Detecting Regular Visit Patterns

Authors:
Bojan Djordjevic;Joachim Gudmundsson;Anh Pham;Thomas Wolle
Affiliations:
School of Information Technologies, University of Sydney, Australia NSW 2006 and NICTA, Sydney, Australia NSW 1435;NICTA, Sydney, Australia NSW 1435;School of Information Technologies, University of Sydney, Australia NSW 2006 and NICTA, Sydney, Australia NSW 1435;NICTA, Sydney, Australia NSW 1435
Venue:
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Year:
2008

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Abstract

We are given a trajectory $\mathcal{T}$ and an area $\mathcal{A}$. $\mathcal{T}$ might intersect $\mathcal{A}$ several times, and our aim is to detect whether $\mathcal{T}$ visits $\mathcal{A}$ 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 mostof the days (weeks or years), but not necessarily on everyday (week or year).During the modelling of such applications, we encounter an elementary problem on bitstrings, that we call LDS (LongestDenseSubstring). The bits of the bitstring correspond to a sequence of regular time points, in which a bit is set to 1 iff the trajectory $\mathcal{T}$ intersects the area $\mathcal{A}$ at the corresponding time point. For the LDS problem, we are given a string sas input and want to output a longest substring of s, such that the ratio of 1's in the substring is at least a certain threshold.In our model, LDS is a core problem for many applications that aim at detecting regularity of $\mathcal{T}$ intersecting $\mathcal{A}$. We propose an optimal algorithm to solve LDS, and also for related problems that are closer to applications, we provide efficient algorithms for detecting regularity.