
Quantum Lower Bounds for 2DGrid and Dyck Language
We show quantum lower bounds for two problems. First, we consider the pr...
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Quantum Query Complexity of Dyck Languages with Bounded Height
We consider the problem of determining if a sequence of parentheses is w...
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Automation Strategies for Unconstrained Crossword Puzzle Generation
An unconstrained crossword puzzle is a generalization of the constrained...
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Quantum algorithm for Dyck Language with Multiple Types of Brackets
We consider the recognition problem of the Dyck Language generalized for...
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Sparsification of Balanced Directed Graphs
Sparsification, where the cut values of an input graph are approximately...
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A Quantum Query Complexity Trichotomy for Regular Languages
We present a trichotomy theorem for the quantum query complexity of regu...
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On Finding Quantum Multicollisions
A kcollision for a compressing hash function H is a set of k distinct i...
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Quantum Lower and Upper Bounds for 2DGrid and Dyck Language
We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyck_k,n problem. We prove a lower bound of Ω(c^k √(n)), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative example of starfree languages for which a surprising Õ(√(n)) query quantum algorithm was recently constructed by Aaronson et al. Their proof does not give rise to a general algorithm. When k is not a constant, Dyck_k,n is not contextfree. We give an algorithm with O(√(n)(logn)^0.5k) quantum queries for Dyck_k,n for all k. This is better than the trival upper bound n for k=o(log(n)/loglog n). Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of Ω(n^1.5ϵ) for the directed 2D grid and Ω(n^2ϵ) for the undirected 2D grid. The directed problem is interesting as a blackbox model for a class of classical dynamic programming strategies including the one that is usually used for the wellknown edit distance problem. We also show a generalization of this result to more than 2 dimensions.
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