Search results for "Computer Science - Data Structures and Algorithms"
showing 10 items of 64 documents
Quantum algorithms for formula evaluation
2010
We survey the recent sequence of algorithms for evaluating Boolean formulas consisting of NAND gates.
Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games
2017
We study quantum algorithms on search trees of unknown structure, in a model where the tree can be discovered by local exploration. That is, we are given the root of the tree and access to a black box which, given a vertex $v$, outputs the children of $v$. We construct a quantum algorithm which, given such access to a search tree of depth at most $n$, estimates the size of the tree $T$ within a factor of $1\pm \delta$ in $\tilde{O}(\sqrt{nT})$ steps. More generally, the same algorithm can be used to estimate size of directed acyclic graphs (DAGs) in a similar model. We then show two applications of this result: a) We show how to transform a classical backtracking search algorithm which exam…
Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs
2018
In this paper, we present a quantum algorithm for dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is $O(\sqrt{\hat{n}m}\log \hat{n})$, and the running time of the best known deterministic algorithm is $O(n+m)$, where $n$ is the number of vertices, $\hat{n}$ is the number of vertices with at least one outgoing edge; $m$ is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non…
Constructing Antidictionaries in Output-Sensitive Space
2021
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 $y_1,y_2,\ldots,y_k$ over an alphabet $\Sigma$, we are asked to compute the set $\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$ of minimal absent words of length at most $\ell$ of word $y=y_1\#y_2\#\ldots\#y_k$, $\#\notin\Sigma$. 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. This computation generally requires $\Omega(n)$ space for $n=|y|$ using any of the plenty available $\mathcal{O}(n)$-time algorithms. This is because a…
String attractors and combinatorics on words
2019
The notion of \emph{string attractor} has recently been introduced in [Prezza, 2017] and studied in [Kempa and Prezza, 2018] to provide a unifying framework for known dictionary-based compressors. A string attractor for a word $w=w[1]w[2]\cdots w[n]$ is a subset $\Gamma$ of the positions $\{1,\ldots,n\}$, such that all distinct factors of $w$ have an occurrence crossing at least one of the elements of $\Gamma$. While finding the smallest string attractor for a word is a NP-complete problem, it has been proved in [Kempa and Prezza, 2018] that dictionary compressors can be interpreted as algorithms approximating the smallest string attractor for a given word. In this paper we explore the noti…
Large-scale compression of genomic sequence databases with the Burrows-Wheeler transform
2012
Motivation The Burrows-Wheeler transform (BWT) is the foundation of many algorithms for compression and indexing of text data, but the cost of computing the BWT of very large string collections has prevented these techniques from being widely applied to the large sets of sequences often encountered as the outcome of DNA sequencing experiments. In previous work, we presented a novel algorithm that allows the BWT of human genome scale data to be computed on very moderate hardware, thus enabling us to investigate the BWT as a tool for the compression of such datasets. Results We first used simulated reads to explore the relationship between the level of compression and the error rate, the leng…
Binary jumbled string matching for highly run-length compressible texts
2012
The Binary Jumbled String Matching problem is defined as: Given a string $s$ over $\{a,b\}$ of length $n$ and a query $(x,y)$, with $x,y$ non-negative integers, decide whether $s$ has a substring $t$ with exactly $x$ $a$'s and $y$ $b$'s. Previous solutions created an index of size O(n) in a pre-processing step, which was then used to answer queries in constant time. The fastest algorithms for construction of this index have running time $O(n^2/\log n)$ [Burcsi et al., FUN 2010; Moosa and Rahman, IPL 2010], or $O(n^2/\log^2 n)$ in the word-RAM model [Moosa and Rahman, JDA 2012]. We propose an index constructed directly from the run-length encoding of $s$. The construction time of our index i…
Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness
2016
We study space and time efficient quantum algorithms for two graph problems -- deciding whether an $n$-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in $\tilde{O}(n^{3/2})$ time and using $O(\log n)$ classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorithms for deciding the two properties in the adjacency array model, which run in time $\tilde{O}(n\sqrt{d_m})$ and also require $O(\log n)$ space, where $d_m$ is the maximum degree of any vertex in the input graph.
Standard Vs Uniform Binary Search and Their Variants in Learned Static Indexing: The Case of the Searching on Sorted Data Benchmarking Software Platf…
2023
Learned Indexes are a novel approach to search in a sorted table. A model is used to predict an interval in which to search into and a Binary Search routine is used to finalize the search. They are quite effective. For the final stage, usually, the lower_bound routine of the Standard C++ library is used, although this is more of a natural choice rather than a requirement. However, recent studies, that do not use Machine Learning predictions, indicate that other implementations of Binary Search or variants, namely k-ary Search, are better suited to take advantage of the features offered by modern computer architectures. With the use of the Searching on Sorted Sets SOSD Learned Indexing bench…
A New Class of Searchable and Provably Highly Compressible String Transformations
2019
The Burrows-Wheeler Transform is a string transformation that plays a fundamental role for the design of self-indexing compressed data structures. Over the years, researchers have successfully extended this transformation outside the domains of strings. However, efforts to find non-trivial alternatives of the original, now 25 years old, Burrows-Wheeler string transformation have met limited success. In this paper we bring new lymph to this area by introducing a whole new family of transformations that have all the "myriad virtues" of the BWT: they can be computed and inverted in linear time, they produce provably highly compressible strings, and they support linear time pattern search direc…