Data Structures, Algorithms, & Applications in Java
Chapter 8, Exercise 9

(a)

Java uses the array of arrays representation. When m = 10 and n = 2, we have 1 one-dimensional array of size 10 (the elements of this array are Java references) and 10 one-dimensional arrays of size 2 (the elements of these arrays are integers). Each of the 11 one-dimensional arrays has a length field which is an integer. So, the total space is
1 x 10 x 4 + 10 x 2 x 4 + 11 x 4 = 164 bytes.

When a single one-dimensional array is used (i.e., when the elements are mapped using (say) row-major order), the space required is 10 x 2 x 4 + 4 = 84 bytes. The ratio is 164 / 84 = 1.95.

For general m and n, the array of arrays representation requires
1 x m x 4 + m x n x 4 + (m + 1) x 4 = 4mn + 8m + 4 bytes.
The one-dimensional mapping takes m x n x 4 + 4 = 4mn + 4 bytes.

(b)

The ratio is
(4mn + 8m + 4) / (4mn + 4) = 1 + 8m/(4mn + 4).
This is maximum when n is least. For a two-dimensional array, n >= 2. So, the maximum value of the ratio is 1 + 8m/(8m + 4). This approaches 2 as m gets large.