Abstract
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 \(\varTheta (b\log b)\) 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 \(\varTheta (\sqrt{n}\log n)\). When a \(\varTheta (b\log b)\) algorithm is used to sort the keys in a bucket, the exponent becomes \(\varTheta (n)\). 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 well-studied 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 \(\varTheta (n)\).
This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and the United States National Science Foundation (NSF).
A full version of this paper can be found at [1].
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Notes
- 1.
Throughout the paper, \(\ln x\) denotes the natural logarithm of x and \(\log x\) denotes the logarithm of base 2 of x.
- 2.
One should not confuse this analysis with concentration bounds that address small deviations from the expectation.
- 3.
The threshold C depends on: (1) the constant that appears in the sorting algorithm used within each bucket, and (2) the constant that appears in the expected running time of \(b^2\)-Bucket Sort.
- 4.
Interestingly, the sum of squares of bin occupancies, i.e., \(f(\mathbf {b})\), also appears in the FKS perfect hashing construction [8].
- 5.
Formally, T(L) may contain a subset of these n nodes. If a node \(v_j\) at depth \(\log n\) is not chosen by any string, then define \(C_j=0\).
- 6.
Note that RVs \(\left\{ \varDelta \right\} _i\) are not independent and probably not even negatively associated. Hence, standard concentration bounds do not apply to \(\sum \varDelta _i\).
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Bercea, I.O., Even, G. (2021). Upper Tail Analysis of Bucket Sort and Random Tries. In: Calamoneri, T., Corò, F. (eds) Algorithms and Complexity. CIAC 2021. Lecture Notes in Computer Science(), vol 12701. Springer, Cham. https://doi.org/10.1007/978-3-030-75242-2_8
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