We describe space-efficient algorithms for solving problems related to finding maxima among points in two and three dimensions. Our algorithms run in optimal $\mathcal{O}(n\log n)$time and occupy only constant extra space in addition to the space needed for representing the input.

/pdf/in-place-algorithms-for-computing-layers-of-maxima-4v97yrc885.pdf

In-Place Algorithms for Computing (Layers of) Maxima

We describe space-efficient algorithms for solving problems related to finding maxima among points in two and three dimensions Our algorithms run in optimal $\mathcal{O}({n\log n})$ time and occupy only constant extra space in addition to the space needed for representing the input

/pdf/in-place-algorithms-for-computing-layers-of-maxima-58xoy0dakh.pdf

In-Place algorithms for computing (layers of) maxima

Suffix arrays and LCP arrays are one of the most fundamental data structures widely used for various kinds of string processing. We consider two problems for a read-only string of length $N$ over an integer alphabet $[1, \dots, \sigma]$ for $1 \leq \sigma \leq N$, the string contains $\sigma$ distinct characters, the construction of the suffix array, and a simultaneous construction of both the suffix array and LCP array. For the word RAM model, we propose algorithms to solve both of the problems in $O(N)$ time by using $O(1)$ extra words, which are optimal in time and space. Extra words means the required space except for the space of the input string and output suffix array and LCP array. Our contribution improves the previous most efficient algorithms, $O(N)$ time using $\sigma+O(1)$ extra words by [Nong, TOIS 2013] and $O(N \log N)$ time using $O(1)$ extra words by [Franceschini and Muthukrishnan, ICALP 2007], for constructing suffix arrays, and it improves the previous most efficient solution that runs in $O(N)$ time using $\sigma + O(1)$ extra words for constructing both suffix arrays and LCP arrays through a combination of [Nong, TOIS 2013] and [Manzini, SWAT 2004].

Optimal Time and Space Construction of Suffix Arrays and LCP Arrays for Integer Alphabets.

We investigate the problem of stable in-place merging from a ratio k = n/m based point of view where m, n are the sizes of the input sequences with m ≤ n. We introduce a novel algorithm for this problem that is asymptotically optimal regarding the number of assignments as well as comparisons. Our algorithm uses knowledge about the ratio of the input sizes to gain optimality and does not stay in the tradition of Mannila and Ukkonen's work [8] in contrast to all other stable in-place merging algorithms proposed so far. It has a simple modular structure and does not demand the additional extraction of a movement imitation buffer as needed by its competitors. For its core components we give concrete implementations in form of Pseudo Code. Using benchmarking we prove that our algorithm performs almost always better than its direct competitor proposed in [6].

As additional sub-result we show that stable in-place merging is a quite simple problem for every ratio k ≥ √m by proving that there exists a primitive algorithm that is asymptotically optimal for such ratios.

Jing-Chao Chen

Papers

Optimizing stable in-place merging

Efficient sample sort and the average case analysis or PEsort