Algorithm: Buying a collection of goods with vouchers

Question

I'm struggling to come up with a solution for this problem for an algorithms course:

You go to a store and want to buy n = {n ₁ , n ₂ , ..., n _n } goods, where the items can be different or not.

The store has the following promotion: "If a customer buys two articles whose prices add up to a value which ends with 11, 33 or 55 cents, he will receive a voucher, worth the corresponding cent value."

The problem is to come up with an algorithm which computes an optimal strategy for buying a given collection of goods, where a minimization of the total cost is wanted.

For example: If you need to buy 3 products (n ₁ , n ₂ , n ₃ ) with prices (1.01$, 2.10$ and 3$), you should buy n ₁ and n ₂ together and buy n ₃ separately, yielding a total cost of: (1.01 + 2.10) + 3 - (0.11) = 6$.

No hint is given, but I think I can use some approach of flow-networks.

Answer 1

Sort items you want to buy by cent prices... ie (25.01$, 1.02$, ..., 0.99$).

And then check... whether any of the cents sums up to 0.99$... This can be solved easily by two sum problem... You take .01$ and look for its compliment 0.98$ using binary search... Delete this pair and repeat until you find no pair of products with cents that sum up to 0.99$.

If this is done, delete found pairs of products... calculate your gain and check whether any of the cents sums up to 0.88$... The same way.

...

If this is done, delete found pairs of products... and check whether any of the cents sums up to 0.11$... The same way.

This is all O(nlogn) complexity ;)