TY - JOUR
T1 - OPTIMALITY OF DISTRIBUTIVE CONSTRUCTIONS OF MINIMUM WEIGHT AND DEGREE RESTRICTED SPANNING TREES IN A COMPLETE NETWORK OF PROCESSORS.
AU - Korach, E.
AU - Moran, S.
AU - Zaks, S.
N1 - Funding Information:
Thanks are owed to Tsubasa Kohyama and two anonymous reviewers for evaluating this manuscript. We acknowledge valuable presubmission comments from David Salstein and are grateful to Richard Ray for supplying his own comprehensive L2 compilation that served as a reference data set throughout this work. Scott Woodruff and Steven Worley kindly allowed us to reprint some of the ICOADS Release 2.5 coverage maps. Financial support of this study was made available by the Austrian Science Fund (FWF) and the Deutsche Forschungsgemeinschaft (DFG) within project I1479-N29. We also thank Deutscher Wetterdienst, Offenbach, Germany, and appreciate the provision of meteorological data by the ECMWF, Reading, U.K., the National Centers of Environmental Information (NOAA), U.S., and the Research Data Archive (RDA) of NCAR, U.S. Data availability. The in situ tidal estimates and the gridded empirical L2 model for the annual mean and the three seasons can be downloaded at http://ggosatm.hg.tuwien.ac.at/L2.html.
PY - 1987/1/1
Y1 - 1987/1/1
N2 - In a previous paper we showed that the distributive construction of a spanning tree in a complete network of processors can be done in O(n log n) messages. We show in this work that if the spanning tree is required to satisfy certain properties, then the complexity of its construction increases: First we show that the construction of a minimum weight spanning tree requires, in the worst case, at least OMEGA (n**2) messages; then we show that the construction of a spanning tree where the maximum degree is at most k may require at least OMEGA (n**2/k) messages in the worst case. Actually, in both cases the lower bounds are shown for the number of edges used in the worst case. Moreover, the results are valid for both asynchronous and synchronous networks, and are independent of the lengths of the messages. On the other hand, there are algorithms for the above tasks which achieve these lower bounds, up to a constant factor, and use messages of O(log n) length.
AB - In a previous paper we showed that the distributive construction of a spanning tree in a complete network of processors can be done in O(n log n) messages. We show in this work that if the spanning tree is required to satisfy certain properties, then the complexity of its construction increases: First we show that the construction of a minimum weight spanning tree requires, in the worst case, at least OMEGA (n**2) messages; then we show that the construction of a spanning tree where the maximum degree is at most k may require at least OMEGA (n**2/k) messages in the worst case. Actually, in both cases the lower bounds are shown for the number of edges used in the worst case. Moreover, the results are valid for both asynchronous and synchronous networks, and are independent of the lengths of the messages. On the other hand, there are algorithms for the above tasks which achieve these lower bounds, up to a constant factor, and use messages of O(log n) length.
UR - http://www.scopus.com/inward/record.url?scp=0023330183&partnerID=8YFLogxK
U2 - 10.1137/0216019
DO - 10.1137/0216019
M3 - Article
AN - SCOPUS:0023330183
SN - 0097-5397
VL - 16
SP - 231
EP - 236
JO - SIAM Journal on Computing
JF - SIAM Journal on Computing
IS - 2
ER -