169 research outputs found
Succinct Indices for Range Queries with applications to Orthogonal Range Maxima
We consider the problem of preprocessing points in 2D, each endowed with
a priority, to answer the following queries: given a axis-parallel rectangle,
determine the point with the largest priority in the rectangle. Using the ideas
of the \emph{effective entropy} of range maxima queries and \emph{succinct
indices} for range maxima queries, we obtain a structure that uses O(N) words
and answers the above query in time. This is a direct
improvement of Chazelle's result from FOCS 1985 for this problem -- Chazelle
required words to answer queries in
time for any constant .Comment: To appear in ICALP 201
Succinct Permutation Graphs
We present a succinct, i.e., asymptotically space-optimal, data structure for permutation graphs that supports distance, adjacency, neighborhood and shortest-path queries in optimal time; a variant of our data structure also supports degree queries in time independent of the neighborhood's size at the expense of an -factor overhead in all running times. We show how to generalize our data structure to the class of circular permutation graphs with asymptotically no extra space, while supporting the same queries in optimal time. Furthermore, we develop a similar compact data structure for the special case of bipartite permutation graphs and conjecture that it is succinct for this class. We demonstrate how to execute algorithms directly over our succinct representations for several combinatorial problems on permutation graphs: Clique, Coloring, Independent Set, Hamiltonian Cycle, All-Pair Shortest Paths, and others. Moreover, we initiate the study of semi-local graph representations; a concept that "interpolates" between local labeling schemes and standard "centralized" data structures. We show how to turn some of our data structures into semi-local representations by storing only bits of additional global information, beating the lower bound on distance labeling schemes for permutation graphs
Succinct Color Searching in One Dimension
In this paper we study succinct data structures for one-dimensional color reporting and color counting problems.
We are given a set of n points with integer coordinates in the range [1,m] and every point is assigned a color from the set {1,...sigma}.
A color reporting query asks for the list of distinct colors that occur in a query interval [a,b] and a color counting query asks for the number of distinct colors in [a,b].
We describe a succinct data structure that answers approximate color counting queries in O(1) time and uses mathcal{B}(n,m) + O(n) + o(mathcal{B}(n,m)) bits,
where mathcal{B}(n,m) is the minimum number of bits required to represent an arbitrary set of size n from a universe of m elements. Thus we show, somewhat counterintuitively,
that it is not necessary to store colors of points in order to answer approximate color counting queries.
In the special case when points are in the rank space (i.e., when n=m), our data structure needs only O(n) bits.
Also, we show that Omega(n) bits are necessary in that case.
Then we turn to succinct data structures for color reporting.
We describe a data structure that uses mathcal{B}(n,m) + nH_d(S) + o(mathcal{B}(n,m)) + o(nlgsigma) bits and answers queries in O(k+1) time,
where k is the number of colors in the answer, and nH_d(S) (d=log_sigma n) is the d-th order empirical entropy of the color sequence. Finally, we consider succinct color reporting under restricted updates. Our dynamic data structure uses nH_d(S)+o(nlgsigma) bits and supports queries in O(k+1) time
Space-Efficient Data-Analysis Queries on Grids
We consider various data-analysis queries on two-dimensional points. We give
new space/time tradeoffs over previous work on geometric queries such as
dominance and rectangle visibility, and on semigroup and group queries such as
sum, average, variance, minimum and maximum. We also introduce new solutions to
queries less frequently considered in the literature such as two-dimensional
quantiles, majorities, successor/predecessor, mode, and various top-
queries, considering static and dynamic scenarios.Comment: 20 pages, 2 figures, submittin
Path Queries on Functions
Let f : [1..n] -> [1..n] be a function, and l : [1..n] -> [1..s] indicate a label assigned to each element of the domain. We design several compact data structures that answer various queries on the labels of paths in f. For example, we can find the minimum label in f^k (i) for a given i and any k >= 0 in a given range [k1..k2], using n lg n + O(n) bits, or the minimum label in f^(-k) (i) for a given i and k > 0, using 2n lg n + O(n) bits, both in time O(lg n/ lg lg n). By using n lg s + o(n lg s) further bits, we can also count, within the same time, the number of elements within a range of labels, and report each such element in O(1 + lg s / lg lg n) additional time. Several other possible queries are considered, such as top-t queries and t-majorities
Succinct Dynamic One-Dimensional Point Reporting
In this paper we present a succinct data structure for the dynamic one-dimensional range reporting problem. Given an interval [a,b] for some a,b in [m], the range reporting query on an integer set S subseteq [m] asks for all points in S cap [a,b]. We describe a data structure that answers reporting queries in optimal O(k+1) time, where k is the number of points in the answer, and supports updates in O(lg^epsilon m) expected time. Our data structure uses B(n,m) + o(B(n,m)) bits where B(n,m) is the minimum number of bits required to represent a set of size n from a universe of m elements. This is the first dynamic data structure for this problem that uses succinct space and achieves optimal query time
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