Probabilistic Analysis of Graph Algorithms

Probabilistic Analysis of Graph Algorithms. We review some of the known results on the average case performance of graph algorithms. The analysis assumes that the problem instances are randomly selected from some reasonable distribution of problems. We consider two types of problem. The first sort is polynomially solvable in the worst case but there are algorithms with better average case performance. In particular we consider the all-pairs shortest path problem, the minimum spanning tree problem, the assignment problem and the cardinality matching problem in sparse graphs. Our second category of problems consists of problems which seem hard in the worst-case but still have algorithms with good average case performance. In particular we consider three NP-Complete problems; the Hamilton cycle problem, the graph bisection problem and graph colouring. In addition we consider the graph isomorphism problem whose exact complexity is still undetermined.

[1]  Béla Bollobás,et al.  Random Graphs , 1985 .

[2]  Vasek Chvátal,et al.  Determining the Stability Number of a Graph , 1976, SIAM J. Comput..

[3]  Alan M. Frieze,et al.  Probabilistic Analysis of a Relaxation for the k-Median Problem , 1986, Math. Oper. Res..

[4]  János Komlós,et al.  Limit distribution for the existence of hamiltonian cycles in a random graph , 1983, Discret. Math..

[5]  G. Grimmett,et al.  On colouring random graphs , 1975 .

[6]  Jonathan S. Turner,et al.  Almost All k-Colorable Graphs are Easy to Color , 1988, J. Algorithms.

[7]  Alan M. Frieze,et al.  On the value of a random minimum spanning tree problem , 1985, Discret. Appl. Math..

[8]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[9]  Eli Upfal,et al.  Sequential and Distributed Graph Coloring Algorithms with Performance Analysis in Random Graph Spaces , 1984, J. Algorithms.

[10]  Béla Bollobás,et al.  The chromatic number of random graphs , 1988, Comb..

[11]  Alan M. Frieze.  An Algorithm for Finding Hamilton Cycles in Random Directed Graphs , 1988, J. Algorithms.

[12]  Colin McDiarmid,et al.  Determining the Chromatic Number of a Graph , 1979, SIAM J. Comput..

[13]  Silvio Micali,et al.  An O(v|v| c |E|) algoithm for finding maximum matching in general graphs , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[14]  Richard M. Karp,et al.  An algorithm to solve the m × n assignment problem in expected time O(mn log n) , 1980, Networks.

[15]  J. Steele.  Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability , 1981 .

[16]  T.I. Fenner,et al.  On the existence of Hamiltonian cycles in a class of random graphs , 1983, Discret. Math..

[17]  Robert E. Tarjan,et al.  Linear Expected-Time Algorithms for Connectivity Problems , 1980, J. Algorithms.

[18]  Frank Thomson Leighton,et al.  A Framework for Solving VLSI Graph Layout Problems , 1983, J. Comput. Syst. Sci..

[19]  Prabhakar Raghavan,et al.  Parallel graph algorithms that are efficient on average , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[20]  Alistair Moffat,et al.  An All Pairs Shortest Path Algorithm with Expected Time O(n² log n) , 1987, SIAM J. Comput..

[21]  P. Erdos,et al.  On the existence of a factor of degree one of a connected random graph , 1966 .

[22]  Ravi B. Boppana,et al.  Eigenvalues and graph bisection: An average-case analysis , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[23]  Alistair Moffat,et al.  An all pairs shortest path algorithm with expected running time O(n 2logn) , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[24]  Robert E. Tarjan,et al.  Self-Adjusting Heaps , 1986, SIAM J. Comput..

[25]  Robert E. Tarjan,et al.  Finding Minimum Spanning Trees , 1976, SIAM J. Comput..

[26]  P. Erdos,et al.  On the strength of connectedness of a random graph , 1964 .

[27]  Richard M. Karp,et al.  Probabilistic Analysis of Partitioning Algorithms for the Traveling-Salesman Problem in the Plane , 1977, Math. Oper. Res..

[28]  C. McDiarmid,et al.  On random minimum length spanning trees , 1989 .

[29]  Frank Thomson Leighton,et al.  Graph bisection algorithms with good average case behavior , 1984, Comb..

[30]  J. Michael Steele,et al.  On Frieze's χ(3) limit for lengths of minimal spanning trees , 1987, Discret. Appl. Math..

[31]  Alan M. Frieze,et al.  The shortest-path problem for graphs with random arc-lengths , 1985, Discret. Appl. Math..

[32]  Jean Vuillemin,et al.  A data structure for manipulating priority queues , 1978, CACM.

[33]  N. Tomizawa,et al.  On some techniques useful for solution of transportation network problems , 1971, Networks.

[34]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[35]  Paul Erdös,et al.  Random Graph Isomorphism , 1980, SIAM J. Comput..

[36]  Martin E. Dyer,et al.  On linear programs with random costs , 1986, Math. Program..

[37]  Richard M. Karp,et al.  Maximum Matchings in Sparse Random Graphs , 1981, FOCS 1981.

[38]  Zvi Galil,et al.  Efficient implementation of graph algorithms using contraction , 1984, JACM.

[39]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[40]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[41]  David W. Walkup,et al.  On the Expected Value of a Random Assignment Problem , 1979, SIAM J. Comput..

[42]  Alan M. Frieze,et al.  Finding hamilton cycles in sparse random graphs , 1987, J. Comb. Theory, Ser. B.

[43]  B. Bollobás,et al.  An algorithm for finding hamilton paths and cycles in random graphs , 1987 .

[44]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[45]  Alan M. Frieze.  Parallel Algorithms for Finding Hamilton Cycles in Random Graphs , 1987, Inf. Process. Lett..

[46]  Philip M. Spira,et al.  A New Algorithm for Finding all Shortest Paths in a Graph of Positive Arcs in Average Time 0(n2 log2n) , 1973, SIAM J. Comput..

[47]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[48]  R. Karp.  An Upper Bound on the Expected Cost of an Optimal Assignment , 1987 .

[49]  Martin E. Dyer,et al.  The Solution of Some Random NP-Hard Problems in Polynomial Expected Time , 1989, J. Algorithms.

[50]  G. Dantzig.  On the Shortest Route Through a Network , 1960 .

[51]  László Babai,et al.  Canonical labelling of graphs in linear average time , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[52]  János Komlós,et al.  The eigenvalues of random symmetric matrices , 1981, Comb..

[53]  Alan M. Frieze,et al.  Probabilistic Analysis of a Parallel Algorithm for Finding Maximal Independent Sets , 1990, Random Struct. Algorithms.

[54]  Alan M. Frieze.  On the Exact Solution of Random Travelling Salesman Problems with Medium Size Integer Coefficients , 1987, SIAM J. Comput..

[55]  B. Bollobás,et al.  Cliques in random graphs , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[56]  T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes , 1970, Journal of Applied Probability.

[57]  Colin McDiarmid.  On some conditioning results in the probabilistic analysis of algorithms , 1985, Discret. Appl. Math..