Efficiently Computing Directed Minimum Spanning Trees

Computing a directed minimum spanning tree, called arborescence, is a fundamental algorithmic problem, although not as common as its undirected counterpart. In 1967, Edmonds discussed an elegant solution. It was refined to run in $O(\min(n^2, m\log n))$ by Tarjan which is optimal for very dense and very sparse graphs. Gabow et al.~gave a version of Edmonds' algorithm that runs in $O(n\log n + m)$, thus asymptotically beating the Tarjan variant in the regime between sparse and dense. Despite the attention the problem received theoretically, there exists, to the best of our knowledge, no empirical evaluation of either of these algorithms. In fact, the version by Gabow et al.~has never been implemented and, aside from coding competitions, all readily available Tarjan implementations run in $O(n^2)$. In this paper, we provide the first implementation of the version by Gabow et al.~as well as five variants of Tarjan's version with different underlying data structures. We evaluate these algorithms and existing solvers on a large set of real-world and random graphs

Fast Parallel Algorithms for Basic Problems

Parallel processing is one of the most active research areas these days. We are interested in one aspect of parallel processing, i.e. the design and analysis of parallel algorithms. Here, we focus on non-numerical parallel algorithms for basic combinatorial problems, such as data structures, selection, searching, merging and sorting. The purposes of studying these types of problems are to obtain basic building blocks which will be useful in solving complex problems, and to develop fundamental algorithmic techniques. In this thesis, we study the following problems: priority queues, multiple search and multiple selection, and reconstruction of a binary tree from its traversals. The research on priority queue was motivated by its various applications. The purpose of studying multiple search and multiple selection is to explore the relationships between four of the most fundamental problems in algorithm design, that is, selection, searching, merging and sorting; while our parallel solutions can be used as subroutines in algorithms for other problems. The research on the last problem, reconstruction of a binary tree from its traversals, was stimulated by a challenge proposed in a recent paper by Berkman et al. ( Highly Parallelizable Problems, STOC 89) to design doubly logarithmic time optimal parallel algorithms because a remarkably small number of such parallel algorithms exist

Union-find-delete-algoritmien vertailua

Union-find-algoritmi on tehokas ratkaisu erillisten joukkojen käsittelyongelmaan. Union-find-delete-algoritmissa siihen on lisätty poiston mahdollisuus. Esittelen tässä tutkielmassa useita union-find-algoritmeja ja union-find-delete-algoritmeja ja vertailen niitä teoreettisesti ja kokeellisesti. Toteutin kolme erilaista union-find-delete-algoritmia, ja testasin niitä satunnaisilla syötteillä. Toteuttamani algoritmit ovat Ben-Amramin ja Yoffen algoritmi, aidon poiston algoritmi ja vajaan poiston algoritmi. Testieni perusteella vaikuttaa siltä, että asymptoottiselta aikavaatimukseltaan huonompi aidon poiston algoritmi toimii keskimäärin yhtä hyvin tai jopa paremmin kuin Ben-Amramin ja Yoffen algoritmi, jossa poisto tehdään vakioajassa

Melding Priority Queues

We show that any priority queue data structure that supports insert, delete, and find-min operations in pq(n) time, when n is an upper bound on the number of elements in the priority queue, can be converted into a priority queue data structure that also supports fast meld operations with essentially no increase in the amortized cost of the other operations. More specifically, the new data structure supports insert, meld and find-min operations in O(1) amortized time, and delete operations in O(pq(n) + α(n, n)) amortized time, where α(m, n) is a functional inverse of the Ackermann function. The construction is very simple, essentially just placing a non-meldable priority queue at each node of a union-find data structure. We also show that when all keys are integers in the range [1, N], we can replace n in the bound stated above by min{n, N}. Applying this result to non-meldable priority queue data structures obtained recently by Thorup, and by Han and Thorup, we obtain meldable RAM priority queues with O(log log n) amortized cost per operation, or O ( √ log log n) expected amortized cost per operation, respectively. As a by-product, we obtain improved algorithms for the minimum directed spanning tree problem in graphs with integer edge weights: A deterministic O(m log log n) time algorithm and a randomized O(m √ log log n) time algorithm. These bounds improve, for sparse enough graphs, on the O(m + n log n) running time of an algorithm by Gabow, Galil, Spencer and Tarjan that works for arbitrary edge weights