198 research outputs found
Compressed Representations of Permutations, and Applications
We explore various techniques to compress a permutation over n
integers, taking advantage of ordered subsequences in , while supporting
its application (i) and the application of its inverse in
small time. Our compression schemes yield several interesting byproducts, in
many cases matching, improving or extending the best existing results on
applications such as the encoding of a permutation in order to support iterated
applications of it, of integer functions, and of inverted lists and
suffix arrays
Smooth heaps and a dual view of self-adjusting data structures
We present a new connection between self-adjusting binary search trees (BSTs)
and heaps, two fundamental, extensively studied, and practically relevant
families of data structures. Roughly speaking, we map an arbitrary heap
algorithm within a natural model, to a corresponding BST algorithm with the
same cost on a dual sequence of operations (i.e. the same sequence with the
roles of time and key-space switched). This is the first general transformation
between the two families of data structures.
There is a rich theory of dynamic optimality for BSTs (i.e. the theory of
competitiveness between BST algorithms). The lack of an analogous theory for
heaps has been noted in the literature. Through our connection, we transfer all
instance-specific lower bounds known for BSTs to a general model of heaps,
initiating a theory of dynamic optimality for heaps.
On the algorithmic side, we obtain a new, simple and efficient heap
algorithm, which we call the smooth heap. We show the smooth heap to be the
heap-counterpart of Greedy, the BST algorithm with the strongest proven and
conjectured properties from the literature, widely believed to be
instance-optimal. Assuming the optimality of Greedy, the smooth heap is also
optimal within our model of heap algorithms. As corollaries of results known
for Greedy, we obtain instance-specific upper bounds for the smooth heap, with
applications in adaptive sorting.
Intriguingly, the smooth heap, although derived from a non-practical BST
algorithm, is simple and easy to implement (e.g. it stores no auxiliary data
besides the keys and tree pointers). It can be seen as a variation on the
popular pairing heap data structure, extending it with a "power-of-two-choices"
type of heuristic.Comment: Presented at STOC 2018, light revision, additional figure
LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations
LRM-Trees are an elegant way to partition a sequence of values into sorted
consecutive blocks, and to express the relative position of the first element
of each block within a previous block. They were used to encode ordinal trees
and to index integer arrays in order to support range minimum queries on them.
We describe how they yield many other convenient results in a variety of areas,
from data structures to algorithms: some compressed succinct indices for range
minimum queries; a new adaptive sorting algorithm; and a compressed succinct
data structure for permutations supporting direct and indirect application in
time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur
A Geometric Form for the Extended Patience Sorting Algorithm
Patience Sorting is a combinatorial algorithm that can be viewed as an
iterated, non-recursive form of the Schensted Insertion Algorithm. In recent
work the authors extended Patience Sorting to a full bijection between the
symmetric group and certain pairs of combinatorial objects (called pile
configurations) that are most naturally defined in terms of generalized
permutation pattern and barred pattern avoidance. This Extended Patience
Sorting Algorithm is very similar to the Robinson-Schensted-Knuth (or RSK)
Correspondence, which is itself built from repeated application of the
Schensted Insertion Algorithm.
In this work we introduce a geometric form for the Extended Patience Sorting
Algorithm that is in some sense a natural dual algorithm to G. Viennot's
celebrated Geometric RSK Algorithm. Unlike Geometric RSK, though, the lattice
paths coming from Patience Sorting are allowed to intersect. We thus also give
a characterization for the intersections of these lattice paths in terms of the
pile configurations associated with a given permutation under the Extended
Patience Sorting Algorithm.Comment: 14 pages, LaTeX, uses pstricks; v2: major revision after section 3;
to be published in Adv. Appl. Mat
Partition into heapable sequences, heap tableaux and a multiset extension of Hammersley's process
We investigate partitioning of integer sequences into heapable subsequences
(previously defined and established by Mitzenmacher et al). We show that an
extension of patience sorting computes the decomposition into a minimal number
of heapable subsequences (MHS). We connect this parameter to an interactive
particle system, a multiset extension of Hammersley's process, and investigate
its expected value on a random permutation. In contrast with the (well studied)
case of the longest increasing subsequence, we bring experimental evidence that
the correct asymptotic scaling is . Finally
we give a heap-based extension of Young tableaux, prove a hook inequality and
an extension of the Robinson-Schensted correspondence
Accounting for outliers and calendar effects in surrogate simulations of stock return sequences
Surrogate Data Analysis (SDA) is a statistical hypothesis testing framework
for the determination of weak chaos in time series dynamics. Existing SDA
procedures do not account properly for the rich structures observed in stock
return sequences, attributed to the presence of heteroscedasticity, seasonal
effects and outliers. In this paper we suggest a modification of the SDA
framework, based on the robust estimation of location and scale parameters of
mean-stationary time series and a probabilistic framework which deals with
outliers. A demonstration on the NASDAQ Composite index daily returns shows
that the proposed approach produces surrogates that faithfully reproduce the
structure of the original series while being manifestations of linear-random
dynamics.Comment: 21 pages, 7 figure
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