Abstract: Using a set of geometric containers to speed up shortest path queries in a weighted graph has been proven a useful tool for dealing with large sparsegraphs. Given a layout of a graph G=(V,E), we store, for each edge (u,v)set membership, variantE, the bounding box of all nodes tset membership, variantV for which a shortest u-t-path starts with (u,v). Shortest path queries can then be answered by DijkstraImage restricted to edges where the corresponding bounding box contains the target.
In this paper, we present new algorithms as well as an empirical study for the dynamic case of this problem, where edge weights are subject to change and the bounding boxes have to be updated. We evaluate the quality and the time for different update strategies that guarantee correct shortest paths in an interesting application to railway information systems, using real-world data from six European countries.
Abstract: The problem of finding an implicit representation for a graph such that vertex adjacency can be tested quickly is fundamental to all graph algorithms. In particular, it is possible to represent sparsegraphs on n vertices using O(n) space such that vertex adjacency is tested in O(l) time. We show here how to construct such a representation efficiently by providing simple and optimal algorithms, both in a sequential and a parallel setting. Our sequential algorithm runs in O(n) time. The parallel algorithm runs in O(log n) time using O(n/log n) CRCW PRAM processors, or inO(log n log* n) time using O(n/log n log* n) EREW PRAM processors. Previous results for this problem are based on matroid partitioning and thus have a high complexity.
Abstract: In this work we present an efficient algorithm which, with high probability, provides
an almost uniform sample from the set of proper $\chi$-colourings of an instance of sparse
random graphs $G_{n,d/n}$, where $\chi=\chi(d)$ is a sufficiently large constant.
This work improves, asymptotically, the result of Dyer, Flaxman Frieze and Vigoda
in \cite{previous-result} where the algorithm proposed there needs at least
$\Theta(\frac{\log \log n}{\log \log \log n})$ colours.
Abstract: We generalize Cuckoo Hashing [16] to d-ary Cuckoo Hashing and show how this yields a simple hash table data structure that stores n elements in (1 + ∈) n memory cells, for any constant ∈ > 0. Assuming uniform hashing, accessing or deleting table entries takes at most d = O(ln 1/∈ ) probes and the expected amortized insertion time is constant. This is the first dictionary that has worst case constant access time and expected constant update time, works with (1+∈) n space, and supports satellite information. Experiments indicate that d = 4 choices suffice for ∈ ≈ 0.03. We also describe a hash table data structure using explicit constant time hash functions, using at most d = O(ln2 1/∈ ) probes in the worst case.
A corollary is an expected linear time algorithm for finding maximum cardinality matchings in a rather natural model of sparse random bipartite graphs.
This work was partially supported by DFG grant SA 933/1-1 and the Future and Emerging Technologies programme of the EU under contract number IST-1999- 14186 (ALCOM-FT).
The present work was initiated while this author was at BRICS, Aarhus University, Denmark.
Part of this work was done while the author was at MPII. 1 In this paper “whp.” will mean “with probability 1 - O(1/n)”.
Abstract: We generalize Cuckoo Hashing [16] to d-ary Cuckoo Hashing and show how this yields a simple hash table data structure that stores n elements in (1 + ∈) n memory cells, for any constant ∈ > 0. Assuming uniform hashing, accessing or deleting table entries takes at most d = O(ln 1/∈ ) probes and the expected amortized insertion time is constant. This is the first dictionary that has worst case constant access time and expected constant update time, works with (1+∈) n space, and supports satellite information. Experiments indicate that d = 4 choices suffice for ∈ ≈ 0.03. We also describe a hash table data structure using explicit constant time hash functions, using at most d = O(ln2 1/∈ ) probes in the worst case.
A corollary is an expected linear time algorithm for finding maximum cardinality matchings in a rather natural model of sparse random bipartite graphs.
This work was partially supported by DFG grant SA 933/1-1 and the Future and Emerging Technologies programme of the EU under contract number IST-1999- 14186 (ALCOM-FT).
The present work was initiated while this author was at BRICS, Aarhus University, Denmark.
Part of this work was done while the author was at MPII. 1 In this paper “whp.” will mean “with probability 1 - O(1/n)”.
