On representations of ternary order relations in numeric strings

Order-preserving matching is a string matching problem of two numeric strings where the relative orders of consecutive substrings are matched instead of the characters themselves. The order relation between two characters is a ternary relation (>,<,=) rather than a binary relation (>,<), but it was not sufficiently studied in previous works [5, 7, 1]. In this paper, we extend the representations of order relations by Kim et al. [5] to ternary order relations, and prove the equivalence of those representations. The extended prefix representation takes log m + 1 bits per character, while the nearest neighbor representation takes 2 log m bits per character. With our extensions, the time complexities of order-preserving matching in binary order relations can be achieved in ternary order relations as well.