Dynamic algorithm

Have you ever played with LEGOs before? Imagine that you have a big pile of colorful blocks, and you want to make a tall tower. Now, you could start building the tower from scratch every time you want to make it, but that would take a lot of time and effort. Instead, you can keep some of the blocks that you have already used in your previous towers, and only add the new blocks that you have just received. This way, you can build your tower quicker and with less effort.

This is the basic idea behind dynamic programming, a technique used in computer programming that can help us solve complex problems more efficiently. Instead of recalculating everything from scratch every time we want to solve a problem, we can reuse some of the previous calculations that we have already done, and only update the parts that have changed.

For example, imagine that you have a list of numbers, and you want to find the maximum sum you can get by selecting a subsequence (a subset of the original list where the order is preserved). One way to solve this problem is to try all possible subsequences and compute their sums, but this would be very inefficient for large lists. With dynamic programming, we can break down the problem into smaller subproblems and solve them incrementally, keeping track of the best solution found so far.

In summary, dynamic programming is a strategy to solve problems by breaking them down into smaller subproblems, and reusing previous calculations to make the computation more efficient. It is like building a LEGO tower, where you don't start from scratch every time, but use some of the blocks that you have already used before.