This can be solved in O(n^2) using dynamic programming. Basically, the problem is about building the longest palindromic subsequence in x[i…j] using the longest subsequence for x[i+1…j], x[i,…j-1] and x[i+1,…,j-1] (if first and last letters are the same). Firstly, the empty string and a single character string is trivially a palindrome. Notice that for a … Read more
Let’s take a look at this python function: def py_fun(i,N,step): res=0.0 while i<N: res+=i i+=step return res and use ipython-magic to time it: In [11]: %timeit py_fun(0.0,1.0e5,1.0) 10 loops, best of 3: 25.4 ms per loop The interpreter will be running through the resulting bytecode and interpret it. However, we could cut out the interpreter … Read more
I think you can solve this using the dynamic programming longest increasing subsequence algorithm: http://www.algorithmist.com/index.php/Longest_Increasing_Subsequence Accounting for the rotations is easy enough: for every tower all you have to check is what happens if you use its height as the length of the base and its width as the height and what happens if you … Read more
Here’s an explanation to go with the posted code. There are two key tricks to make this work efficiently: (I) Kadane’s algorithm and (II) using prefix sums. You also need to (III) apply the tricks to the matrix. Part I: Kadane’s algorithm Kadane’s algorithm is a way to find a contiguous subsequence with maximum sum. … Read more
Dynamic programming is when you use past knowledge to make solving a future problem easier. A good example is solving the Fibonacci sequence for n=1,000,002. This will be a very long process, but what if I give you the results for n=1,000,000 and n=1,000,001? Suddenly the problem just became more manageable. Dynamic programming is used … 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
Relevant article on Programming.Guide: Dynamic programming vs memoization vs tabulation What is difference between memoization and dynamic programming? Memoization is a term describing an optimization technique where you cache previously computed results, and return the cached result when the same computation is needed again. Dynamic programming is a technique for solving problems of recursive nature, … Read more
rev4: A very eloquent comment by user Sammaron has noted that, perhaps, this answer previously confused top-down and bottom-up. While originally this answer (rev3) and other answers said that “bottom-up is memoization” (“assume the subproblems”), it may be the inverse (that is, “top-down” may be “assume the subproblems” and “bottom-up” may be “compose the subproblems”). … Read more
Getting the elements you packed from the matrix can be done using the data from the matrix without storing any additional data. Pseudo code: line <- W i <- n while (i > 0): if dp[line][i] – dp[line – weight(i)][i-1] == value(i): // the element ‘i’ is in the knapsack i <- i-1 // only … Read more
Subset Sum is the first NP-complete problem I learned at Macalester. This question is viewed 36000+ times yet I don’t see a sufficient answer that explains the algorithm in detail with logic. So I thought I make an attempt to do so. Assumption: For the sake of simplicity first I made the assumption that the … Read more