On the k-coloring of intervals
The problem of coloring a set of n intervals (from the real line) with a set of k colors is studied. In such a coloring, two intervals may have the same color if and only if those intervals do not overlap. Two versions of the problem are considered. For the first, we provide an O(k+n) time algorithm for k-coloring a maximum cardinality subset of the intervals. The best previous algorithm for this problem required time O(kn). In the second version, we assume that each interval has a weight, and provide an O(knlogn) algorithm for k-coloring a set of intervals of maximum total weight. The best previous algorithm for this problem required time O(n2logn). These results provide improved solutions to problems of local register allocation, task scheduling, and the routing of nets on a chip.
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