A trace bound for the hereditary discrepancy

Let A be the incidence matrix of a set system with m sets and n points, m ≤ n, and let t = tr M, where M = AT A. Finally, let σ = tr M2 be the sum of squares of the elements of M. We prove that the hereditary discrepancy of the set system is at least 1 4 cnσ/t2 √ t/n, with c = 1 324 . This general trace bound allows us to resolve discrepancy-type questions for which spectral methods had previously failed. Also, by using this result in conjunction with the spectral lemma for linear circuits, we derive new complexity bounds for range searching. • We show that the (red–blue) discrepancy of the set system formed by n points and n lines in the plane is (n1/6 ) in the worst case and always1 ˜O(n1/6 ). • We give a simple explicit construction of n points and n halfplanes with hereditary discrepancy ˜ (n1/4 ). • We show that in any dimension d = (log n/log log n), there is a set system of n points and n axis-parallel boxes in Rd with discrepancy n (1/log log n) . • Applying t...

• Type: Conference
• Year: 2000
• Where: COMPGEOM
• Authors: Bernard Chazelle, Alexey Lvov
• Keywords: COMPGEOM 2000, Discrete Geometry, Hereditary Discrepancy, Log N, N/log Log N