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 [Cho et al. (Fast order-preserving pattern matching, 2013), Kim et al. (Theor Comput Sci 525:68—79, 2014), Kubica et al. (Inform Process Lett 113:430–433, 2013)]. In this paper, we extend the representations of order relations by Kim et al. (Theor Comput Sci 525:68—79, 2014) to ternary order relations, and prove the equivalence of those representations. With our extensions, the time complexities of order-preserving matching in binary order relations can be achieved in ternary order relations as well.