# Algorithms for dense graphs and networks on the random access computer

@article{Cheriyan2005AlgorithmsFD, title={Algorithms for dense graphs and networks on the random access computer}, author={Joseph Cheriyan and Kurt Mehlhorn}, journal={Algorithmica}, year={2005}, volume={15}, pages={521-549} }

We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size λ=Ω (logn) a maximal matching in ann-vertex bipartite graph in timeO(n2+n2.5/λ)=O(n2.5/logn), how to compute the transitive closure of a digraph withn vertices andm edges in timeO(n2+nm/λ), how to solve the uncapacitated transportation problem with integer costs in the range [O.C] and integer demands in the range [−U.U] in timeO…

## 58 Citations

Matching Algorithms Are Fast in Sparse Random Graphs

- Computer Science, MathematicsSTACS
- 2004

An improved average case analysis of the maximum cardinality matching problem shows that in a bipartite or general random graph on n vertices, with high probability every non-maximum matching has an augmenting path of length O(log n), and holds, if only the average degree is a large enough constant.

Matching Algorithms Are Fast in Sparse Random Graphs

- Computer Science, MathematicsTheory of Computing Systems
- 2005

AbstractWe present an improved average case analysis of the maximum cardinality
matching problem. We show that in a bipartite or general random graph on n
vertices, with high probability every…

Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance

- Computer Science, MathematicsACM Trans. Algorithms
- 2012

We present two online algorithms for maintaining a topological order of a directed n-vertex acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm handles m…

Finding All Allowed Edges in a Bipartite Graph

- Computer Science, MathematicsArXiv
- 2011

The algorithm, apart from being deterministic, improves upon that time complexity for bipartite graphs when $m=O(n^r)$ and $r<1.876$ and is elementary, conceptually simple, and easy to implement.

Finding all maximally-matchable edges in a bipartite graph

- Computer Science, MathematicsTheor. Comput. Sci.
- 2012

The algorithm, apart from being deterministic, improves upon that time complexity for bipartite graphs when m=O(n^r) and r<1.876, and is elementary, conceptually simple, and easy to implement.

Speeding up Graph Algorithms via Switching Classes

- Mathematics, Computer ScienceIWOCA
- 2014

It is shown that switching classes can be used to asymptotically speed up several super-linear unweighted graph algorithms and achieve better bounds for diameter, transitive closure, bipartite maximum matching, and general maximum matching.

Efficient algorithms for maximum weight matchings in general graphs with small edge weights

- Computer Science, MathematicsSODA
- 2012

This work presents a simple iterative algorithm that uses a maximum cardinality matching algorithm as a subroutine to solve the maximum weight matching problem in O(W √nm logn(n2/m) time, or in O (Wnω) time with high probability.

Efficient algorithms for path problems in weighted graphs

- Mathematics
- 2008

Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the…

A New Combinatorial Approach for Sparse Graph Problems

- Mathematics, Computer ScienceICALP
- 2008

A new combinatorial data structure for representing arbitrary Boolean matrices that can perform fast vector multiplications with a given matrix, where the runtime depends on the sparsity of the input vector.

A scaling algorithm for maximum weight matching in bipartite graphs

- Computer Science, MathematicsSODA
- 2012

A new scaling algorithm is presented that runs in O(m√n log N) time, when the weights are integers within the range of [0,N], which improves the previous bounds of O(Nm √n) by Gabow and O(n√m log (nN) byGabow and Tarjan over 20 years ago.

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