An Implementation of a Backtracking Algorithm for the Turnpike Problem in Membranes
An Implementation of a Backtracking Algorithm for the Turnpike Problem in Membranes
Maria Cristina Albores, Richelle Ann Juayong, and Henry Adorna
Department of Computer Science (Algorithms and Complexity) Velasquez Ave., UP Diliman Quezon City 1101
∗
[maalbores, rbjuayong, hnadorna]@up.edu.ph ABSTRACT
The goal of the Turnpike Problem is to reconstruct those point sets that arise from a given distance multiset. Although the Turnpike Problem itself is of unknown complexity, variants of it have been proven to be NP-complete, and there are no existing polynomial algorithms for it. P systems with active membranes and P systems with membrane creation are parallel computing models based on the characteristics of living cells; both have been used to solve NPcomplete problems in polynomial time or better by trading time for an exponential workspace. In this paper we present a P system with active membranes and membrane creation that implements an O(2n n log n)-time backtracking algorithm for the Turnpike Problem in linear time. multiset; according to [9], when these point sets are unique (that is, none of them is a reflection of another), they are called homometric sets. TP first appeared in the 1930’s as a problem in X-ray crystallography, and reappeared in DNA sequencing as the Partial Digest Problem (PDP). The exact computational complexity of TP remains an open problem, although certain variants of it, as well as the decision problem of whether n points in R realize a multiset of n distances, have been proven to be NP-complete 2 in [9]. (Similarly, PDP’s own computational complexity is an open problem; variants of it are proven to be NP-hard or NP-complete in [2].) However, no polynomial-time algorithm has been found that solves TP. Among the algorithms that have been proposed is a polynomial factorization algorithm presented by Rosenblatt and Seymour in [8], and a backtracking algorithm presented by Skiena et al in [9]. The polynomial factorization algorithm
Maria Cristina Albores, Richelle Ann Juayong, and Henry Adorna
Department of Computer Science (Algorithms and Complexity) Velasquez Ave., UP Diliman Quezon City 1101
∗
[maalbores, rbjuayong, hnadorna]@up.edu.ph ABSTRACT
The goal of the Turnpike Problem is to reconstruct those point sets that arise from a given distance multiset. Although the Turnpike Problem itself is of unknown complexity, variants of it have been proven to be NP-complete, and there are no existing polynomial algorithms for it. P systems with active membranes and P systems with membrane creation are parallel computing models based on the characteristics of living cells; both have been used to solve NPcomplete problems in polynomial time or better by trading time for an exponential workspace. In this paper we present a P system with active membranes and membrane creation that implements an O(2n n log n)-time backtracking algorithm for the Turnpike Problem in linear time. multiset; according to [9], when these point sets are unique (that is, none of them is a reflection of another), they are called homometric sets. TP first appeared in the 1930’s as a problem in X-ray crystallography, and reappeared in DNA sequencing as the Partial Digest Problem (PDP). The exact computational complexity of TP remains an open problem, although certain variants of it, as well as the decision problem of whether n points in R realize a multiset of n distances, have been proven to be NP-complete 2 in [9]. (Similarly, PDP’s own computational complexity is an open problem; variants of it are proven to be NP-hard or NP-complete in [2].) However, no polynomial-time algorithm has been found that solves TP. Among the algorithms that have been proposed is a polynomial factorization algorithm presented by Rosenblatt and Seymour in [8], and a backtracking algorithm presented by Skiena et al in [9]. The polynomial factorization algorithm
References: [1] A. Atanasiu. Arithmetic with membranes. In Pre-proceedings of the Workshop on Multiset Processing (Curtea de Arges, Romania), pages 1–17, 2000. (see also CDMTCS Research Report No. 140, 2000, Auckland Univ., New Zealand, www.cs.auckland.ac.nz/CDMTCS). [2] M. Cieliebak. Algorithms and Hardness Results for DNA Physical Mapping, Protein Identification, and