111 research outputs found
A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem
The deterministic space complexity of approximating the length of the longest increasing subsequence of a stream of N integers is known to be Theta~(sqrt N). However, the randomized complexity is wide open. We show that the technique used in earlier work to establish the Omega(sqrt N) deterministic lower bound fails strongly under randomization: specifically, we show that the communication problems on which the lower bound is based have very efficient randomized protocols. The purpose of this note is to guide and alert future researchers working on this very interesting problem
Private Decayed Sum Estimation under Continual Observation
In monitoring applications, recent data is more important than distant data.
How does this affect privacy of data analysis? We study a general class of data
analyses - computing predicate sums - with privacy. Formally, we study the
problem of estimating predicate sums {\em privately}, for sliding windows (and
other well-known decay models of data, i.e. exponential and polynomial decay).
We extend the recently proposed continual privacy model of Dwork et al.
We present algorithms for decayed sum which are \eps-differentially
private, and are accurate. For window and exponential decay sums, our
algorithms are accurate up to additive 1/\eps and polylog terms in the range
of the computed function; for polynomial decay sums which are technically more
challenging because partial solutions do not compose easily, our algorithms
incur additional relative error. Further, we show lower bounds, tight within
polylog factors and tight with respect to the dependence on the probability of
error
Space-Efficient Algorithms for Longest Increasing Subsequence
Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in O(n log n) time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For sqrt(n) <= s <= n, we present algorithms that use O(s log n) bits and O(1/s n^2 log n) time for computing the length of a longest increasing subsequence, and O(1/s n^2 log^2 n) time for finding an actual subsequence. We also show that the time complexity of our algorithms is optimal up to polylogarithmic factors in the framework of sequential access algorithms with the prescribed amount of space
Streaming and Query Once Space Complexity of Longest Increasing Subsequence
Longest Increasing Subsequence (LIS) is a fundamental problem in
combinatorics and computer science. Previously, there have been numerous works
on both upper bounds and lower bounds of the time complexity of computing and
approximating LIS, yet only a few on the equally important space complexity.
In this paper, we further study the space complexity of computing and
approximating LIS in various models. Specifically, we prove non-trivial space
lower bounds in the following two models: (1) the adaptive query-once model or
read-once branching programs, and (2) the streaming model where the order of
streaming is different from the natural order.
As far as we know, there are no previous works on the space complexity of LIS
in these models. Besides the bounds, our work also leaves many intriguing open
problems.Comment: This paper has been accepted to COCOON 202
On Fully Dynamic Graph Sparsifiers
We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a -spectral sparsifier with amortized update time . Second, we give a fully dynamic algorithm for maintaining a -cut sparsifier with \emph{worst-case} update time . Both sparsifiers have size . Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a -approximation to the value of the maximum flow in an unweighted, undirected, bipartite graph with amortized update time
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