Sparse language

In computational complexity theory, a sparse language is a formal language (a set of strings) such that the number of strings of length n in the language is bounded by a polynomial function of n. They are used primarily in the study of the relationship of the complexity class NP with other classes. The complexity class of all sparse languages is called SPARSE.

Sparse languages are called sparse because there are a total of 2ⁿ strings of length n, and if a language only contains polynomially many of these, then the proportion of strings of length n that it contains rapidly goes to zero as n grows. All unary languages are sparse. An example of a nontrivial sparse language is the set of binary strings containing exactly k 1 bits for some fixed k; for each n, there are only n choose k strings in the language, which is bounded by n^k.

Relationships to other complexity classes

SPARSE contains TALLY, the class of unary languages, since these have at most one string of any one length. If any sparse language is NP-complete, then P = NP, as shown by Mahaney in 1982.^[1] A simpler proof of this based on left-sets was given by Ogihara and Osamu in 1991.^[2] EXPTIME ≠ NEXPTIME if and only if there exist sparse languages in NP that are not in P.^[3] In 1999, Jin-Yi Cai and D. Sivakumar, building on work by Ogihara, showed that if there exists a sparse P-complete problem, then L = P.^[4]

References

^ S. R. Mahaney. Sparse complete sets for NP: Solution of a conjecture by Berman and Hartmanis. Journal of Computer and System Sciences 25:130-143. 1982.
^ M. Ogiwara and O. Watanabe. On polynomial time bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal on Computing volume 20, pp.471–483. 1991.
^ Juris Hartmanis, Neil Immerman, Vivian Sewelson. Sparse Sets in NP-P: EXPTIME versus NEXPTIME. Information and Control, volume 65, issue 2/3, pp.158–181. 1985. At ACM Digital Library
^ Jin-Yi Cai and D. Sivakumar. Sparse hard sets for P: resolution of a conjecture of Hartmanis. Journal of Computer and System Sciences, volume 58, issue 2, pp.280–296. 1999. ISSN:0022-0000. At Citeseer

Lance Fortnow. Favorite Theorems: Small Sets. April 18, 2006. http://weblog.fortnow.com/2006/04/favorite-theorems-small-sets.html
Bill Gasarch. Sparse Sets (Tribute to Mahaney). June 29, 2007. http://weblog.fortnow.com/2007/06/sparse-sets-tribute-to-mahaney.html
SPARSE at Complexity Zoo

[1] S. R. Mahaney. Sparse complete sets for NP: Solution of a conjecture by Berman and Hartmanis. Journal of Computer and System Sciences 25:130-143. 1982.

[2] M. Ogiwara and O. Watanabe. On polynomial time bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal on Computing volume 20, pp.471–483. 1991.

[3] Juris Hartmanis, Neil Immerman, Vivian Sewelson. Sparse Sets in NP-P: EXPTIME versus NEXPTIME. Information and Control, volume 65, issue 2/3, pp.158–181. 1985. At ACM Digital Library

[4] Jin-Yi Cai and D. Sivakumar. Sparse hard sets for P: resolution of a conjecture of Hartmanis. Journal of Computer and System Sciences, volume 58, issue 2, pp.280–296. 1999. ISSN:0022-0000. At Citeseer

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