A parallel algorithm for computing minimum spanning trees

TLDR: A simple and implementable algorithm that computes a minimum spanning tree of an undirected weighted graph G = (V;E) of n = jV j vertices andm = jEj edges on an EREW PRAM in O(log3=2n) time using n+m processors is presented.

Abstract: We present a simple and implementable algorithm that computes a minimum spanning tree of an undirected weighted graph G = (V;E) of n = jV j vertices andm = jEj edges on an EREW PRAM in O(log3=2n) time using n+m processors. This represents a substantial improvement in the running time over the previous results for this problem using at the same time the weakest of the PRAM models. It also implies the existence of algorithms having the same complexity bounds for the EREW PRAM, for connectivity, ear decomposition, biconnectivity, strong orientation, st-numbering and Euler tours problems.

Full text

AParallel Algorithm for Computing Minimum
Spanning Trees
Donald B. Johnson

Panagiotis Metaxas
y
Dartmouth College
z
Wellesley College
x

Email address: djohnson@dartmouth.edu, telephone: (603) 646-3385
y
Part of this work was done while this author was with the Mathematics and Computer Science
department, Dartmouth College. Email address: pmetaxas@wellesley.edu, telephone: (617) 283-
3054
z
6211 Sudiko Lab oratory for Computer Science, Hanover, NH 03755
x
Department of Computer Science, Wellesley, MA 02181-8289
1

Running Head:
Parallel Minimum Spanning Tree Algorithm.
Keywords:
Parallel Algorithms, Graph Algorithms, Minimum Spanning Tree,
EREW PRAM model.
Corresp onding Address:
Panagiotis T. Metaxas, Department of Computer
Science, Wellesley College, Wellesley, MA 02181-8289.
2

Abstract
We present a simple and implementable algorithm that computes a min-
imum spanning tree of an undirected weighted graph
G
=(
V E
)of
n
=
j
V
j
vertices and
m
=
j
E
j
edges on an EREW PRAM in
O
(log
3
=
2
n
) time using
n
+
m
processors. This represents a substantial improvement in the running time over
the previous results for this problem using at the same time the weakest of the
PRAM models. It also implies the existence of algorithms having the same
complexity bounds for the EREW PRAM, for connectivity, ear decomp osition,
biconnectivity, strong orientation,
st
-numb ering and Euler tours problems.
3

List of Symbols
O
() Capital Oh, slanted (math)
O Capital Oh
0 zero
l ell
1 one
o
() Lowercase oh, slanted (math)
o lowercase oh

greek alpha
4

1 Intro duction
This pap er describes a new parallel algorithm for computing the minimum spanning
tree (MST) of a graph in the EREW PRAM model of parallel computation, the
weakest of the PRAM mo dels. This algorithm is faster by a factor of
q
log
j
V
j
than
any deterministic algorithm previously known for any model that do es not make
use of concurrent writing. The algorithm uses the growth-control scheduling of the
connectivity algorithm describ ed in JM91] it also makes use of an observation by
GGS89].
A ma jor innovation is our discovery that necessary information ab out comp o-
nents can be extracted without ever explicitly shrinking the comp onents. Comp onent
shrinking is a feature of every other parallel MST and connectivity algorithm known
to us.
Two of our ob jectives while designing the algorithm were simplicityand im-
plementability, that is, to be able to implement the algorithm using simple, well-
understoo d routines (like sorting and list ranking) that are likely to be found on most
parallel machines. We feel that wehave succeeded in b oth. In fact, the complexity
of our solution is in the proof | not in the algorithm itself.
Even though the connectivity algorithm of JM91] improved the running time of
several other graph-theoretic problems it seemed that there was no obvious wayto
create a MST algorithm from the connectivity algorithm having comparable complex-
ity with the latter. The diculty, of course, is that the selection of minimum weight
edges from edge-lists seems to require either a p owerful concurrent-write model of
computation or some other minimization process, which thereby takes time logarith-
mic in the length of the list. A connectivity algorithm mayselectany edge, not
the one with minimum weight, and that makes the selection simpler. Thus, a new
approachwas needed to achievean
o
(log
2
j
V
j
) running time for this problem. As
we will explain, we maintain a subset of edges that contains all the edges that must
be considered in any one phase of the algorithm in order to control the number of
candidates that must b e tested. Maintaining this subset is essential to the bound on
the running time.
Our results.
We present an algorithm that computes a minimum spanning tree
(MST) of an undirected weighted graph
G
=(
V E
)of
n
=
j
V
j
vertices and
m
=
j
E
j
edges on an EREW PRAM in
O
(log
3
=
2
n
) time using
n
+
m
processors. (If
G
is not
connected, our algorithm nds a minimum spanning tree for each connected com-
ponent.) This represents a substantial improvement in the running time over the
previous results for this problem using at the same time the weakest of the PRAM
models. It also implies the existence of a connectivity algorithm with the same com-
plexity b ounds for the EREW PRAM, therefore improving on previous work JM91].
Furthermore, we note that the numb er of pro cessors used can b e reduced by a factor
5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "A parallel algorithm for computing minimum spanning trees" ?

The authors present a simple and implementable algorithm that computes a min imum spanning tree of an undirected weighted graph G V E of n jV j vertices andm jEj edges on an EREW PRAM in O log n time using n m processors 

Q2. how to compute b list for each component?

To compute B list v for each v in parallel the authors may use a selection algorithm Col a Vis CY Using the algorithm by Cole Col a the authors can select the B st least weight element b in time O log n using almost n log n EREW PRAM processors 

Q3. how do you enumerate the edges of the B lists?

The E C list created by removing the edges unlabeled in the picture from the B lists Running list rank on E C the authors can enumerate and identify the components all but x that formed C Before starting a new sub phase B C is formed by including the B counterC s least weight outgoing useful edges 

Q4. how many edges are used for hooking?

Then the copy of the edge used for hooking by component Ci is assigned value counterCi s and the remaining edges are assigned value Using pointer jumping on ptr for dlogBe steps over the edges the authors can compact each B list if the new component contains up to B s edges 

Q5. What is the ect of the edge plugging of the B lists?

The authors note that any plugging that is prevented by this condition is deferred until the end of the phase so it is not lost Step of procedure phase will take care of thatUsing the plugged B lists the authors try to enumerate components of trees into counterr C s where r C is the root of the newly created component C 

Q6. how many processors can be used to derive a connectivity algorithm?

It also implies the existence of a connectivity algorithm with the same complexity bounds for the EREW PRAM therefore improving on previous work JM Furthermore the number of processors used can be reduced by a factor ofO p log n provided that there exists an practical integer sorting subroutine whichruns in O log n time using n p log n EREW PRAM processors 

Q7. how many parallel algorithms are reported in kr?

Other parallel algorithms are reported in KRS KR Ben SJ Recently CL have improved the running time of JM to O log n log log n mainly by providing a recursive version of the growth control schedule 

Q8. How is the merged edge list augmented?

Met to perform the augmentation of r s edge list in constant time and without memory access con icts Finally housekeeping is performed on the merged edge list to remove internal and multiple edges 

Q9. What is the shortest running time for the algorithm?

As the authors will explain the authors maintain a subset of edges that contains all the edges that must be considered in any one phase of the algorithm in order to control the number of candidates that must be tested Maintaining this subset is essential to the bound on the running time 

Q10. what is the known sequential algorithm for a weighted graph?

The authors note that among the problems having running times depending on the connectivity algorithm are ear decomposition MR biconnectivity TV strong orientation Vis st numbering MSV and Euler tours AV Computing the MST of a weighted graph has attracted much attention in both the sequential and parallel settings