Recursion formula in dynamic programming

Question

Where l_1 = 1, l_2 = 4, l_3 = 5 are blocks with different length and I need to make one big block with the length of l = 8 using the formula.

Can someone explain me the following formula:

The formula is in LaTeX, with array L = [l + 1]

Sorry about the formatting, but I can`t upload images.

Answer 1

The question seems to be about finding what is the minimum number of blocks needed to make a bigger block. Also, there seems to be no restriction on the number of individual blocks available.

Assuming you have blocks of n different lengths. l1, l2 .. ln . What is the minimum number of blocks you can use to make one big block of length k ?

The idea behind the recursive formula is that you can make a block of length i by adding one block of length l1 to a hypothetical big block of length i-l1 that you might already have made using the minimum number of blocks (because that is what your L array holds. For any index j , it holds the minimum number of blocks needed to make a block of size j ). Say the i-l1 block was built using 4 blocks. Using those 4 blocks and 1 more block of size l1 , you created a block of size i using 5 blocks.

But now, say a block of size i-l2 was made only using 3 blocks. Then you could easily add another block of size l2 to this block of size i-l2 and make a block of size i using only 4 blocks!

That is the idea behind iterating over all possible block lengths and choosing the minimum of them all (mentioned in the third line of your latex image).

Hope that helps.