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Link List are good for implementing sparse arrays because you need not store values that are not in the sparse arrays.

Binary tree. Key value is the index and do inorder traversal.

Can use hash table.

What if there is sparse matrix. What is the suitable data structure for it.
Any other idea.

What does it mean by multiplying arrays? Is it like multiplying two vectors?

I think a simple 1-D array would do. In that array just store the locations and their values of the sparse arrays as objects. If you like to multiply, we can use standard multiplication and can be cleverly designed to implement minimum number of steps, because of the sparseness of the array.

Implement using Tries