Description
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 <= 10001 <= k <= n-1
Solutions
Solution 1
PythonJavaC++GoTypeScript
class Solution:
def numberOfSets(self, n: int, k: int) -> int:
mod = 10**9 + 7
f = [[0] * (k + 1) for _ in range(n + 1)]
g = [[0] * (k + 1) for _ in range(n + 1)]
f[1][0] = 1
for i in range(2, n + 1):
for j in range(k + 1):
f[i][j] = (f[i - 1][j] + g[i - 1][j]) % mod
g[i][j] = g[i - 1][j]
if j:
g[i][j] += f[i - 1][j - 1]
g[i][j] %= mod
g[i][j] += g[i - 1][j - 1]
g[i][j] %= mod
return (f[-1][-1] + g[-1][-1]) % mod(code-box)
class Solution {
private static final int MOD = (int) 1e9 + 7;
public int numberOfSets(int n, int k) {
int[][] f = new int[n + 1][k + 1];
int[][] g = new int[n + 1][k + 1];
f[1][0] = 1;
for (int i = 2; i <= n; ++i) {
for (int j = 0; j <= k; ++j) {
f[i][j] = (f[i - 1][j] + g[i - 1][j]) % MOD;
g[i][j] = g[i - 1][j];
if (j > 0) {
g[i][j] += f[i - 1][j - 1];
g[i][j] %= MOD;
g[i][j] += g[i - 1][j - 1];
g[i][j] %= MOD;
}
}
}
return (f[n][k] + g[n][k]) % MOD;
}
}(code-box)
class Solution {
public:
int f[1010][1010];
int g[1010][1010];
const int mod = 1e9 + 7;
int numberOfSets(int n, int k) {
memset(f, 0, sizeof(f));
memset(g, 0, sizeof(g));
f[1][0] = 1;
for (int i = 2; i <= n; ++i) {
for (int j = 0; j <= k; ++j) {
f[i][j] = (f[i - 1][j] + g[i - 1][j]) % mod;
g[i][j] = g[i - 1][j];
if (j > 0) {
g[i][j] += f[i - 1][j - 1];
g[i][j] %= mod;
g[i][j] += g[i - 1][j - 1];
g[i][j] %= mod;
}
}
}
return (f[n][k] + g[n][k]) % mod;
}
};(code-box)
func numberOfSets(n int, k int) int {
f := make([][]int, n+1)
g := make([][]int, n+1)
for i := range f {
f[i] = make([]int, k+1)
g[i] = make([]int, k+1)
}
f[1][0] = 1
var mod int = 1e9 + 7
for i := 2; i <= n; i++ {
for j := 0; j <= k; j++ {
f[i][j] = (f[i-1][j] + g[i-1][j]) % mod
g[i][j] = g[i-1][j]
if j > 0 {
g[i][j] += f[i-1][j-1]
g[i][j] %= mod
g[i][j] += g[i-1][j-1]
g[i][j] %= mod
}
}
}
return (f[n][k] + g[n][k]) % mod
}(code-box)
function numberOfSets(n: number, k: number): number {
const f = Array.from({ length: n + 1 }, _ => new Array(k + 1).fill(0));
const g = Array.from({ length: n + 1 }, _ => new Array(k + 1).fill(0));
f[1][0] = 1;
const mod = 10 ** 9 + 7;
for (let i = 2; i <= n; ++i) {
for (let j = 0; j <= k; ++j) {
f[i][j] = (f[i - 1][j] + g[i - 1][j]) % mod;
g[i][j] = g[i - 1][j];
if (j) {
g[i][j] += f[i - 1][j - 1];
g[i][j] %= mod;
g[i][j] += g[i - 1][j - 1];
g[i][j] %= mod;
}
}
}
return (f[n][k] + g[n][k]) % mod;
}(code-box)
