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Depth-first search is inherently sequential. (English) Zbl 0572.68051
This paper concerns the computational complexity of depth-first search. Suppose we are given a rooted graph G with fixed adjacency lists and vertices u,v. We wish to test if u is first visited before v in depth- first search order of G. We show that this problem, for undirected and directed graphs, is complete in deterministic polynomial time with respect to deterministic log-space reductions. This gives strong evidence that depth-first search ordering can be done neither in deterministic space (log n)\({}^ c\) nor in parallel time (log n)\({}^ c\), for any constant \(c>0\).

MSC:
68R10 Graph theory (including graph drawing) in computer science
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[1] Aleliunas, R.; Karp, R.M.; Lipton, R.H.; Lovasz, L.; Rackoff, C., Random walks, universal traversal sequences, and complexity of maze problems, (), 218-223
[2] Chin, F.; Lam, J.; Chen, I., Optimal parallel algorithms for the connected components problem, (1981), Foundations of Computer Science (FOCS)
[3] Cook, S.A., An observation on time-storage trade off, J. comput. system sci., 9, 3, 308-316, (1974) · Zbl 0306.68026
[4] Cook, S.A., Towards a complexity theory of synchronous parallel computation, () · Zbl 0484.68036
[5] Dobkin, D.; Lipton, R.J.; Reiss, S., Linear programming is log-space hard for P, Inform. process. lett., 9, 2, 96-97, (1979) · Zbl 0402.68042
[6] Dymond, P.W., Speed-up of multi-take Turing machines by synchronous parallel machines, Tech. Rept., Dept. of EE and Computer Science, Univ. of California, San Diego, CA
[7] Dymond, P.W.; Tompa, M., Speed-ups of deterministic machines by synchronous parallel machines, (), 336-346
[8] Even, S.; Tarjan, R.E., Network flow and testing graph connectivity, J. SIAM comput., 4, 4, 507-512, (1975) · Zbl 0328.90031
[9] Fortune, S.; Wyllie, J.C., Parallelism in random access machines, (), 114-118 · Zbl 1282.68104
[10] Goldschlager, L., A unified approach to models of synchronous parallel machines, (), 89-94 · Zbl 1282.68105
[11] Goldschlager, L.M.; Shaw, R.A.; Staples, J., The maximum flow problem is log-space complete for P, Theoret. comput. sci., 21, 105-111, (1982) · Zbl 0486.68035
[12] Hopcroft, J.E.; Karp, R.M., An \(n\^{}\{52\}\) algorithm for maximum matching in bipartite graphs, J. SIAM comput., 2, 225-231, (1973) · Zbl 0266.05114
[13] Hopcroft, J.E.; Tarjan, R.E., Efficient planarity testing, J. ACM, 21, 549-568, (1974) · Zbl 0307.68025
[14] Hopcroft, J.E.; Tarjan, R.E., Dividing a graph into triconnected components, SIAM J. comput., 2, 3, (1973) · Zbl 0281.05111
[15] Hopcroft, J.E.; Tarjan, R.E., Efficient algorithms for graph manipulation, Comm. ACM, 16, 6, 372-378, (1973) · Zbl 0281.05111
[16] Hopcroft, J.E.; Ullman, J.D., Introduction to automata theory, languages and computation, (1979), Addison-Wesley Reading, MA · Zbl 0196.01701
[17] Ja’ja’, J., Graph connectivity problems on parallel computers, ()
[18] Ja’ja’, J.; Simon, J., Parallel algorithms in graph theory: planarity testing, SIAM J. comput., 11, 2, 372-378, (1982)
[19] Jones, N.D.; Laaser, W.T., Complete problems for deterministic polynomial time, Theoret. comput. sci., 3, 1, 105-117, (1976) · Zbl 0352.68068
[20] Ladner, R.E., The circuit value problem is log-space complete for P, SIGACT news, 7, 1, 18-20, (1975)
[21] Miklail, A.J.; Kosaraju, S.R., Graph problems on a mesh-connected processor array, (), 345-353
[22] Reif, J.H., Symmetric complementation, J. ACM, 31, 2, 401-421, (1984) · Zbl 0632.68062
[23] Reif, J.H., On the power of probabilistic choice in synchronous parallel computations, SIAM J. comput., 13, 1, 46-55, (1984)
[24] Savage, C.; Ja’ja’, J., Fast, efficient parallel algorithms for some graph problems, SIAM J. comput., 10, 4, 682-691, (1981) · Zbl 0476.68036
[25] Tarjan, R.E., Depth-first search and linear graph algorithms, SIAM J. comput., 1, 2, 146-160, (1972) · Zbl 0251.05107
[26] Wyllie, J.C., The complexity of parallel computations, ()
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