On Representations of Ternary Order Relations in Numeric Strings

Abstract

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 su ciently 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.

Fast Multiple Order-Preserving Matching Algorithms

“…The time complexity can be reduced to O(|x|) if the characters can be sorted in O(|x|) time. [10] for strings with duplicate characters. We omit the details here.…”

“…Using this representation, we can check if two strings match in time linear to the size of the input, even when the strings have duplicate characters. 10,11,13] Given two strings x and y where |x| = |y|, assume NN(x) is computed. Then we can determine whether x matches y in O(|x|) time.…”

Fast Cartesian Tree Matching

“…These problems are characterized by the way of defining a match, which depends on the application domains of the problems. In particular, order-preserving matching [18,17,20] and Cartesian tree matching [21] deal with the order relations between numbers.…”

Transitivity properties of ternary fuzzy relations

“…In applications, (fuzzy) ternary relations can be found in many different areas, for instance in social sciences (e.g., philosophy [3]), in biology (e.g., modelling of phylogenies [28]), in computer science (e.g., the Resource Description Framework (RDF) [27]). Some classes of ternary relations came to play an important role in specific applications, e.g., betweenness relations in models for decision making [25] and aggregation [24], ternary order relation in string matching [16], cyclic orders in qualitative spatial reasoning [15] and particular fuzzy ternary relations in models of choice behavior [22].…”