117,946 research outputs found
A Generalization of the {\L}o\'s-Tarski Preservation Theorem over Classes of Finite Structures
We investigate a generalization of the {\L}o\'s-Tarski preservation theorem
via the semantic notion of \emph{preservation under substructures modulo
-sized cores}. It was shown earlier that over arbitrary structures, this
semantic notion for first-order logic corresponds to definability by
sentences. In this paper, we identify two properties of
classes of finite structures that ensure the above correspondence. The first is
based on well-quasi-ordering under the embedding relation. The second is a
logic-based combinatorial property that strictly generalizes the first. We show
that starting with classes satisfying any of these properties, the classes
obtained by applying operations like disjoint union, cartesian and tensor
products, or by forming words and trees over the classes, inherit the same
property. As a fallout, we obtain interesting classes of structures over which
an effective version of the {\L}o\'s-Tarski theorem holds.Comment: 28 pages, 1 figur
Bounded repairability for regular tree languages
We study the problem of bounded repairability of a given restriction tree language R into a target tree language T. More precisely, we say that R is bounded repairable w.r.t. T if there exists a bound on the number of standard tree editing operations necessary to apply to any tree in R in order to obtain a tree in T. We consider a number of possible specifications for tree languages: bottom-up tree automata (on curry encoding of unranked trees) that capture the class of XML Schemas and DTDs. We also consider a special case when the restriction language R is universal, i.e., contains all trees over a given alphabet. We give an effective characterization of bounded repairability between pairs of tree languages represented with automata. This characterization introduces two tools, synopsis trees and a coverage relation between them, allowing one to reason about tree languages that undergo a bounded number of editing operations. We then employ this characterization to provide upper bounds to the complexity of deciding bounded repairability and we show that these bounds are tight. In particular, when the input tree languages are specified with arbitrary bottom-up automata, the problem is coNEXPTIME-complete. The problem remains coNEXPTIME-complete even if we use deterministic non-recursive DTDs to specify the input languages. The complexity of the problem can be reduced if we assume that the alphabet, the set of node labels, is fixed: the problem becomes PSPACE-complete for non-recursive DTDs and coNP-complete for deterministic non-recursive DTDs. Finally, when the restriction tree language R is universal, we show that the bounded repairability problem becomes EXPTIME-complete if the target language is specified by an arbitrary bottom-up tree automaton and becomes tractable (PTIME-complete, in fact) when a deterministic bottom-up automaton is used
Succinct Representations of Permutations and Functions
We investigate the problem of succinctly representing an arbitrary
permutation, \pi, on {0,...,n-1} so that \pi^k(i) can be computed quickly for
any i and any (positive or negative) integer power k. A representation taking
(1+\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in
constant time, for any positive constant \epsilon <= 1. A representation taking
the optimal \ceil{\lg n!} + o(n) bits can be used to compute arbitrary powers
in O(lg n / lg lg n) time.
We then consider the more general problem of succinctly representing an
arbitrary function, f: [n] \rightarrow [n] so that f^k(i) can be computed
quickly for any i and any integer power k. We give a representation that takes
(1+\epsilon) n lg n + O(1) bits, for any positive constant \epsilon <= 1, and
computes arbitrary positive powers in constant time. It can also be used to
compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time.
We place emphasis on the redundancy, or the space beyond the
information-theoretic lower bound that the data structure uses in order to
support operations efficiently. A number of lower bounds have recently been
shown on the redundancy of data structures. These lower bounds confirm the
space-time optimality of some of our solutions. Furthermore, the redundancy of
one of our structures "surpasses" a recent lower bound by Golynski [Golynski,
SODA 2009], thus demonstrating the limitations of this lower bound.Comment: Preliminary versions of these results have appeared in the
Proceedings of ICALP 2003 and 2004. However, all results in this version are
improved over the earlier conference versio
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
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