dynamic-programming

We are given an array with n values. Example: [1,4,5,6,6] For each index i of the array a ,we construct a new element of array b such that, b[i]= [a[i]/1] + [a[i+1]/2] + [a[i+2]/3] + ⋯ + [a[n]/(n−i+1)] where [.] denotes the greatest integer function. We are given an integer k as well. We have to find the minimum i such that b[i] ≤ k . I know the brute-force O(n^2) algorithm (to create the array - 'b'), can anybody suggest a better time complexity and way solve it? For example, for the input [1,2,3] , k=3 , the output is 1(minimum-index) . Here, a[1]=1; a[2]=2; a[3]=3; Now, b[1] = [a[1]/1] + [a

I have a question which asks us to reduce the string as follows. The input is a string having only A , B or C . Output must be length of the reduced string The string can be reduced by the following rules If any 2 different letters are adjacent, these two letters can be replaced by the third letter. Eg ABA -> CA -> B . So final answer is 1 (length of reduced string) Eg ABCCCCCCC This doesn't become CCCCCCCC , as it can be reduced alternatively by ABCCCCCCC -> AACCCCCC -> ABCCCCC -> AACCCC -> ABCCC -> AACC -> ABC -> AA as here length is 2 < (length of CCCCCCCC ) How do you go about this problem

I found the following problem on the internet, and would like to know how I would go about solving it: Problem: Integer Partition without Rearrangement Input: An arrangement S of non negative numbers {s 1 , . . . , s n } and an integer k. Output: Partition S into k or fewer ranges, to minimize the maximum of the sums of all k or fewer ranges, without reordering any of the numbers.* Please help, seems like interesting question... I actually spend quite a lot time in it, but failed to see any solution.. Let's try to solve the problem using dynamic programming. Note: If k > n we can use only n

I was given the following problem in an interview: Given a staircase with N steps, you can go up with 1 or 2 steps each time. Output all possible way you go from bottom to top. For example: N = 3 Output : 1 1 1 1 2 2 1 When interviewing, I just said to use dynamic programming. S(n) = S(n-1) +1 or S(n) = S(n-1) +2 However, during the interview, I didn't write very good code for this. How would you code up a solution to this problem? Thanks indeed! Nikita Rybak You can generalize your recursive function to also take already made moves. void steps(n, alreadyTakenSteps) { if (n == 0) { print

I am trying to come up with a dynamic programming algorithm that finds the largest sub matrix within a matrix that consists of the same number: example: {5 5 8} {5 5 7} {3 4 1} Answer : 4 elements due to the matrix 5 5 5 5 TMS This is a question I already answered here (and here , modified version). In both cases the algorithm was applied to binary case (zeros and ones), but the modification for arbitrary numbers is quite easy (but sorry, I keep the images for the binary version of the problem ). You can do this very efficiently by two pass linear O(n) time algorithm - n being number of

Can anyone please help me understand the core logic behind the solution to a problem mentioned at http://www.topcoder.com/stat?c=problem_statement&pm=1259&rd=4493 A zig zag sequence is one that alternately increases and decreases. So, 1 3 2 is zig zag, but 1 2 3 is not. Any sequence of one or two elements is zig zag. We need to find the longest zig zag subsequence in a given sequence. Subsequence means that it is not necessary for elements to be contiguous, like in the longest increasing subsequence problem. So, 1 3 5 4 2 could have 1 5 4 as a zig zag subsequence. We are interested in the

In the book I am using Introduction to the Design & Analysis of Algorithms , dynamic programming is said to focus on the Principle of Optimality , "An optimal solution to any instance of an optimization problem is composed of optimal solutions to its subinstances". Whereas, the greedy technique focuses on expanding partially constructed solutions until you arrive at a solution for a complete problem. It is then said, it must be "the best local choice among all feasible choices available on that step". Since both involve local optimality, isn't one a subset of the other? Dynamic programming is