33,123 research outputs found
Derandomization with Minimal Memory Footprint
Existing proofs that deduce BPL = ? from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE[S] ? DSPACE[c ? S] for c ? 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ? 1.
Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC?, that were not known before
The Quantum Complexity of Set Membership
We study the quantum complexity of the static set membership problem: given a
subset S (|S| \leq n) of a universe of size m (m \gg n), store it as a table of
bits so that queries of the form `Is x \in S?' can be answered. The goal is to
use a small table and yet answer queries using few bitprobes. This problem was
considered recently by Buhrman, Miltersen, Radhakrishnan and Venkatesh, where
lower and upper bounds were shown for this problem in the classical
deterministic and randomized models. In this paper, we formulate this problem
in the "quantum bitprobe model" and show tradeoff results between space and
time.In this model, the storage scheme is classical but the query scheme is
quantum.We show, roughly speaking, that similar lower bounds hold in the
quantum model as in the classical model, which imply that the classical upper
bounds are more or less tight even in the quantum case. Our lower bounds are
proved using linear algebraic techniques.Comment: 19 pages, a preliminary version appeared in FOCS 2000. This is the
journal version, which will appear in Algorithmica (Special issue on Quantum
Computation and Quantum Cryptography). This version corrects some bugs in the
parameters of some theorem
Element Distinctness, Frequency Moments, and Sliding Windows
We derive new time-space tradeoff lower bounds and algorithms for exactly
computing statistics of input data, including frequency moments, element
distinctness, and order statistics, that are simple to calculate for sorted
data. We develop a randomized algorithm for the element distinctness problem
whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than
previous lower bounds for comparison-based algorithms, showing that element
distinctness is strictly easier than sorting for randomized branching programs.
This algorithm is based on a new time and space efficient algorithm for finding
all collisions of a function f from a finite set to itself that are reachable
by iterating f from a given set of starting points. We further show that our
element distinctness algorithm can be extended at only a polylogarithmic factor
cost to solve the element distinctness problem over sliding windows, where the
task is to take an input of length 2n-1 and produce an output for each window
of length n, giving n outputs in total. In contrast, we show a time-space
tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to
compute the number of distinct elements over sliding windows. The same lower
bound holds for computing the low-order bit of F_0 and computing any frequency
moment F_k, k neq 1. This shows that those frequency moments and the decision
problem F_0 mod 2 are strictly harder than element distinctness. We complement
this lower bound with a T in O(n^2/S) comparison-based deterministic RAM
algorithm for exactly computing F_k over sliding windows, nearly matching both
our lower bound for the sliding-window version and the comparison-based lower
bounds for the single-window version. We further exhibit a quantum algorithm
for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider
the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437
Time lower bounds for nonadaptive turnstile streaming algorithms
We say a turnstile streaming algorithm is "non-adaptive" if, during updates,
the memory cells written and read depend only on the index being updated and
random coins tossed at the beginning of the stream (and not on the memory
contents of the algorithm). Memory cells read during queries may be decided
upon adaptively. All known turnstile streaming algorithms in the literature are
non-adaptive.
We prove the first non-trivial update time lower bounds for both randomized
and deterministic turnstile streaming algorithms, which hold when the
algorithms are non-adaptive. While there has been abundant success in proving
space lower bounds, there have been no non-trivial update time lower bounds in
the turnstile model. Our lower bounds hold against classically studied problems
such as heavy hitters, point query, entropy estimation, and moment estimation.
In some cases of deterministic algorithms, our lower bounds nearly match known
upper bounds
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