
A LinearTime n^0.4Approximation for Longest Common Subsequence
We consider the classic problem of computing the Longest Common Subseque...
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A Subquadratic Algorithm for 3XOR
Given a set X of n binary words of equal length w, the 3XOR problem asks...
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The Generalized Trust Region Subproblem: solution complexity and convex hull results
We consider the Generalized Trust Region Subproblem (GTRS) of minimizing...
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Sorting Lists with Equal Keys Using Mergesort in Linear Time
This article introduces a new optimization method to improve mergesort's...
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Large deviations for the largest eigenvalue of Gaussian networks with constant average degree
Large deviation behavior of the largest eigenvalue λ_1 of Gaussian netwo...
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A Tail Estimate with Exponential Decay for the Randomized Incremental Construction of Search Structures
We revisit the randomized incremental construction of the Trapezoidal Se...
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The Fitness Level Method with Tail Bounds
The fitnesslevel method, also called the method of fbased partitions, ...
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Upper Tail Analysis of Bucket Sort and Random Tries
Bucket Sort is known to run in expected linear time when the input keys are distributed independently and uniformly at random in the interval [0,1). The analysis holds even when a quadratic time algorithm is used to sort the keys in each bucket. We show how to obtain linear time guarantees on the running time of Bucket Sort that hold with very high probability. Specifically, we investigate the asymptotic behavior of the exponent in the upper tail probability of the running time of Bucket Sort. We consider large additive deviations from the expectation, of the form cn for large enough (constant) c, where n is the number of keys that are sorted. Our analysis shows a profound difference between variants of Bucket Sort that use a quadratic time algorithm within each bucket and variants that use a Θ(blog b) time algorithm for sorting b keys in a bucket. When a quadratic time algorithm is used to sort the keys in a bucket, the probability that Bucket Sort takes cn more time than expected is exponential in Θ(√(n)log n). When a Θ(blog b) algorithm is used to sort the keys in a bucket, the exponent becomes Θ(n). We prove this latter theorem by showing an upper bound on the tail of a random variable defined on tries, a result which we believe is of independent interest. This result also enables us to analyze the upper tail probability of a wellstudied trie parameter, the external path length, and show that the probability that it deviates from its expected value by an additive factor of cn is exponential in Θ(n).
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