arXiv subjects

Carlos Ochoa

Publications and source records attributed to Carlos Ochoa.

4 recordsLinked to original sources

arXiv2019

On the Approximation Ratio of Ordered Parsings

Shannon's entropy is a clear lower bound for statistical compression. The situation is not so well understood for dictionary-based compression. A plausible lower bound is $b$, the least number of phrases of a general bidirectional parse of a text, where phrases can be copied from anywhere else in the text. Since computing $b$ is NP-complete, a popular gold standard is $z$, the number of phrases in the Lempel-Ziv parse of the text, which is the optimal one when phrases can be copied only from the left. While $z$ can be computed in linear time with a greedy algorithm, almost nothing has been known for decades about its approximation ratio with respect to $b$. In this paper we prove that $z=O(b\log(n/b))$, where $n$ is the text length. We also show that the bound is tight as a function of $n$, by exhibiting a text family where $z = Ω(b\log n)$. Our upper bound is obtained by building a run-length context-free grammar based on a locally consistent parsing of the text. Our lower bound is obtained by relating $b$ with $r$, the number of equal-letter runs in the Burrows-Wheeler transform of the text. We proceed by observing that Lempel-Ziv is just one particular case of greedy parses, meaning that the optimal value of $z$ is obtained by scanning the text and maximizing the phrase length at each step, and of ordered parses, meaning that there is an increasing order between phrases and their sources. As a new example of ordered greedy parses, we introduce {\em lexicographical} parses, where phrases can only be copied from lexicographically smaller text locations. We prove that the size $v$ of the optimal lexicographical parse is also obtained greedily in $O(n)$ time, that $v=O(b\log(n/b))$, and that there exists a text family where $v = Ω(b\log n)$.

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arXiv2017

Synergistic Computation of Planar Maxima and Convex Hull

Refinements of the worst case complexity over instances of fixed input size consider the input order or the input structure, but rarely both at the same time. Barbay et al. [2016] described ``synergistic'' solutions on multisets, which take advantage of the input order and the input structure, such as to asymptotically outperform any comparable solution which takes advantage only of one of those features. We consider the extension of their results to the computation of the \textsc{Maxima Set} and the \textsc{Convex Hull} of a set of planar points. After revisiting and improving previous approaches taking advantage only of the input order or of the input structure, we describe synergistic solutions taking optimally advantage of various notions of the input order and input structure in the plane. As intermediate results, we describe and analyze the first adaptive algorithms for \textsc{Merging Maxima} and \textsc{Merging Convex Hulls}.

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arXiv2016

Synergistic Sorting, MultiSelection and Deferred Data Structures on MultiSets

Karp et al. (1988) described Deferred Data Structures for Multisets as "lazy" data structures which partially sort data to support online rank and select queries, with the minimum amount of work in the worst case over instances of size $n$ and number of queries $q$ fixed (i.e., the query size). Barbay et al. (2016) refined this approach to take advantage of the gaps between the positions hit by the queries (i.e., the structure in the queries). We develop new techniques in order to further refine this approach and to take advantage all at once of the structure (i.e., the multiplicities of the elements), the local order (i.e., the number and sizes of runs) and the global order (i.e., the number and positions of existing pivots) in the input; and of the structure and order in the sequence of queries. Our main result is a synergistic deferred data structure which performs much better on large classes of instances, while performing always asymptotically as good as previous solutions. As intermediate results, we describe two new synergistic sorting algorithms, which take advantage of the structure and order (local and global) in the input, improving upon previous results which take advantage only of the structure (Munro and Spira 1979) or of the local order (Takaoka 1997) in the input; and one new multiselection algorithm which takes advantage of not only the order and structure in the input, but also of the structure in the queries. We described two compressed data structures to represent a multiset taking advantage of both the local order and structure, while supporting the operators rank and select on the multiset.

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arXiv2015

Refining the Analysis of Divide and Conquer: How and When

Divide-and-conquer is a central paradigm for the design of algorithms, through which some fundamental computational problems, such as sorting arrays and computing convex hulls, are solved in optimal time within $Θ(n\log{n})$ in the worst case over instances of size $n$. A finer analysis of those problems yields complexities within $O(n(1 + \mathcal{H}(n_1, \dots, n_k))) \subseteq O(n(1{+}\log{k})) \subseteq O(n\log{n})$ in the worst case over all instances of size $n$ composed of $k$ "easy" fragments of respective sizes $n_1, \dots, n_k$ summing to $n$, where the entropy function $\mathcal{H}(n_1, \dots, n_k) = \sum_{i=1}^k{\frac{n_i}{n}}\log{\frac{n}{n_i}}$ measures the "difficulty" of the instance. We consider whether such refined analysis can be applied to other algorithms based on divide-and-conquer, such as polynomial multiplication, input-order adaptive computation of convex hulls in 2D and 3D, and computation of Delaunay triangulations.

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