Sublinear Space Research Articles

Estimating Entropy and Entropy Norm on Data Streams

We consider the problem of computing information-theoretic functions, such as entropy, on a data stream, using sublinear space. Our first result deals with a measure we call the _entropy norm_ of an input stream: it is closely related to entropy but is structurally similar to the well-studied notion of frequency moments. We give a polylogarithmic-space, one-pass algorithm for estimating this norm under certain conditions on the input stream. We also prove a lower bound that rules out such an algorithm if these conditions do not hold. Our second group of results is for estimating the empirical entropy of an input stream. We first present a sublinear-space, one-pass algorithm for this problem. For a stream of _m_ items and a given real parameter α, our algorithm uses space _Õ_(_m_^2α) and provides an approximation of 1/α in the worst case and (1+ε) in "most" cases. We then present a two-pass, polylogarithmic-space, (1+ε)-approximation algorithm. All our algorithms are quite simple.

An improved data stream summary: the count-min sketch and its applications

We introduce a new sublinear space data structure—the count-min sketch—for summarizing data streams. Our sketch allows fundamental queries in data stream summarization such as point, range, and inner product queries to be approximately answered very quickly; in addition, it can be applied to solve several important problems in data streams such as finding quantiles, frequent items, etc. The time and space bounds we show for using the CM sketch to solve these problems significantly improve those previously known—typically from 1/ ε 2 to 1/ ε in factor.

Compact Routing with Minimum Stretch

We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n2/3log4/3n) and achieves paths that are most 3 times the length of the shortest path distances for all nodes in an arbitrary weighted undirected network. This answers an open question of Gavoille and Gengler who showed that any universal compact routing algorithm with maximum stretch strictly less than 3 must use Ω(n) local space at some vertex.

Time–Space Tradeoffs for Satisfiability

We give the first nontrivial model-independent time–space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved in n1+o(1) time and n1−ε space for any ε>0 general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and n space. We also give lower bounds for log-space uniform NC1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnjaščiı that shows that a nondeterministic computation of superlinear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL and NP. We give some possibilities and limitations of this approach.

A Sublinear Space, Polynomial Time Algorithm for Directed s-t Connectivity

Directed s-t connectivity is the problem of detecting whether there is a path from vertex s to vertex t in a directed graph. We present the first known deterministic sublinear space, polynomial time algorithm for directed s-t connectivity. For $n$-vertex graphs, our algorithm can use as little as $n/2^{\Theta(\sqrt{\log n})}$ space while still running in polynomial time.

Saving comparisons in the Crochemore-Perrin string-matching algorithm

Crochemore and Perrin discovered an elegant linear-time constant-space string-matching algorithm that makes at most 2 n − m symbol comparisons. This paper shows how to modify their algorithm to use fewer comparisons. Given any fixed ε > 0, the new algorithm takes linear time, uses constant space and makes at most n + [ 1 + ε 2 (n − m)] symbol comparisons. If O( log m) space is available, then the algorithm makes at most n + [ 1 2 (n − m)] symbol comparisons. The pattern preprocessing step also takes linear time and uses constant space. These are the first string-matching algorithms that make fewer than 2 n − m symbol comparisons and use sub-linear space.

Undirecteds-t connectivity in polynomial time and sublinear space

Thes-t connectivity problem for undirected graphs is to decide whether two designated vertices,s andt, are in the same connected component. This paper presents the first known deterministic algorithms solving undirecteds-t connectivity using sublinear space and polynomial time. Our algorithms provide a nearly smooth time-space tradeoff between depth-first search and Savitch's algorithm. Forn vertex,m edge graphs, the simplest of our algorithms uses spaceO(s),n 1/2log2 n≤s≤nlog2 n, and timeO(((m+n)n 2 log2 n)/s). We give a variant of this method that is faster at the higher end of the space spectrum. For example, with space θ(nlogn), its time bound isO((m+n)logn), close to the optimal time for the problem. Another generalization uses less space, but more time: spaceO(λn 1/λlogn), for 2≤λ≤log2 n, and timen O(λ). For constant λ the time remains polynomial.

Relations among simultaneous complexity classes of nondeterministic and alternating Turing machines

Ruzzo [Tree-size bounded alternation, J. Comput. Syst. Sci. 21] introduced the notion of tree-size for alternating Turing machines (ATMs) and showed that it is a reasonable measure for classification of complexity classes. We establish in this paper that computations by tree-size and space simultaneously bounded ATMs roughly correspond to computations by time and space simultaneously bounded nondeterministic TMs (NTMs). We also show that not every polynomial time bounded and sublinear space simultaneously bounded NTM can be simulated by any deterministic TM with a slightly increased time bound and a slightly decreased space bound simultaneously.