Structure in Complexity Theory pp 219-233 | Cite as

# The power of the queue

## Abstract

Queues, stacks (pushdown stores), and tapes are storage models which have direct applications in compiler design and the general design of algorithms. Whereas stacks (pushdown store or last-in-first-out storage) have been thoroughly investigated and are well understood, this is much less the case for queues (first-in-first-out storage). In this paper we present a comprehensive study comparing queues to stacks and tapes. We address off-line machines with a one-way input. In particular, 1 queue and 1 tape (or stack) are not comparable:

(1) Simulating 1 stack (and hence 1 tape) by 1 queue requires Ω(*n*^{4/3}/log*n*) time in both the deterministic and the nondeterministic cases.

(2) Simulating 1 queue by 1 tape requires Ω(*n*^{2}) time in the deterministic case, and Ω(*n*^{4/3}/log*n*) in the nondeterministic case;

We further compare the relative power between different numbers of queues:

(3) Nondeterministically simulating 2 queues (or 2 tapes) by 1 queue requires Ω(*n*^{2}/(log^{2}*n* loglog*n*)) time and deterministically simulating 2 queues (or 2 tapes) by 1 queue requires Ω(*n*^{2}) time. The second bound is tight. The first is almost tight.

(4) We also obtain the simulation results for queues: 2 nondeterministic queues (or 3 pushdown stores) can simulate *k* queues in linear time. One queue can simulate *k* queues in quadratic time.

## Keywords

Turing Machine Input Tape Length Prefix Input Head Work Tape## Preview

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## References

- [Aa]Aanderaa, S.O., "On k-tape versus (k-1)-tape real-time computation," in
*Complexity of Computation*, ed. R.M. Karp, SIAM-AMS Proceedings, vol. 7, pp. 75–96, American Math. Society, Providence, R.I., 1974.Google Scholar - [BGW]Book, R., S. Greibach, and B. Wegbreit, "Time-and tape-bound Turing acceptors and AFL's,"
*J. Computer and System Sciences*, vol. 4, pp. 606–621, 1970.Google Scholar - [Ch]Chaitin, G.J., "Algorithmic Information Theory,"
*IBM J. Res. Dev.*, vol. 21, pp. 350–359, 1977.Google Scholar - [DGPR]Duris, P., Z. Galil, W. Paul, and R. Reischuk, "Two nonlinear lower bounds for on-line computations,"
*Information and Control*, vol. 60, pp. 1–11, 1984.Google Scholar - [GKS]Galil, Z., R. Kannan, E. Szemeredi, “On nontrivial separators for k-page graphs and simulations by non-deterministic one-tape Turing machines,” in Proceedings 18th Annual ACM Symposium on Theory of Computing, 1986.Google Scholar
- [HS1]Hartmanis, J. and R.E. Stearns, "On the computational complexity of algorithms,"
*Trans. Amer. Math. Soc.*, vol. 117, pp. 285–306, 1969.Google Scholar - [HS2]Hennie, F.C. and R.E. Stearns, "Two tape simulation of multitape Turing machines,"
*J. Ass. Comp. Mach.*, vol. 4, pp. 533–546, 1966.Google Scholar - [HU]Hopcroft, J.E. and J.D. Ullman,
*Formal Languages and their Relations to Automata*, Addison-Wesley, 1969.Google Scholar - [Kl]Klawe, M., "Limitations on explicit construction of expanding graphs,"
*SIAM J. Comp.*, vol. 13, no. 4, pp. 156–166, 1984.Google Scholar - [Ko]Kolmogorov, A.N., "Three approaches to the quantitative definition of information,"
*Problems in Information Transmission*, vol. 1, no. 1, pp. 1–7, 1965.Google Scholar - [Li1]Li, M., "Simulating two pushdowns by one tape in O(n**1.5 (log n)**0.5) time," 26th Annual IEEE Symposium on the Foundations of Computer Science, 1985.Google Scholar
- [Li2]Li, M., "Lower Bounds in Computational Complexity," Ph.D. Thesis, Report TR-85-663, Computer Science Department, Cornell University, march 1985.Google Scholar
- [Li3]Li, M., "Lower bounds by Kolmogorov-complexity", 12th ICALP, Lecture Notes in Computer Science, 194, pp. 383–393, 1985.Google Scholar
- [LV]Li, M. and P.M.B. Vitanyi, "Tape versus queue and stacks: The lower bounds," Submitted for publication.Google Scholar
- [LS]Leong, B.L. and J.I. Seiferas, "New real-time simulations of mul-tihead tape units,"
*J. Ass. Comp. Mach.*, vol. 28, pp. 166–180, 1981.Google Scholar - [Ma]Maass, W., "Combinatorial lower bound arguments for deterministic and nondeterministic Turing machines,"
*Trans. Amer. Math. Soc.*, 292,2, pp. 675–693, 1985. (Preliminary Version “Quadratic lower bounds for deterministic and nondeterminstic one-tape Turing machines,” pp 401–408 in Proceedings 16th ACM Symposium on Theory of Computing, 1984.)Google Scholar - [PSS]Paul, W.J., J.I. Seiferas, and J. Simon, "An information theoretic approach to time bounds for on-line computation,"
*J. Computer and System Sciences*, vol. 23, pp. 108–126, 1981.Google Scholar - [Pa]Paul, W.J., "On-line simulation of k+1 tapes by k tapes requires nonlinear time,"
*Information and Control*, pp. 1–8, 1982.Google Scholar - [So]Solomonov, R.,
*Information and Control*, vol. 7, pp. 1–22, 1964.CrossRefGoogle Scholar - [Vi1]Vitányi, P.M.B., "One queue or two pushdown stores take square time on a one-head tape unit," Computer Science Technical Report CS-R8406, CWI, Amsterdam, March 1984.Google Scholar
- [Vi2]Vitányi, P.M.B., "An N**1.618 lower bound on the time to simulate one queue or two pushdown stores by one tape,"
*Information Processing Letters*, vol. 21, pp. 147–152, 1985.Google Scholar - [Vi3]Vitányi, P.M.B., "On two-tape real-time computation and queues,"
*J. Computer and System Sciences*, vol. 29, pp. 303–311, 1984.Google Scholar