Thomas H. Spencer

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log(i)n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log(m/n+2)n). Both algorithms can be extended to allow a degree constraint at one vertex.

Efficient implementation of graph algorithms using contraction

The (<italic>component</italic>) <italic>merging problem</italic> is a new graph problem. Versions of this problem appear as bottlenecks in various graph algorithms. A new data structure solves this problem efficiently, and two special cases of the problem have even more efficient solutions based on other data structures. The performance of the data structures is sped up by introducing a new algorithmic tool called <italic>packets</italic>. The algorithms that use these solutions to the component merging problem also exploit new properties of two existing data structures. Specifically, Β-trees can be used simultaneously as a priority queue and a concatenable queue. Similarly, F-heaps support some kinds of split operations with no loss of efficiency. An immediate application of the solution to the simplest version of the merging problem is an &Ogr;(<italic>t</italic>(<italic>m</italic>, <italic>n</italic>)) algorithm for finding minimum spanning trees in undirected graphs <italic>without</italic> using F-heaps, where <italic>t</italic>(<italic>m</italic>, <italic>n</italic>) = <italic>m</italic>log<subscrpt>2</subscrpt>log<subscrpt>2</subscrpt>log<subscrpt>d</subscrpt><italic>n</italic>, the graph has <italic>n</italic> vertices and <italic>m</italic> edges, and <italic>d</italic> = max(<italic>m</italic>/<italic>n</italic>, 2). Packets also improve the F-heap minimum spanning tree algorithm, giving the fastest algorithm currently known for this problem. The efficient solutions to the merging problem and the new observation about F-heaps lead to an &Ogr;(<italic>n</italic>(<italic>t</italic>(<italic>m</italic>, <italic>n</italic>) + <italic>n</italic>log<italic>n</italic>)) algorithm for finding a maximum weighted matching in general graphs. This settles an open problem posed by Tarjan [ 15, p. 123], where the weaker bound of <italic>O</italic>(<italic>nm</italic> log (<italic>n</italic><supscrpt>2</supscrpt>/<italic>m</italic>)) was conjectured.

Constructing a maximal independent set in parallel

The problem of constructing in parallel a maximal independent set of a given graph is considered. A new deterministic NC-algorithm implemented in the EREW PRAM model is presented. On graphs with n vertices and m edges, it uses

Separator based sparsification for dynamic planar graph algorithms

O((n + m)/\log n)

Efficient Implementation Of Graph Algorithms Using Contraction

processors and runs in

Separator Based Sparsification

O(\log^3 n)

Time-Work Tradeoffs of the Single-Source Shortest Paths Problem

time. This reduces by a factor of

A new parallel algorithm for the maximal independent set problem

\log n

Solving a bilevel linear program when the inner decision maker controls few variables

both the running time and the processor count of the previously fastest deterministic algorithm that solves the problem using a linear number of processors.

Separator-Based Sparsification II: Edge and Vertex Connectivity

We describe algorithms and data structures for maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We give a fully dynamic planarity testing algorithm that maintains a graph subject to edge insertions and deletions, and allows queries that test whether the graph is currently planar, or whether a potential new edge would violate planarity, in amortized time O(nl 12) per update or query. We maintain the 2and 3-vertex-connected components, and the 3and 4-edge-connected components of a planar graph in O(n.llz ) time per insertion, deletion or query. We give fully dynamic algorithms for maintaining the connected components, the 2-edge-connected components, and the minimum spanning forest of a planar graph in time (9(log n) per insertion and 0(log2 n) per deletion, assuming that insertions keep the graph planar. All our algorithms improve previous bounds: the improvements are based upon a new type of sparsification combined wit h several properties of separators in planar graphs.