Efficient on-line algorithm for maintaining k-cover of sparse bit-strings

We consider the on-line problem of representing a sparse bit string by a set of k intervals, where k is much smaller than the length of the string. The goal is to minimize the total length of these intervals under the condition that each 1-bit must be in one of these intervals. We give an efficient greedy algorithm which takes time O(log k) per update (an update involves converting a 0-bit to a 1-bit), which is independent of the size of the entire string. We prove that this greedy algorithm is 2-competitive. We use a natural linear programming relaxation for this problem, and analyze the algorithm by finding a dual feasible solution whose value matches the cost of the greedy algorithm.