Order-preserving matching

Abstract

We introduce a new string matching problem called order-preserving matching on numeric strings where a pattern matches a text if the text contains a substring whose relative orders coincide with those of the pattern. Order-preserving matching is applicable to many scenarios such as stock price analysis and musical melody matching in which the order relations should be matched instead of the strings themselves. Solving order-preserving matching has to do with representations of order relations of a numeric string. We define prefix representation and nearest neighbor representation, which lead to efficient algorithms for order-preserving matching. We present efficient algorithms for single and multiple pattern cases. For the single pattern case, we give an O(n log m) time algorithm and optimize it further to obtain O(n + m log m) time. For the multiple pattern case, we give an O(n log m) time algorithm.

A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

“…If the pattern is required to be matched to consecutive elements in the text, a O(n + k) algorithm has been found [28]. A similar result has been found independently in [24]. This work has been extended to the cases where some mismatches are tolerated [19]; also suffix trees have recently been generalized to be applicable in this setting [12].…”

On Representations of Ternary Order Relations in Numeric Strings

“…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. With the extended prefix representation, order-preserving matching can be done in O(n log m) time, and the representation of order relations takes (log m + 1)-bit per character.…”

“…It has many practical applications such as stock price analysis and musical melody matching. The study on this field was introduced by Kubica et al [7] and Kim et al [5] where Kubica et al [7] defined order relations by order isomorphism of two strings, while Kim et al [5] defined them explicitly by the sequence Copyright c by the paper's authors. Copying permitted only for private and academic purposes.…”

“…Kim et al [5] considered only binary order relations by assuming that all the characters in a string are distinct, while Kubica et al [7] considered ternary order relations but their match condition on equal characters was faulty and it was fixed later by Cho et al [1]. As there was no integrated study on the relationship between the representations of order relations, previous works [1,2,3] individually handled the cumbersome details of ternary order relations on their own works.…”

“…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…”