24 research outputs found
Reconstruction of Integers from Pairwise Distances
Given a set of integers, one can easily construct the set of their pairwise
distances. We consider the inverse problem: given a set of pairwise distances,
find the integer set which realizes the pairwise distance set. This problem
arises in a lot of fields in engineering and applied physics, and has
confounded researchers for over 60 years. It is one of the few fundamental
problems that are neither known to be NP-hard nor solvable by polynomial-time
algorithms. Whether unique recovery is possible also remains an open question.
In many practical applications where this problem occurs, the integer set is
naturally sparse (i.e., the integers are sufficiently spaced), a property which
has not been explored. In this work, we exploit the sparse nature of the
integer set and develop a polynomial-time algorithm which provably recovers the
set of integers (up to linear shift and reversal) from the set of their
pairwise distances with arbitrarily high probability if the sparsity is
O(n^{1/2-\eps}). Numerical simulations verify the effectiveness of the
proposed algorithm.Comment: 14 pages, 4 figures, submitted to ICASSP 201
Approximation algorithms for the shortest common superstring problem
AbstractThe object of the shortest common superstring problem (SCS) is to find the shortest possible string that contains every string in a given set as substrings. As the problem is NP-complete, approximation algorithms are of interest. The value of an aproximate solution to SCS is normally taken to be its length, and we seek algorithms that make the length as small as possible. A different measure is given by the sum of the overlaps between consecutive strings in a candidate solution. When considering this measure, the object is to find solutions that make it as large as possible. These two measures offer different ways of viewing the problem. While the two viewpoints are equivalent with respect to optimal solutions, they differ with respect to approximate solutions. We describe several approximation algorithms that produce solutions that are always within a factor of two of optimum with respect to the overlap measure. We also describe an efficient implementation of one of these, using McCreight's compact suffix tree construction algorithm. The worstcase running time is O(m log n) for small alphabets, where m is the sum of the lengths of all the strings in the set and n is the number of strings. For large alphabets, the algorithm can be implemented in O(m log m) time by using Sleator and Tarjan's lexicographic splay tree data structure