50 research outputs found
On optimal language compression for sets in PSPACE/poly
We show that if DTIME[2^O(n)] is not included in DSPACE[2^o(n)], then, for
every set B in PSPACE/poly, all strings x in B of length n can be represented
by a string compressed(x) of length at most log(|B^{=n}|)+O(log n), such that a
polynomial-time algorithm, given compressed(x), can distinguish x from all the
other strings in B^{=n}. Modulo the O(log n) additive term, this achieves the
information-theoretic optimum for string compression. We also observe that
optimal compression is not possible for sets more complex than PSPACE/poly
because for any time-constructible superpolynomial function t, there is a set A
computable in space t(n) such that at least one string x of length n requires
compressed(x) to be of length 2 log(|A^=n|).Comment: submitted to Theory of Computing System
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs
We investigate the space complexity of certain perfect matching problems over
bipartite graphs embedded on surfaces of constant genus (orientable or
non-orientable). We show that the problems of deciding whether such graphs have
(1) a perfect matching or not and (2) a unique perfect matching or not, are in
the logspace complexity class \SPL. Since \SPL\ is contained in the logspace
counting classes \oplus\L (in fact in \modk\ for all ), \CeqL, and
\PL, our upper bound places the above-mentioned matching problems in these
counting classes as well. We also show that the search version, computing a
perfect matching, for this class of graphs is in \FL^{\SPL}. Our results
extend the same upper bounds for these problems over bipartite planar graphs
known earlier. As our main technical result, we design a logspace computable
and polynomially bounded weight function which isolates a minimum weight
perfect matching in bipartite graphs embedded on surfaces of constant genus. We
use results from algebraic topology for proving the correctness of the weight
function.Comment: 23 pages, 13 figure
New Algorithms for Distributed Sliding Windows
Computing functions over a distributed stream of data is a significant problem with practical applications. The distributed streaming model is a natural computational model to deal with such scenarios. The goal in this model is to maintain an approximate value of a function of interest over a data stream distributed across several computational nodes. These computational nodes have a two-way communication channel with a coordinator node that maintains an approximation of the function over the entire data stream seen so far. The resources of interest, which need to be minimized, are communication (primary), space, and update time. A practical variant of this model is that of distributed sliding window (dsw), where the computation is limited to the last W items, where W is the window size. Important problems such as sampling and counting have been investigated in this model. However, certain problems including computing frequency moments and metric clustering, that are well studied in other streaming models, have not been considered in the distributed sliding window model.
We give the first algorithms for computing the frequency moments and metric clustering problems in the distributed sliding window model. Our algorithms for these problems are a result of a general transfer theorem we establish that transforms any algorithm in the distributed infinite window model to an algorithm in the distributed sliding window model, for a large class of functions. In particular, we show an efficient adaptation of the smooth histogram technique of Braverman and Ostrovsky, to the distributed streaming model. Our construction allows trade-offs between communication and space. If we optimize for communication, we get algorithms that are as communication efficient as their infinite window counter parts (upto polylogarithmic factors)
Complete Problems for Multi-Pseudodeterministic Computations
We exhibit several computational problems that are complete for multi-pseudodeterministic computations in the following sense: (1) these problems admit 2-pseudodeterministic algorithms (2) if there exists a pseudodeterministic algorithm for any of these problems, then any multi-valued function that admits a k-pseudodeterministic algorithm for a constant k, also admits a pseudodeterministic algorithm. We also show that these computational problems are complete for Search-BPP: a pseudodeterministic algorithm for any of these problems implies a pseudodeterministic algorithm for all problems in Search-BPP
List and Certificate Complexities in Replicable Learning
We investigate replicable learning algorithms. Ideally, we would like to
design algorithms that output the same canonical model over multiple runs, even
when different runs observe a different set of samples from the unknown data
distribution. In general, such a strong notion of replicability is not
achievable. Thus we consider two feasible notions of replicability called list
replicability and certificate replicability. Intuitively, these notions capture
the degree of (non) replicability. We design algorithms for certain learning
problems that are optimal in list and certificate complexity. We establish
matching impossibility results
New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs
We obtain the following new simultaneous time-space upper bounds for the directed reachability problem. (1) A polynomial-time, O(n^{2/3} * g^{1/3})-space algorithm for directed graphs embedded on orientable surfaces of genus g. (2) A polynomial-time, O(n^{2/3})-space algorithm for all H-minor-free graphs given the tree decomposition, and (3) for K_{3,3}-free and K_5-free graphs, a polynomial-time, O(n^{1/2 + epsilon})-space algorithm, for every epsilon > 0.
For the general directed reachability problem, the best known simultaneous time-space upper bound is the BBRS bound, due to Barnes, Buss, Ruzzo, and Schieber, which achieves a space bound of O(n/2^{k * sqrt(log(n))}) with polynomial running time, for any constant k. It is a significant open question to improve this bound for reachability over general directed graphs. Our algorithms beat the BBRS bound for graphs embedded on surfaces of genus n/2^{omega(sqrt(log(n))}, and for all H-minor-free graphs. This significantly broadens the class of directed graphs for which the BBRS bound can be improved