Efficient algorithms for finding minimum spanning trees in undirected and directed graphs Harold N. Gabow Zvi Galil Thomas Spencer Robert E. Tarjan Article

Received: 23 January 1985 Revised: 01 December 1985 DOI :
10.1007/BF02579168

Cite this article as: Gabow, H.N., Galil, Z., Spencer, T. et al. Combinatorica (1986) 6: 109. doi:10.1007/BF02579168 Abstract 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.

AMS subject classification (1980) 68 B 15 68 C 05 Research supported in part by National Science Foundation Grant MCS-8302648.

Research supported in part by National Science Foundation Grant MCS-8303139.

Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.

References [1]

F. Bock , An algorithm to construct a minimum directed spanning tree in a directed network, in:Developments in Operations Research , Gordon and Breach, New York, 1971, 29–44.

[2]

O. Boruvka , O jistem problemu minimálni’mPráca Moravské Priŕodovědecké Spolěcnosti
3 (1926), 37–58. (In Czech).

[3]

P. M. Camerini, L. Fratta and

F. Maffioli , A note on finding optimum branchings,

Networks
9 (1979), 309–312.

MATH MathSciNet [4]

D. Cheriton and

R. E. Tarjan , Finding minimum spanning trees,

SIAM J. Comput.
5 (1976), 724–742.

MATH CrossRef MathSciNet [5]

Y. J. Chu and

T. H. Liu , On the shortest arborescence of a directed graph,

Sci. Sinica
14 (1965), 1396–1400.

MATH MathSciNet [6]

J. Edmonds , Optimum branchings,

J. Res. Nat. Bur. Standards
71B (1967), 233–240.

MathSciNet [7]

M. L. Fredman andR. E. Tarjan , Fibonacci heaps and their uses in network optimization algorithms,J. Assoc. Comput. Mach., to appear; also Proc.
25
th Annual IEEE Symp. on Found. of Comp. Sci. (1984), 338–346.

[8]

H. N. Gabow, Z. Galil andT. H. Spencer , Efficient implementation of graph algorithms using contraction,Proc.
25
th Annual IEEE Symp. on Found. of Comp. Sci. (1984), 347–357.

[9]

H. N. Gabow, Z. Galil andT. Spencer , Efficient implementation of graph algorithms using contraction,J. Assoc. Comput. Mach., submitted .

[10]

H. N. Gabow and

R. E. Tarjan , Efficient algorithms for a family of matroid intersection problems,

J. Algorithms
5 (1984), 80–131.

MATH CrossRef MathSciNet [11]

R. L. Graham and

P. Hell , On the history of the minimum spanning tree problem,

Ann. History of Computing
7 (1985), 43–57.

MATH MathSciNet CrossRef [12]

R. M. Karp , A simple derivation of Edmonds’ algorithm for optimum branchings,

Networks
1 (1971), 265–272.

MathSciNet [13]

J. Komlós , Linear verification for spanning trees,

Combinatorica
5 (1985), 57–65.

MATH MathSciNet [14]

R. E. Tarjan , Efficiency of a good but not linear set union algorithm,

J. Assoc. Comput. Mach.
22 (1975), 215–225.

MATH MathSciNet [15]

R. E. Tarjan , Finding optimum branchings,

Networks
7 (1977), 25–35.

MATH MathSciNet [16]

R. E. Tarjan , Applications of path compression on balanced trees,

J. Assoc. Comput. Mach.
26 (1979), 690–715.

MATH MathSciNet [17]

R. E. Tarjan ,Data Structures and Network Algorithms , CBMS 44, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983.

[18]

R. E. Tarjan and

J. Van Leeuwen , Worst-case analysis of set union algorithms,

J. Assoc. Comput. Mach.
31 (1984), 245–281.

MATH MathSciNet [19]

R. E. Tarjan , Amortized computational complexity,

SIAM J. Alg. Disc. Meth.
6 (1985), 306–318.

MATH MathSciNet [20]

A. Yao , An

O (|

E |log log|

V |) algorithm for finding minimum spanning trees,

Inform. Process. Lett.
4 (1975), 21–23.

MATH CrossRef Authors and Affiliations Harold N. Gabow Zvi Galil Thomas Spencer Robert E. Tarjan 1. University of Colorado Boulder USA 2. Columbia University New York USA 3. Tel Aviv University Tel Aviv Israel 4. Rensselaer Polytechnic Inst. Troy USA 5. AT&T Bell Laboratories Murray Hill USA