1,160 research outputs found
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
Efficient Distributed Quantum Computing
We provide algorithms for efficiently addressing quantum memory in parallel.
These imply that the standard circuit model can be simulated with low overhead
by the more realistic model of a distributed quantum computer. As a result, the
circuit model can be used by algorithm designers without worrying whether the
underlying architecture supports the connectivity of the circuit. In addition,
we apply our results to existing memory intensive quantum algorithms. We
present a parallel quantum search algorithm and improve the time-space
trade-off for the Element Distinctness and Collision problems.Comment: Some material rearranged and references adde
The quantum complexity of approximating the frequency moments
The 'th frequency moment of a sequence of integers is defined as , where is the number of times that occurs in the
sequence. Here we study the quantum complexity of approximately computing the
frequency moments in two settings. In the query complexity setting, we wish to
minimise the number of queries to the input used to approximate up to
relative error . We give quantum algorithms which outperform the best
possible classical algorithms up to quadratically. In the multiple-pass
streaming setting, we see the elements of the input one at a time, and seek to
minimise the amount of storage space, or passes over the data, used to
approximate . We describe quantum algorithms for , and
in this model which substantially outperform the best possible
classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio
Quantum walk-based search algorithms with multiple marked vertices
The quantum walk is a powerful tool to develop quantum algorithms, which
usually are based on searching for a vertex in a graph with multiple marked
vertices, Ambainis's quantum algorithm for solving the element distinctness
problem being the most shining example. In this work, we address the problem of
calculating analytical expressions of the time complexity of finding a marked
vertex using quantum walk-based search algorithms with multiple marked vertices
on arbitrary graphs, extending previous analytical methods based on Szegedy's
quantum walk, which can be applied only to bipartite graphs. Two examples based
on the coined quantum walk on two-dimensional lattices and hypercubes show the
details of our method.Comment: 12 pages, 1 table, 2 fig
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