Exercises for Complexity Theory of Polynomial-Time Problems
نویسنده
چکیده
Exercise 1 (8 points) Show a sub-quadratic reduction from Convolution 3SUM to 3SUM. More precisely, show that if 3SUM can be solved in time O(n2− ) for some > 0, then Convolution 3SUM can be solved in time O(n2−δ) for some δ > 0. Solution : We are given a set A = {a1, a2, . . . , an} having n integers, we want to test whether there exist i, j ∈ [n] such that ai + aj = ai+j. We create a new set B = {b1, b2, . . . , bn}, where we set bi = 2nai + i. Now we use the 3SUM algorithm on set B. If in the original set A, there exist i, j ∈ [n] such that ai+aj = ai+j, then bi+j = 2nai+j+(i+j) = 2n(ai + aj) + (i+ j) = (2nai + i) + (2naj + j) = bi + bj. Conversely, if in the set B there is a triple with bk = bi + bj, then 2nak + k = (2nai + i) + (2naj + j) = 2n(ai + aj) + (i + j). Since we have i + j < 2n, we must have ak = ai + aj and k = i+ j.
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