Constant Time Sorting and Searching

Title from PDF of title page, viewed January 4, 2023Thesis advisor: Yijie HanVitaIncludes bibliographical references (pages 25-28)Thesis (M.S.)--Department of Computer Science and Electrical Engineering. University of Missouri--Kansas City, 2022To study the sorting of real numbers into a linked list on Parallel Random Access Machine model. To show that input array of n real numbers can be sorted into a linked list in constant time using n²/logᶜn processors for any positive constant c. The searching problem studied is locating the interval of n sorted real numbers for inserting a query real number. Taking into account an input of n real numbers and organize them in the sorted order to facilitate searching. Initially, sorting the n input real numbers and then convert these real numbers into integers such that their relative order is preserved. Convert the query input real number to a query integer and then search the interval among these n integers for the insertion point of this query real number in constant time.Introduction -- Sorting in constant time -- Searching in constant time -- Theorem -- Conclusion

Self-Improving Algorithms

We investigate ways in which an algorithm can improve its expected performance by fine-tuning itself automatically with respect to an unknown input distribution D. We assume here that D is of product type. More precisely, suppose that we need to process a sequence I_1, I_2, ... of inputs I = (x_1, x_2, ..., x_n) of some fixed length n, where each x_i is drawn independently from some arbitrary, unknown distribution D_i. The goal is to design an algorithm for these inputs so that eventually the expected running time will be optimal for the input distribution D = D_1 * D_2 * ... * D_n. We give such self-improving algorithms for two problems: (i) sorting a sequence of numbers and (ii) computing the Delaunay triangulation of a planar point set. Both algorithms achieve optimal expected limiting complexity. The algorithms begin with a training phase during which they collect information about the input distribution, followed by a stationary regime in which the algorithms settle to their optimized incarnations.Comment: 26 pages, 8 figures, preliminary versions appeared at SODA 2006 and SoCG 2008. Thorough revision to improve the presentation of the pape

Fast Deterministic Selection

The Median of Medians (also known as BFPRT) algorithm, although a landmark theoretical achievement, is seldom used in practice because it and its variants are slower than simple approaches based on sampling. The main contribution of this paper is a fast linear-time deterministic selection algorithm QuickselectAdaptive based on a refined definition of MedianOfMedians. The algorithm's performance brings deterministic selection---along with its desirable properties of reproducible runs, predictable run times, and immunity to pathological inputs---in the range of practicality. We demonstrate results on independent and identically distributed random inputs and on normally-distributed inputs. Measurements show that QuickselectAdaptive is faster than state-of-the-art baselines.Comment: Pre-publication draf

Improved Bounds for 3SUM, kk-SUM, and Linear Degeneracy

Given a set of $n$ real numbers, the 3SUM problem is to decide whether there are three of them that sum to zero. Until a recent breakthrough by Gr{\o}nlund and Pettie [FOCS'14], a simple $\Theta(n^2)$-time deterministic algorithm for this problem was conjectured to be optimal. Over the years many algorithmic problems have been shown to be reducible from the 3SUM problem or its variants, including the more generalized forms of the problem, such as $k$-SUM and $k$-variate linear degeneracy testing ($k$-LDT). The conjectured hardness of these problems have become extremely popular for basing conditional lower bounds for numerous algorithmic problems in P. In this paper, we show that the randomized $4$-linear decision tree complexity of 3SUM is $O(n^{3/2})$, and that the randomized $(2k-2)$-linear decision tree complexity of $k$-SUM and $k$-LDT is $O(n^{k/2})$, for any odd $k\ge 3$. These bounds improve (albeit randomized) the corresponding $O(n^{3/2}\sqrt{\log n})$ and $O(n^{k/2}\sqrt{\log n})$ decision tree bounds obtained by Gr{\o}nlund and Pettie. Our technique includes a specialized randomized variant of fractional cascading data structure. Additionally, we give another deterministic algorithm for 3SUM that runs in $O(n^2 \log\log n / \log n )$ time. The latter bound matches a recent independent bound by Freund [Algorithmica 2017], but our algorithm is somewhat simpler, due to a better use of word-RAM model