TY - GEN

T1 - Bi-criteria linear-time approximations for generalized k-mean/median/center

AU - Feldman, Dan

AU - Fiat, Amos

AU - Sharir, Micha

AU - Segev, Danny

PY - 2007

Y1 - 2007

N2 - We consider the problem of approximating a set P of n points in R d by a collection of j-dimensional flats, andextensions thereof, under the standard median / mean / centermeasures, in which we wish to minimize, respectively, the sum of thedistances from each point of P to its nearest flat, the sum of thesquares of these distances, or the maximal such distance.Such problems cannot be approximated unless P=NP but do allowbi-criteria approximations where one allows some leeway in both the numberof flats and the quality of the objective function.We give a very simple bi-criteria approximation algorithm, which producesat most α(k,j,n) = (k j log n) O(j) flats, which exceeds the optimalobjective value for any k j-dimensional flats by a factor of nomore than Β(j)= 2O(j). Given this bi-criteria approximation, wecan use it to reduce the approximation factor arbitrarily, at the costof increasing the number of flats. Our algorithm hasmany advantages over previous work, in that it is muchmore widely applicable (wider set of objective functions and classes ofclusters) and much more efficient - reducing the running time bound from O(nPoly(k,j)) to nd (jk)O(j).Our algorithm is randomized and successful with probability 1/2(easily boosted to probabilities arbitrary close to 1).

AB - We consider the problem of approximating a set P of n points in R d by a collection of j-dimensional flats, andextensions thereof, under the standard median / mean / centermeasures, in which we wish to minimize, respectively, the sum of thedistances from each point of P to its nearest flat, the sum of thesquares of these distances, or the maximal such distance.Such problems cannot be approximated unless P=NP but do allowbi-criteria approximations where one allows some leeway in both the numberof flats and the quality of the objective function.We give a very simple bi-criteria approximation algorithm, which producesat most α(k,j,n) = (k j log n) O(j) flats, which exceeds the optimalobjective value for any k j-dimensional flats by a factor of nomore than Β(j)= 2O(j). Given this bi-criteria approximation, wecan use it to reduce the approximation factor arbitrarily, at the costof increasing the number of flats. Our algorithm hasmany advantages over previous work, in that it is muchmore widely applicable (wider set of objective functions and classes ofclusters) and much more efficient - reducing the running time bound from O(nPoly(k,j)) to nd (jk)O(j).Our algorithm is randomized and successful with probability 1/2(easily boosted to probabilities arbitrary close to 1).

KW - Approximation

KW - Bi-criteria approximation

KW - Geometric optimization

KW - K-center

KW - K-mean

KW - K-median

UR - http://www.scopus.com/inward/record.url?scp=35348816131&partnerID=8YFLogxK

U2 - 10.1145/1247069.1247073

DO - 10.1145/1247069.1247073

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AN - SCOPUS:35348816131

SN - 1595937056

SN - 9781595937056

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 19

EP - 26

BT - Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07

T2 - 23rd Annual Symposium on Computational Geometry, SCG'07

Y2 - 6 June 2007 through 8 June 2007

ER -