A Simple Approach to the Reconstruction of a Set of Points from the Multiset of n² Pairwise Distances in n² Steps for the Sequencing Problem: II. Algorithm

Abstract

A new uniform algorithm based on sequential removal of redundancy from inputs is proposed to solve the turnpike and beltway problems. For error-free inputs that simulate experimental data with high accuracy, the size of inputs decreases from [Formula: see text] to [Formula: see text], which permits one to eliminate exhaustive search almost completely and reconstruct sequences in [Formula: see text] steps. Computational experiments show high efficiency of the algorithm for both the turnpike and beltway cases, with the reconstruction time for sequences of lengths up to several thousand elements being within 1 second on a modern PC.

Reconstructing Point Sets from Distance Distributions

“…It is also impossible for the backtracking approach to rely on the remaining largest unassigned distance to find the point locations progressively [6]. For small problems, Fomin [35,36] proposed to avoid an exhaustive search in the noiseless case by further removing the redundant distances from H sequentially, and later extended it to handle noisy measurements [37]. To the best of our knowledge, our work in this paper offers an alternative by providing the first practical approach to solve the large-scale noisy beltway problem.…”

Speeding Up the Partial Digest Algorithm

“…Different algorithms are proposed to find the exact solution of PD problem [1, 2, 4-7, 9, 11-13, 15, 18]. Some of these algorithms are impractical algorithms, while the other algorithms are practical such as [1,9,18]. The first practical algorithm was designed by Skiena, Smith and Lemke [18] and ran in n 2 log n in the average case.…”

“…In the worst case (Zhang instances [19]), the algorithm was run in order of 2 n−1 time. Also, Fomin [9] introduced the second practical algorithm for PD. In the case of Zhang instances, Skiena, Smith and Lemke algorithm is faster than Formin's algorithm, while in some other case Formin's algorithm is faster than the Skiena, Smith and Lemke algorithm.…”