README.md

1621. Number of Sets of K Non-Overlapping Line Segments

Given n points on a 1-D plane, where the ith point (from 0 to n-1) is at x = i, find the number of ways we can draw exactly k non-overlapping line segments such that each segment covers two or more points. The endpoints of each segment must have integral coordinates. The k line segments do not have to cover all n points, and they are allowed to share endpoints.

Return the number of ways we can draw k non-overlapping line segments. Since this number can be huge, return it modulo 109 + 7.

Example 1:

Input: n = 4, k = 2
Output: 5
Explanation: The two line segments are shown in red and blue.
The image above shows the 5 different ways {(0,2),(2,3)}, {(0,1),(1,3)}, {(0,1),(2,3)}, {(1,2),(2,3)}, {(0,1),(1,2)}.

Example 2:

Input: n = 3, k = 1
Output: 3
Explanation: The 3 ways are {(0,1)}, {(0,2)}, {(1,2)}.

Example 3:

Input: n = 30, k = 7
Output: 796297179
Explanation: The total number of possible ways to draw 7 line segments is 3796297200. Taking this number modulo 109 + 7 gives us 796297179.

Constraints:

• 2 <= n <= 1000
• 1 <= k <= n-1

Solutions (Python)

1. Solution

class Solution:
    def numberOfSets(self, n: int, k: int) -> int:
        dp = [[0] * (n + 1) for _ in range(k + 1)]

        for j in range(2, n + 1):
            dp[1][j] = dp[1][j - 1] + j - 1

        for i in range(2, k + 1):
            for j in range(i + 1, n + 1):
                dp[i][j] = dp[i - 1][j - 1] + 2 * dp[i][j - 1] - dp[i][j - 2]

        return dp[k][n] % 1000000007