Time-space trade-off lower bounds for randomized computation of decision problems

Abstract

We prove the first time-space lower bound trade-offs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are extension of those used by Ajtai and by Beame, Jayram, and Saks that applied to deterministic branching programs. Our results also give a quantitative improvement over the previous results.Previous time-space trade-off results for decision problems can be divided naturally into results for functions with Boolean domain, that is, each input variable is {0, 1}-valued, and the case of large domain, where each input variable takes on values from a set whose size grows with the number of variables.In the case of Boolean domain, Ajtai exhibited an explicit class of functions, and proved that any deterministic Boolean branching program or RAM using space S = o(n) requires superlinear time T to compute them. The functional form of the superlinear bound is not given in his paper, but optimizing the parameters in his arguments gives T = (n log log n/ log log log n) for S = O(n 1− ). 155For the same functions considered by Ajtai, we prove a time-space trade-off (for randomized branching programs with error) of the form T = (n √ log(n/S)/ log log(n/S)). In particular, for space O(n 1− ), this improves the lower bound on time to (n √ log n/ log log n). In the large domain case, we prove lower bounds of the form T = (n √ log(n/S)/ log log(n/S)) for randomized computation of the element distinctness function and lower bounds of the form T = (n log(n/S)) for randomized computation of Ajtai's Hamming closeness problem and of certain functions associated with quadratic forms over large fields.

Lower bounds for randomized read/write stream algorithms

“…The following is a generalization of an argument of Babai, Frankl, and Simon [2]. Its appears in a somewhat different form in [4]. Let k = p n/m.…”

“…[4]) could be used to obtain tradeoffs between reversals and space in the read/write stream model by bounding the time used as a function of the number of reversals. However, those lower bounds only apply with readonly inputs and sub-linear space bounds; with the ability of read/write stream algorithms to write to linear (or even significantly super-linear) numbers of memory locations on the external tapes, neither the time nor the space bounds are small enough for the time-space tradeoffs to apply.…”

Time-space lower bounds for satisfiability

“…We should point out that Beame et al [BSSV03], building on earlier work by Ajtai [Ajt99], establish nonuniform time-space lower bounds for a problem in P based on binary quadratic forms. They show that any branching program for that problem that uses only space n 1−ǫ for some positive constant ǫ takes time Ω(n · log n/ log log n),…”

A Survey of Lower Bounds for Satisfiability and Related Problems

“…We point out that some of the more recent work in circuit complexity does not seem to have implications for satisfiability. In particular, the non-uniform time-space lower bounds by Ajtai [Ajt99] and their improvements by Beame et al [BSSV03] do not yield time-space lower bounds for satisfiability. These authors consider a problem in P based on a binary quadratic form, and showed that any branching program for it that uses only n 1−ǫ space for some positive constant ǫ takes time Ω(n · log n/ log log n).…”