Geometric partitioning made easier, even in parallel
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Almost optimal set covers in finite VC-dimension: (preliminary version)
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Concept learning with geometric hypotheses
COLT '95 Proceedings of the eighth annual conference on Computational learning theory
On linear-time deterministic algorithms for optimization problems in fixed dimension
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
When crossings count — approximating the minimum spanning tree
Proceedings of the sixteenth annual symposium on Computational geometry
Catching elephants with mice: sparse sampling for monitoring sensor networks
Proceedings of the 5th international conference on Embedded networked sensor systems
Catching elephants with mice: Sparse sampling for monitoring sensor networks
ACM Transactions on Sensor Networks (TOSN)
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Let (X, R) be a set system on an n-point set X. For a two-coloring on X, its discrepancy is defined as the maximum number by which the occurrences of the two colors differ in any set in R. It is shown that if for any m-point subset Y contained in X the number of distinct subsets induced by R on Y is bounded by O(m/sup d/) for a fixed integer d is a coloring with discrepancy bounded by O(n/sup 1/2-1/2d/ (log n)/sup 1+1/2d/). Also, if any subcollection of m sets of R partitions the points into at most O(m/sup d/) classes, then there is a coloring with discrepancy at most O(n/sup 1/2-1/2d/ n). These bounds imply improved upper bounds on the size of in -approximations for (X, R). All of the bounds are tight up to polylogarithmic factors in the worst case. The results allow the generalization of several results of J. Beck (1984) bounding the discrepancy in certain geometric settings to the case when the discrepancy is taken relative to an arbitrary measure.