Data Structures for Traveling Salesmen

Abstract

The choice of data structure for tour representation plays a critical role in the efficiency of local improvement heuristics for the traveling salesman problem. The tour data structure must permit queries about the relative order of cities in the current tour and must allow sections of the tour to be reversed. The traditional array-based representation of a tour permits the relative order of cities to be determined in small constant time, but requires worst-case Ω(N) time (where N is the number of cities) to implement a reversal, which renders it impractical for large instances. This paper considers alternative tour data structures, examining them from both a theoretical and experimental point of view. The first alternative we consider is a data structure based on splay trees, where all queries and updates take amortized time O(log N). We show that this is close to the best possible, because in the cell probe model of computation any data structure must take worst-case amortized time Ω(log N/log log N) per operation. Empirically (for random Euclidean instances), splay trees overcome their large constant-factor overhead and catch up to arrays by N = 10,000, pulling ahead by a factor of 4-10 (depending on machine) when N = 100,000. Two alternative tree-based data structures do even better in this range, however. Although both are asymptotically inferior to the splay tree representations, the latter does not appear to pull even with them until N ∼ to 1,000,000.