By Gene Myers Ph.D (auth.), Oscar H. Ibarra, Louxin Zhang (eds.)

ISBN-10: 354043996X

ISBN-13: 9783540439967

ISBN-10: 3540456554

ISBN-13: 9783540456551

This ebook constitutes the refereed complaints of the eighth Annual overseas Computing and Combinatorics convention, COCOON 2002, held in Singapore in August 2002.

The 60 revised complete papers awarded including 3 invited contributions have been rigorously reviewed and chosen from 106 submissions. The papers are geared up in topical sections on complexity thought, discrete algorithms, computational biology and studying thought, radio networks, automata and formal languages, web networks, computational geometry, combinatorial optimization, and quantum computing.

**Read Online or Download Computing and Combinatorics: 8th Annual International Conference, COCOON 2002 Singapore, August 15–17, 2002 Proceedings PDF**

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**Extra resources for Computing and Combinatorics: 8th Annual International Conference, COCOON 2002 Singapore, August 15–17, 2002 Proceedings**

**Example text**

I) ΣkZPP Σl = Σk+l , ii) BP · ΣkZPP = BP · Σk+l . 22 Jin-Yi Cai et al. Next, we consider a hierarchy with interleaving levels of BP · Σl and Σk . The upper bound in this theorem is an improvement of n levels over the trivial bounds. Theorem 2. 1. For k1 , . . , kn ≥ 1 and l1 , · · · ln ≥ 0, where n ≥ 1 ·· BP·Σl· 1 k1 Σ BP·Σl n kn ⊆ ZPPΣk1 +k2 +···+kn +l1 +l2 +···+ln . Σ 2. For k1 ≥ 2, k2 , . . , kn ≥ 1, and l1 , · · · ln ≥ 0, where n ≥ 1 ·· BP·Σl· 1 k1 Σ BP·Σl n kn Σ = Σk1 +k2 +···+kn +l1 +l2 +···+ln .

In this work, we further explore the worst case complexity boundary of P and N PC when p is further reduced (not a constant but a function of n). Somewhat surprisingly, such an extension allows us to suggest another candidate for natural problems in N PI under N P = P. In fact, we present a natural problem in N PI under ET H . In Section 2, we present the necessary deﬁnitions and the related important properties for our study. In Section 3, we present a candidate for natural problems in N PI and prove it not in N PC under N P = P.

S. Goldwasser and M. Sipser, Private coins versus public coins in interactive proof systems, STOC 18:59–68(1986). 14. Zuckerman, Another Proof that BPP ⊆ PH (and more), ECCC, TR97-045, October 1997. 15. J. K¨ obler and O. Watanabe, New collapse consequences of NP having small circuits ICALP, LNCS 944:196–207(1995). 16. C. Lund, L. Fortnow, H. Karloﬀ and N. Nisan, Algebraic Methods for Interactive Proof Systems, Journal of the ACM, 39(4):859-868, October 1992. 17. A. Shamir, IP = PSPACE, Journal of the ACM, 39(4):869-877, October 1992.

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