Assignment of Starting Offsets for Periodic Activities with Occupation Time Unit 1 and a Restricted Sufficient Condition for Some Practical Cases

We handle a problem to assign the starting offsets of a collision-free schedule in a certain form called a well-formed gcd notation. Each activity in an activity set recurs with its own period. If more than one activity occurs at the same time slot, it is a collision. If a schedule can be constructed as a well-formed gcd notation, the schedule is known to be collision-free. In this paper, we introduce a well-formed gcd notation and present an algorithm to assign the starting offsets of a collision-free schedule. We also propose a restricted form of a sufficient condition for detecting a collision-free periodic schedule, which can be decided in a tractable time constraint.