73,500 research outputs found
The Space Complexity of 2-Dimensional Approximate Range Counting
We study the problem of -dimensional orthogonal range counting with
additive error. Given a set of points drawn from an grid
and an error parameter \eps, the goal is to build a data structure, such that
for any orthogonal range , it can return the number of points in
with additive error \eps n. A well-known solution for this problem is the
{\em \eps-approximation}, which is a subset that can estimate
the number of points in with the number of points in . It is
known that an \eps-approximation of size O(\frac{1}{\eps} \log^{2.5}
\frac{1}{\eps}) exists for any with respect to orthogonal ranges, and the
best lower bound is \Omega(\frac{1}{\eps} \log \frac{1}{\eps}). The
\eps-approximation is a rather restricted data structure, as we are not
allowed to store any information other than the coordinates of the points in
. In this paper, we explore what can be achieved without any restriction on
the data structure. We first describe a simple data structure that uses
O(\frac{1}{\eps}(\log^2\frac{1} {\eps} + \log n) ) bits and answers queries
with error \eps n. We then prove a lower bound that any data structure that
answers queries with error \eps n must use
\Omega(\frac{1}{\eps}(\log^2\frac{1} {\eps} + \log n) ) bits. Our lower bound
is information-theoretic: We show that there is a collection of
point sets with large {\em union combinatorial
discrepancy}, and thus are hard to distinguish unless we use
bits.Comment: 19 pages, 5 figure
Planar Visibility: Testing and Counting
In this paper we consider query versions of visibility testing and visibility
counting. Let be a set of disjoint line segments in and let
be an element of . Visibility testing is to preprocess so that we can
quickly determine if is visible from a query point . Visibility counting
involves preprocessing so that one can quickly estimate the number of
segments in visible from a query point .
We present several data structures for the two query problems. The structures
build upon a result by O'Rourke and Suri (1984) who showed that the subset,
, of that is weakly visible from a segment can be
represented as the union of a set, , of triangles, even though
the complexity of can be . We define a variant of their
covering, give efficient output-sensitive algorithms for computing it, and
prove additional properties needed to obtain approximation bounds. Some of our
bounds rely on a new combinatorial result that relates the number of segments
of visible from a point to the number of triangles in that contain .Comment: 22 page
Linear-Space Data Structures for Range Mode Query in Arrays
A mode of a multiset is an element of maximum multiplicity;
that is, occurs at least as frequently as any other element in . Given a
list of items, we consider the problem of constructing a data
structure that efficiently answers range mode queries on . Each query
consists of an input pair of indices for which a mode of must
be returned. We present an -space static data structure
that supports range mode queries in time in the worst case, for
any fixed . When , this corresponds to
the first linear-space data structure to guarantee query time. We
then describe three additional linear-space data structures that provide
, , and query time, respectively, where denotes the
number of distinct elements in and denotes the frequency of the mode of
. Finally, we examine generalizing our data structures to higher dimensions.Comment: 13 pages, 2 figure
Approximate Set Union Via Approximate Randomization
We develop an randomized approximation algorithm for the size of set union
problem \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert, which given a list
of sets with approximate set size for with , and biased random generators
with Prob(x=\randomElm(A_i))\in \left[{1-\alpha_L\over |A_i|},{1+\alpha_R\over
|A_i|}\right] for each input set and element where . The approximation ratio for \arrowvert A_1\cup A_2\cup...\cup
A_m\arrowvert is in the range for any , where
. The complexity of the algorithm
is measured by both time complexity, and round complexity. The algorithm is
allowed to make multiple membership queries and get random elements from the
input sets in one round. Our algorithm makes adaptive accesses to input sets
with multiple rounds. Our algorithm gives an approximation scheme with
O(\setCount\cdot(\log \setCount)^{O(1)}) running time and rounds,
where is the number of sets. Our algorithm can handle input sets that can
generate random elements with bias, and its approximation ratio depends on the
bias. Our algorithm gives a flexible tradeoff with time complexity
O\left(\setCount^{1+\xi}\right) and round complexity for any
Upper and lower bounds for dynamic data structures on strings
We consider a range of simply stated dynamic data structure problems on
strings. An update changes one symbol in the input and a query asks us to
compute some function of the pattern of length and a substring of a longer
text. We give both conditional and unconditional lower bounds for variants of
exact matching with wildcards, inner product, and Hamming distance computation
via a sequence of reductions. As an example, we show that there does not exist
an time algorithm for a large range of these problems
unless the online Boolean matrix-vector multiplication conjecture is false. We
also provide nearly matching upper bounds for most of the problems we consider.Comment: Accepted at STACS'1
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