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Additional info for A 3/4-Approximation Algorithm for Multiple Subset Sum
Example text
Assume that for each i < k, we have found the optimal ( − 1)-bucket histogram on the prefix of data indexed by [0, i). To find the best -bucket histogram on [0, k), try all possibilities m < k for the final boundary, and form a histogram by joining the best ( − 1)-bucket histogram on [0, m) with the 1-bucket histogram on [m, k). The time for this step is therefore at most N to try all values of m. Since this has to be repeated for all k < N and all ≤ B, the total time is O(N 2 B). The space is O(BN ), to store a table of a B-bucket histogram for each i.
Consider the hashed sequence T [1, . . , N ] where T [i] = h(S[i]) for a random hash function. Define the monotonic chain C to be the minimum element and successive minimums, each to the right of the predecessor. What is the expected length of C? Consider building a treap (See [173, 172]) on S: the list C is the right spine of this treap. If the hash values are a fully independent, random permutation, then it is well known that [172, 173, 198]: Theorem 20. With high probability, |C| = Θ(HN ), where HN is the N th Harmonic number, given by 1 + 1/2 + 1/3 + · · · + 1/N = Θ(log N ).
We maintain the following two sets of items: (1) Highest B-wavelet basis coefficients for the signal seen thus far. (2) log N straddling coefficients, one for each level of the Haar wavelet transform tree. At level j, the wavelet basis vector that straddles i is ψj,k where k(N/2j ) ≤ i ≤ k(N/2j ) + N/2j − 1, and there is at most one such vector per level. When the following data item (i + 1, A[i + 1]) is read, we update each of the straddling coefficients at each level if they get affected. Some of the straddling coefficients may no longer remain straddling.