Derandomizing Non-uniform Color-Coding I

Color-coding, as introduced by Alon, Yuster, and Zwick, is a wellknown tool for algorithm design and can often be efficiently derandomized using universal hash functions. In the special case of only two colors, one can use (n, k)-universal sets for the derandomization. In this paper, we introduce (n, k, l)universal sets that are typically smaller and can be constructed faster. Nevertheless, for some problems they are still sufficient for derandomization and faster deterministic algorithms can be obtained. This particularly works well when the color-coding does not use a uniform probability distribution. To exemplify the concept, we present an algorithm for the Unique Coverage problem introduced by Demaine, Feige, Hajiaghayi, and Salavatipour. The example also shows how to extend the concept to multiple colors.