### lvisbl_'s blog

By lvisbl_, history, 5 weeks ago,

Recently, I encounter this 1741E - Sending a Sequence Over the Network that combine both forward and backward DP style, that make me feel surprised. I thought that we could only use either forward and backward DP in a problem to solve it. But in this problem we use both? Let call dp[i] is can we separate array from [1,i] into valid segments. Then:

The backward DP: if from range [1,i] we can not separate array into valid segments then we try to make the current number at index i as the length of the last segment => Check backward the range [1,i — b[i] — 1].

The forward DP: if we can separate array into valid segments in range [1,i-1], then the current number at index i can possibly length of the next segments => Update dp[i + b[i]] = true.

The doubt is can I combine both forward and backward DP style in all DP problems? If not, in what case/what type of DP properties the combination of forward and backward style will work? Do we have any related problems that require to combine both styles like this?

• -1

By lvisbl_, history, 6 weeks ago,

Hello mates, I'm trying to solve this DP question Coin Combinations II from CSES: https://cses.fi/problemset/task/1636/

Initially, I define an array dp[i][j]: meaning number of ways to form sum i, starting choosing coins from j-th index. And this solution using this DP array get me TLE.

But if I switch the two dimension, array dp[i][j]: number of ways to form sum j, starting choosing coins from i-th index. This solution give me accepted. But why?

Note:

• Two solution is 99% the same the only different is that I switch two dimensions.

• Sum i at most 10^6 and there are at most 10^2 coins.