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Program to find number of sets of k-non-overlapping line segments in Python
Suppose we have n points on a line, where the ith point (from 0 to n-1) is at position x = i, we have to find the number of ways we can draw exactly k different non-overlapping line segments such that each segment covers two or more points. The endpoints of each line segment must have integral coordinates. The k line segments do not have to cover all given n points, and they can share endpoints. If the answer is too large, then return result mod 10^9+7.
So, if the input is like n = 4 k = 2, then the output will be 5 because we can make five possibilities [(0 to 2),(2 to 3)], [(0 to 1),(1 to 3)], [(0 to 1),(2 to 3)], [(1 to 2),(2 to 3)] and [(0 to 1),(1 to 2)]
To solve this, we will follow these steps −
- m := 10^9 + 7
- n := n - 1
- Define a function dp() . This will take i, covered, j
- if i is same as n, then
- return true if j is same as k otherwise false
- if j > k, then
- ans := dp(i + 1, False, j) + dp(i + 1, True, j + 1)
- if covered is true, then
- ans := ans + dp(i + 1, True, j)
- return ans mod m
- From the main method return dp(0, False, 0)
Example
Let us see the following implementation to get better understanding −
def solve(n, k): m = 10 ** 9 + 7 n -= 1 def dp(i, covered, j): if i == n: return j == k if j > k: return 0 ans = dp(i + 1, False, j) + dp(i + 1, True, j + 1) if covered: ans += dp(i + 1, True, j) return ans % m return dp(0, False, 0) n = 4 k = 2 print(solve(n, k))
Input
4, 2
Output
5
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