dynamic-programming

I found this challenge problem which states the following : Suppose that there are n rectangles on the XY plane. Write a program to calculate the maximum possible number of rectangles that can be crossed with a single straight line drawn on this plane. I have been brainstorming for quite a time but couldn't find any solution. Maybe at some stage, we use dynamic programming steps but couldn't figure out how to start. Here is a sketch of an O(n^2 log n) solution. First, the preliminaries shared with other answers. When we have a line passing through some rectangles, we can translate it to any of

I'm looking at Problem thirty one on Project Euler, which asks, how many different ways are there of making £2 using any number of coins of 1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p). There are recursive solutions, such as this one in Scala (credit to Pavel Fatin) def f(ms: List[Int], n: Int): Int = ms match { case h :: t => if (h > n) 0 else if (n == h) 1 else f(ms, n - h) + f(t, n) case _ => 0 } val r = f(List(1, 2, 5, 10, 20, 50, 100, 200), 200) and although it runs fast enough, it's relatively inefficient, calling the f function around 5.6 million times. I saw someone else's

Given any number n, and three operations on n: add 1 subtract 1 divide by 2 if the number is even I want to find the minimum number of the above operations to reduce n to 1. I have tried dynamic progamming approach, also BFS with pruning, but n can be very large (10^300) and I do not know how to make my algorithm faster. The greedy approach (divide by 2 if even and subtract 1 if odd) also does not give the optimal result. Is another solution exists? 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

The building bridges problem is stated as follows: There is a river that runs horizontally through an area. There are a set of cities above and below the river. Each city above the river is matched with a city below the river, and you are given this matching as a set of pairs. You are interested in building a set of bridges across the river to connect the largest number of the matching pairs of cities, but you must do so in a way that no two bridges intersect one another. Devise an algorithm to solve this problem as efficiently as possible. I have heard that this problem is related to the

Here is another spoj problem that asks how to find the number of distinct subsequences of a string ? For example, Input AAA ABCDEFG CODECRAFT Output 4 128 496 How can I solve this problem ? 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 = strlen(a) for i = 1 to n dp[i] = sum[i - 1] - sum[last[a[i]] - 1] sum[i] = sum[i - 1] + dp[i] last[a[i]] = i

In a book I encountered following question: Given N step stair, in how many number of ways can you climb if you use either 1, 2 or 3 steps at a time? Following is the code that book has given: int countWays(int n){ if(n<0) return 0; if(n == 0) return 1; else return countWays(n-1) + countWays(n-2) + countWays(n-3); } I have the following concerns in understanding this code: I do not understand why 1 is being returned for n=0. If there are 0 steps then obviously we do not have to climb any and 0 should be returned. For n=3 function returns 4 but i can see only 3 cases i.e. (1,1,1), (1,2), (3). I