In this paper, we consider the problem of efficiently representing a set @math of @math items out of a universe @math while supporting a number of operations on it. Let @math be the gap stream associated with @math , @math its bit-size when encoded with , and @math its empirical zero-order entropy. We prove that (1) @math if @math is highly compressible, and (2) @math . Let @math be the number of gap lengths between elements in @math . We firstly propose a new space-efficient zero-order compressed representation of @math taking @math bits of space. Then, we describe a fully-indexable dictionary that supports and queries in @math time while requiring asymptotically the same space as the proposed compressed representation of @math .
The on asks to maintain a (space-efficient) data structure called over a set @math , @math , supporting efficiently a range of queries on @math . In this problem, @math is an ordered set and is called . As showed by Jacobson in his doctoral thesis @cite_1 , a set of just two operations, and , is sufficient and powerful enough in order to derive other fundamental functionalities desired from such a structure: , , and . @math , with @math , is the number of elements in @math that are smaller than or equal to @math . @math , where @math , is the @math -th smallest element in @math . In this paper, we focus on , i.e. data structures supporting both rank and select operations efficiently.
@cite_1
2073214264 2141957180 1782245179 1985174631
On a stream of two dimensional data items @math where @math is an item identifier, and @math is a numerical attribute, a correlated aggregate query requires us to first apply a selection predicate along the second ( @math ) dimension, followed by an aggregation along the first ( @math ) dimension. For selection predicates of the form @math , where parameter @math is provided at query time, we present new streaming algorithms and lower bounds for estimating statistics of the resulting sub stream of elements that satisfy the predicate. We provide the first sub linear space algorithms for a large family of statistics in this model, including frequency moments. We experimentally validate our algorithms, showing that their memory requirements are significantly smaller than existing linear storage schemes for large datasets, while simultaneously achieving fast per-record processing time. We also study the problem when the items have weights. Allowing negative weights allows for analyzing values which occur in the symmetric difference of two datasets. We give a strong space lower bound which holds even if the algorithm is allowed up to a logarithmic number of passes over the data(before the query is presented). We complement this with a small space algorithm which uses a logarithmic number of passes. Given a set @math of @math strings of total length @math , our task is to report the "most relevant"strings for a given query pattern @math . This involves somewhat more advanced query functionality than the usual pattern matching, as some notion of "most relevant" is involved. In information retrieval literature, this task is best achieved by using inverted indexes. However, inverted indexes work only for some predefined set of patterns. In the pattern matching community, the most popular pattern-matching data structures are suffix trees and suffix arrays. However, a typical suffix tree search involves going through all the occurrences of the pattern over the entire string collection, which might be a lot more than the required relevant documents. The first formal framework to study such kind of retrieval problems was given by [Muthukrishnan, 2002]. He considered two metrics for relevance: frequency and proximity. He took a threshold-based approach on these metrics and gave data structures taking @math words of space. We study this problem in a slightly different framework of reporting the top @math most relevant documents (in sorted order) under similar and more general relevance metrics. Our framework gives linear space data structure with optimal query times for arbitrary score functions. As a corollary, it improves the space utilization for the problems in [Muthukrishnan, 2002] while maintaining optimal query performance. We also develop compressed variants of these data structures for several specific relevance metrics. We study the aggregate group nearest neighbor searching for the MAX operator in the plane. For a set @math of @math points and a query set @math of @math points, the query asks for a point of @math whose maximum distance to the points in @math is minimized. We present data structures for answering such queries for both @math and @math distance measures. Previously, only heuristic and approximation algorithms were given for both versions. For the @math version, we build a data structure of O(n) size in @math time, such that each query can be answered in @math time. For the @math version, we build a data structure in @math time and @math space, such that each query can be answered in @math time, and alternatively, we build a data structure in @math time and space for any @math , such that each query can be answered in @math time. Further, we extend our result for the @math version to the top- @math queries where each query asks for the @math points of @math whose maximum distances to @math are the smallest for any @math with @math : We build a data structure of O(n) size in @math time, such that each top- @math query can be answered in @math time. We consider the indexable dictionary problem which consists in storing a set S ⊆ 0,…, m - 1 for some integer m, while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x e S, and -1 otherwise; and select(i) which returns the i-th smallest element in S.We give a structure that supports both operations in O(1) time on the RAM model and requires B(n,m) + o(n) + O(lg lg m) bits to store a set of size n, where B(n,m) = ⌈lg ( n m)⌉ is the minimum number of bits required to store any n-element subset from a universe of size m. Previous dictionaries taking this space only supported (yes no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh.We also present two applications of our dictionary structure:• An information-theoretically optimal representation for k-ary cardinal trees (aka k-ary tries). Our structure uses C(n,k) + o(n + lg k) bits to store a k-ary tree with n nodes and can support parent, i-th child, child labeled i, and the degree of a node in constant time, where C(n,k) is the minimum number of bits to store any n-node k-ary tree. Previous space efficient representations for cardinal k-ary trees required C(n,k) + Ω(n) bits.• An optimal representation for multisets where (appropriate generalisations of) the select and rank operations can be supported in O(1) time. Our structure uses B(n, m + n) + o(n) + O(lg lg m) bits to represent a multiset of size n from an m element set; the first term is the minimum number of bits required to represent such a multiset.