In a sparse array, most elements are zero or empty.

A two-dimensional array of dimension n takes O(n²) storage; that gets expensive fast as n increases. However, a sparse array might have only O(n) nonzero elements.

What we would like to do is to store only the nonzero elements in a lookup structure; if we look up an element and don't find it, we can return zero.

For example, we could have an array of size n for the first index; this could contain pointers to a linked list or tree of values using the second index as the search key.

(defun sparse-aref (arr i j)
  (or (second (assoc j (aref arr i)))
      0))

(setq myarr (make-array '(10) :initial-contents
'(nil nil ((3 77) (2 13)) nil ((5 23))
  nil nil ((2 47) (6 52)) nil nil)))

>(sparse-aref myarr 7 6)
52

>(sparse-aref myarr 7 4)
0

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