Ultrafast Randomized Parallel Construction and Approximation Algorithms for Spanning Forests in Dense Graphs

@inproceedings{Dessmark1998UltrafastRP,
  title={Ultrafast Randomized Parallel Construction and Approximation Algorithms for Spanning Forests in Dense Graphs},
  author={Anders Dessmark and Carsten Dorgerloh and Andrzej Lingas and Juergen Wirtgen},
  booktitle={IPPS/SPDP Workshops},
  year={1998}
}
We present a first randomized $\O$$(\log^{(k)} n)$ time and $\O$$(n + m)$ work CRCW-PRAM algorithm for finding a spanning forest of an undirected dense graph with $n$ vertices. Furthermore we construct a randomized $\O$$(\log \log n)$ time and $\O$$(n \log n)$ work CREW-PRAM algorithm for finding spanning trees in random graphs. Our algorithm is optimal with respect to time, work and space. 
1 Citations
Approximation Algorithms for Bandwidth Problems on Some Large Graph Classes
TLDR
This paper presents the rst PTAS for the topological bandwidth of trees and constructs n-approximation algorithms for the bandwidth of graphs with minimum degree n, for any ; > 0.

References

SHOWING 1-10 OF 45 REFERENCES
Optimal randomized EREW PRAM algorithms for finding spanning forests and for other basic graph connectivity problems
We present the first randomized O(logn) time and O(m+n) work EREW PRAM algorithm for finding a spanning forest of an undirected graph G=(V,E) with n vertices and m edges. Our algorithm is optimal
Approximating dense cases of covering problems
TLDR
The dense set cover problem can be approximated with the performance ratio $c\log n$ for any $c<0$ and it is unlikely to be NP-hard, and the vertex cover problem in $\epsilon$-dense graphs is studied.
On the Parallel Complexity of Hamiltonian Cycle and Matching Problem on Dense Graphs
TLDR
It is proved that finding an NC algorithm for perfect matching in slightly less dense graphs (minimum degree is at least (12 ? ?)|V|) is as hard as the same problem for all graphs, and interestingly the problem of finding a Hamiltonian cycle becomes NP-complete.
The matching problem for bipartite graphs with polynomially bounded permanents is in NC
  • D. Grigoriev, M. Karpinski
  • Mathematics, Computer Science
    28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
  • 1987
TLDR
An NC3 algorithm for the problem of constructing all perfect matchings in a graph G with a permanent bounded by O(nk) is designed, which entails also among other things an efficient NC3-algorithm for computing small (polynomially bounded) arithmetic permanents, and a sublinear parallel time algorithm for enumerating all the perfect matching in graphs with permanents up to 2nε.
An Approximation Algorithm for the Bandwidth Problem on Dense Graphs
TLDR
This paper presents a randomized 3-approximation algorithm for the bandwidth problem restricted to dense graphs and a randomized 2-app approximator for the same problem on directed dense graphs.
Polynomial time approximation schemes for dense instances of NP-hard problems
We present a unified framework for designing polynomial time approximation schemes (PTASs) for “dense” instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph
Parallelism in Comparison Problems
TLDR
The worst-case time complexity of algorithms for multiprocessor computers with binary comparisons as the basic operations is investigated and the algorithm for finding the maximum is shown to be optimal for all values of k and n.
The regularity lemma and approximation schemes for dense problems
  • A. Frieze, R. Kannan
  • Mathematics
    Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
TLDR
The central point here is that the Regularity Lemma provides an explanation of why these Max-SNP hard problems turn out to be easy in dense graphs.
...
...