features can be decided in exponential time. Our logic being closed under negation, it may be used to decide typical problems on XPath queries such as satisfiability, type checking with relation to regular types, containment, or equivalence.
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The study of node selection query languages for (finite) trees has been a major topic in the recent research on query languages for Web documents. On one hand, there has been an extensive study of XPath and its various extensions. On the other hand, query languages based on classical logics, such as first-order logic (FO) or Monadic Second-Order Logic (MSO), have been considered. Results in this area typically relate an XPath-based language to a classical logic. What has yet to emerge is an XPath-related language that is as expressive as MSO, and at the same time enjoys the computational properties of XPath, which are linear time query evaluation and exponential time query-containment test. In this paper we propose muXPath, which is the alternation-free fragment of XPath extended with fixpoint operators. Using two-way alternating automata, we show that this language does combine desired expressiveness and computational properties, placing it as an attractive candidate for the definite node-selection query language for trees.
Efficient large-scale annotation of genomic intervals is essential for personal genome interpretation in the realm of precision medicine. There are 13 possible relations between two intervals according to Allens interval algebra. Conventional interval trees are routinely used to identify the genomic intervals satisfying a coarse relation with a query interval, but cannot support efficient query for more refined relations such as all Allens relations. We design and implement a novel approach to address this unmet need. Through rewriting Allens interval relations, we transform an interval query to a range query, then adapt and utilize the range trees for querying. We implement two types of range trees: a basic 2-dimensional range tree (2D-RT) and an augmented range tree with fractional cascading (RTFC) and compare them with the conventional interval tree (IT). Theoretical analysis shows that RTFC can achieve the best time complexity for interval queries regarding all Allens relations among the three trees. We also perform comparative experiments on the efficiency of RTFC, 2D-RT and IT in querying noncoding element annotations in a large collection of personal genomes. Our experimental results show that 2D-RT is more efficient than IT for interval queries regarding most of Allens relations, RTFC is even more efficient than 2D-RT. The results demonstrate that RTFC is an efficient data structure for querying large-scale datasets regarding Allens relations between genomic intervals, such as those required by interpreting genome-wide variation in large populations.
We use a sign-reversing involution to show that trees on the vertex set [n], considered to be rooted at 1, in which no vertex has exactly one child are counted by 1/n sum_{k=1}^{n} (-1)^(n-k) {n}-choose-{k} (n-1)!/(k-1)! k^(k-1). This result corrects a persistent misprint in the Encyclopedia of Integer Sequences.
In recent years, emerging hardware storage technologies have focused on divergent goals: better performance or lower cost-per-bit of storage. Correspondingly, data systems that employ these new technologies are optimized either to be fast (but expensive) or cheap (but slow). We take a different approach: by combining multiple tiers of fast and low-cost storage technologies within the same system, we can achieve a Pareto-efficient balance between performance and cost-per-bit. This paper presents the design and implementation of PrismDB, a novel log-structured merge tree based key-value store that exploits a full spectrum of heterogeneous storage technologies (from 3D XPoint to QLC NAND). We introduce the notion of read-awareness to log-structured merge trees, which allows hot objects to be pinned to faster storage, achieving better tiering and hot-cold separation of objects. Compared to the standard use of RocksDB on flash in datacenters today, PrismDBs average throughput on heterogeneous storage is 2.3$times$ faster and its tail latency is more than an order of magnitude better, using hardware than is half the cost.
A clean lattice triangle in ${mathbb R}^2$ is a triangle that does not contain any lattice points on its sides other than its vertices. The central goal of this paper is to count the number of clean triangles of a given area up to unimodular equivalence. In doing so we use a variant of the Euler phi function which we call $imph(n)$ (imitation phi).
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