Update I have made an improvement of the algorithm that it takes an average of O(M + N^2) and memory needs of O(M+N). Mainly is the same that the protocol described below, but to calculate the possible factors A,K for ech diference D, I preload a table. This table takes less than a second to … Read more
I don’t think you need dynamic programming at all for this problem. It can basically be expressed by binary calculations. If you convert the number of a station to binary it tells you right away how to get there from station 0, e.g., station 6 = 110 tells you that you need to take the … Read more
Dynamic programming is a useful type of algorithm that can be used to optimize hard problems by breaking them up into smaller subproblems. By storing and re-using partial solutions, it manages to avoid the pitfalls of using a greedy algorithm. There are two kinds of dynamic programming, bottom-up and top-down. In order for a problem … Read more
It’s a classic dynamic programming problem. Let: dp[i] = number of distinct subsequences ending with a[i] sum[i] = dp[1] + dp[2] + … + dp[i]. So sum[n] will be your answer. last[i] = last position of character i in the given string. A null string has one subsequence, so dp[0] = 1. read a n … Read more
Here is a sketch of the solution: For each of the cells we will keep a counter of how big a square can be made using that cell as top left. Clearly all cells with 0 will have 0 as the count. Start iterating from bottom right cell and go to bottom left, then go … Read more
Dr Jon Harrop, whom I contacted over e-mail, explained what is going on: The problem is simply that the program has been optimized for Python. This is common when the programmer is more familiar with one language than the other, of course. You just have to learn a different set of rules that dictate how … Read more
Dynamic programming is applicable to problems exhibiting the properties of: overlapping subproblems, and optimal substructure. Optimal substructure means that you can greedily solve subproblems and combine the solutions to solve the larger problem. The difference between dynamic programming and greedy algorithms is that with dynamic programming, there are overlapping subproblems, and those subproblems are solved … Read more
The common way to do this is via lazy memoization. In some sense, the recursive fibonacci function can be considered dynamic programming, because it computes results out of overlapping subproblems. I realize this is a tired example, but here’s a taste. It uses the data-memocombinators library for lazy memoization. import qualified Data.MemoCombinators as Memo fib … Read more
I’m not sure who is still paying attention to this thread, but here goes anyway. First, in the official-looking linked version, it only has to be 1000 factorial, not 10000 factorial. Also, when this problem was reused in another programming contest, the time limit was 3 seconds, not 1 second. This makes a huge difference … Read more
There is a pattern which allows you to know the optimal next step in constant time. In fact, there can be cases where there are two equally optimal choices — in that case one of them can be derived in constant time. If you look at the binary representation of n, and its least significant … Read more