#### Title

Optimal Algorithms for the Single and Multiple Vertex Updating Problems of a Minimum Spanning Tree

#### Department

Computer Science

#### Document Type

Article

#### Publication Date

1996

#### Abstract

The vertex updating problem for a minimum spanning tree (MST) is defined as follows: Given a graph *G* = (*V, E*_{G}) and an MST *T* for *G*, find a new MST for *G* to which a new vertex *z* has been added along with weighted edges that connect *z* with the vertices of *G*. We present a set of rules that produce simple optimal parallel algorithms that run in *O*(lg *n*) time using *n*/lg *n* EREW PRAM processors, where *n* = |V|. These algorithms employ any valid tree-contraction schedule that can be produced within the stated resource bounds. These rules can also be used to derive simple linear-time sequential algorithms for the same problem. The previously best-known parallel result was a rather complicated algorithm that used *n* processors in the more powerful CREW PRAM model. Furthermore, we show how our solution can be used to solve the multiple vertex updating problem: Update a given MST when *k* new vertices are introduced simultaneously. This problem is solved in *O*(lg *k *· lg *n*) parallel time using (*k* · *n*)/ (lg *k* · lg *n*) EREW PRAM processors. This is optimal for graphs having Ω (*kn*) edges.

#### Recommended Citation

Optimal Algorithms for the Single and Multiple Vertex Updating Problems of a Minimum Spanning Tree, with D.B. Johnson. Algorithmica, 16 (6):633-648 (1996).