Detecting Areas Visited Regularly

We are given a trajectory T and an area A. T might intersect A several times, and our aim is to detect whether T visits A with some regularity, e.g. what is the longest time span that a GPS-GSM equipped elephant visited a specific lake on a daily (weekly or yearly) basis, where the elephant has to visit the lake most of the days (weeks or years), but not necessarily on every day (week or year). We call this a regular pattern with period of one day (week or year, respectively). We consider the most general version of the problem defined in [8], the case where we are not given the period length of the regular pattern but have to find the longest regular pattern over all possible period lengths. We give an exact algorithm with O(n log n) running time and an approximate algorithm with O( 1 ε n log n) running time. We also consider the problem of finding a region that is visited regularly if one is not given. We give exact and approximate algorithms for this problem when the period length is fixed.