TY - GEN

T1 - Computing the discrete Fréchet distance with imprecise input

AU - Ahn, Hee Kap

AU - Knauer, Christian

AU - Scherfenberg, Marc

AU - Schlipf, Lena

AU - Vigneron, Antoine

N1 - Funding Information:
Work by Ahn was supported by the Korea Research Foundation Grant funded by the Korean Government(KRF-2008-614-D00008). Work by Knauer and Scher-fenberg was supported by the German Science Foundation (DFG) under grant Al 253/5-3. Work by Schlipf was supported by the Deutsche Forschungsgemein-schaft within the research training group ’Methods for Discrete Structures’(GRK 1408).

PY - 2010

Y1 - 2010

N2 - We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2)m2n 2 log2(mn) the Fréchet distance lower bound between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 2(mn) + (m2 + n2)log(mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log(dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the Fréchet distance upper bound, as well as the smallest possible Fréchet distance lower/upper bound that can be achieved between two imprecise point sequences when one is allowed to translate them. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

AB - We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2)m2n 2 log2(mn) the Fréchet distance lower bound between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 2(mn) + (m2 + n2)log(mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log(dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the Fréchet distance upper bound, as well as the smallest possible Fréchet distance lower/upper bound that can be achieved between two imprecise point sequences when one is allowed to translate them. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

UR - http://www.scopus.com/inward/record.url?scp=78650861075&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-17514-5_36

DO - 10.1007/978-3-642-17514-5_36

M3 - Conference contribution

AN - SCOPUS:78650861075

SN - 3642175163

SN - 9783642175169

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 422

EP - 433

BT - Algorithms and Computation - 21st International Symposium, ISAAC 2010, Proceedings

T2 - 21st Annual International Symposium on Algorithms and Computations, ISAAC 2010

Y2 - 15 December 2010 through 17 December 2010

ER -