514 research outputs found

    Adversary Lower Bound for Element Distinctness with Small Range

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    The Element Distinctness problem is to decide whether each character of an input string is unique. The quantum query complexity of Element Distinctness is known to be Θ(N2/3)\Theta(N^{2/3}); the polynomial method gives a tight lower bound for any input alphabet, while a tight adversary construction was only known for alphabets of size Ω(N2)\Omega(N^2). We construct a tight Ω(N2/3)\Omega(N^{2/3}) adversary lower bound for Element Distinctness with minimal non-trivial alphabet size, which equals the length of the input. This result may help to improve lower bounds for other related query problems.Comment: 22 pages. v2: one figure added, updated references, and minor typos fixed. v3: minor typos fixe

    Element Distinctness, Frequency Moments, and Sliding Windows

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    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

    Quantum Algorithms for the Triangle Problem

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    We present two new quantum algorithms that either find a triangle (a copy of K3K_{3}) in an undirected graph GG on nn nodes, or reject if GG is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes O~(n10/7)\tilde{O}(n^{10/7}) queries. The second algorithm uses O~(n13/10)\tilde{O}(n^{13/10}) queries, and it is based on a design concept of Ambainis~\cite{amb04} that incorporates the benefits of quantum walks into Grover search~\cite{gro96}. The first algorithm uses only O(logn)O(\log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in~\cite{bdhhmsw01}, where an algorithm with O(n+nm)O(n+\sqrt{nm}) query complexity was presented, where mm is the number of edges of GG.Comment: Several typos are fixed, and full proofs are included. Full version of the paper accepted to SODA'0
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