Exercises on Reductions
1. Prove that the following two problems have the same complexity by giving a linear-time reductions between the two.
3-SUM: given n integers x1, …, xn, are there three distinct integers i, j, and k such that xi + xj + xk = 0.
3-SUM-PLUS: given n integers x1, …, xn, and an integer b are there three distinct integers i, j, and k such that xi + xj + xk = b.
a. Give a linear-time reduction from 3-SUM to 3-SUM-PLUS. To demonstrate your reduction, give the 3-SUM-PLUS instance that you would construct to solve the following 3-SUM instance: x1, …, xn.
b. Give a linear-time reduction from 3-SUM-PLUS to 3-SUM. To demonstrate your reduction, give the 3-SUM instance that you would construct to solve the following 3-SUM-PLUS instance: b, x1, …, xn.
