It can be proved that 1/N ≤ DN ≤ 1 and 1/(2N) ≤ D∗ N ≤ DN ≤ 2D∗ N . The sequence X is uniformly distributed in [0, 1) if and only if limN→∞DN(X) = 0. We are interested in low-discrepancy sequences, that is, sequences X for which DN(X) is small for all N . The efficient construction of such X is essential in quasi-Monte Carlo algorithms used, for example, to approximate a multivariate integral o...

The linear discrepancy of a partially ordered set P = (X;), denoted by ld(P), is the least integer k for which there exists an injection f : X ! Z satisfying (i) if x y, then f (x) < f (y); and (ii) if x and y are incomparable, then jf(x)?f(y)j k. There is an apparent connection between ld(P) and the bandwidth of the incomparability graph of P. We prove that, in fact, these two quantities are a...

Beam search reduces the memory consumption of bestfirst search at the cost of finding longer paths but its memory consumption can still exceed the given memory capacity quickly. We therefore develop BULB (Beam search Using Limited discrepancy Backtracking), a complete memory-bounded search method that is able to solve more problem instances of large search problems than beam search and does so ...

Lovász, Spencer and Vesztergombi proved that the linear discrepancy of a hypergraph H is bounded above by the hereditary discrepancy of H, and conjectured a sharper bound that involves the number of vertices in H. In this paper we give a short proof of this conjecture for hypergraphs of hereditary discrepancy 1. For hypergraphs of higher hereditary discrepancy we give some partial results and p...

The aim of this note is to prove a new discrepancy principle. The advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem (see problem (2) below) approximately, rather than exactly, and in the proof of a stability result. To explain this in more detail, let us recall the usual discrepancy principle, which can be stated as follows. Consid...

Spencer’s theorem asserts that, for any family of n subsets of ground set of size n, the elements of the ground set can be “colored” by the values ±1 such that the sum of every set is O( √ n) in absolute value. All existing proofs of this result recursively construct “partial colorings”, which assign ±1 values to half of the ground set. We devise the first algorithm for Spencer’s theorem that d...