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The paper addresses the problem of defining families of ordered sequences$\{x_i\}_{i\in N}$ of elements of a compact subset $X$ of $R^d$ whose prefixes$X_n=\{x_i\}_{i=1}^{n}$, for all orders $n$, have good space-filling propertiesas measured by the dispersion (covering radius) criterion. Our ultimate aim isthe definition of incremental algorithms that generate sequences $X_n$ withsmall optimality gap, i.e., with a small increase in the maximum distancebetween points of $X$ and the elements of $X_n$ with respect to the optimalsolution $X_n^\star$. The paper is a first step in this direction, presentingincremental design algorithms with proven optimality bound for one-parameterfamilies of criteria based on coverings and spacings that both converge todispersion for large values of their parameter. The examples presented showthat the covering-based method outperforms state-of-the-art competitors,including coffee-house, suggesting that it inherits from its guaranteed 50\%optimality gap.

Incremental space-filling design based on coverings and spacings: improving upon low discrepancy sequences