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Asked: 2026-05-21T07:23:12+00:00

Given two arrays of integers, B and A, how can we rearrange their elements

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Given two arrays of integers, B and A, how can we rearrange their elements such that ∏ A[i]B[i] for all i is maximized?

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  1. Editorial Team

    If the arrays contain non-negative numbers, you just sort both A and B in descending order. To see that this gives the maximum product, consider that once A and B are sorted in this order, you can try to swap two items of A, say A[i] and A[j] s.t. i

                  B[j]      B[i]
          A[i]      A[j]
          ------------------
              B[i]      B[j]
          A[i]      A[j]

    i.e. A[i]B[j]-B[i] A[j]B[i]-B[j], which equals (A[i]/A[j])(B[j]-B[i]) where the exponent is zero or negative because B[i] ≥ B[j]. By assumption A[i] ≥ A[j] so A[i]/A[j] ≥ 1. Therefore the ratio is 1 or less as the exponent is either 0 or negative. This shows that the new product is smaller in value than the old product. Note: this is just an illustration, not a formal proof because it considers only the swapping of two elements.

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