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QUESTION

# An addition chain for an integer n is an increasing sequence of integers that starts with 1 and ends with n, such that each entry after the first is...

An addition chain for an integer n is an increasing sequence of integers that starts with 1 and ends with n, such that each entry after the first is the sum of two earlier entries. More formally, the integer sequence x0 < x1 < x2 < ··· < x` is an addition chain for n if and only if • x0 = 1, • x` = n, and • for every index k > 0, there are indices i ≤ j < k such that xk = xi + x j . The `ength of an addition chain is the number of elements minus 1; we don't bother to count the first entry. For example, 〈1, 2, 3, 5, 10, 20, 23, 46, 92, 184, 187, 374〉 is an addition chain for 374 of length 11.

a. Describe a recursive backtracking algorithm to compute a minimum-length addition chain for a given positive integer n. Don't analyze or optimize your algorithm's running time, except to satisfy your own curiosity. A correct algorithm whose running time is exponential in n is sufficient for full credit. [Hint: This problem is a lot more like n Queens than text segmentation.]