Two Algorithms for Deciding Coincidence In Double Temporal Recurrence of Eventuality Sequences

Abstract

Two disjoint sequences of eventualities x and y both recur over the same time interval and each of the eventualities (xp from x say) have a fixed duration when they occur and it is required to determine whether or not a subinterval exists within the interval of double recurrence, for any intervals of occurrence of eventualities xp from x and yq from y. This paper presents two algorithms for solving the problem. One based on the existing idea of temporal projection over a cycle of double recurrence with a worst case running time of Ο(duration(x)*duration(y)) and the other based on the notion of gcd-partitions runs in linear time, i.e. Ο(max(duration(x), duration(y)). The gcd-partition of a double recurrence of a pair of eventuality sequence of (x, y) is another eventuality sequence pair (w, z) such that each a double recurrence of the pair (x, y) holds exactly when the double recurrence of (w, z) also holds, the duration of each eventuality in w and z is the greatest common divisor of the durations of x and y. A key property of gcd partitions is that for any wr and zs in w and z respectively, an interval exists within any cycle of the recurrence of x and y, over which both wr and zs occur. The algorithm then explores the eventualities from w that are naturally non-disjoint with xp and those from z that are naturally non-disjoint with yq in order to determine the coincidence of xp and yq.