Description
You are given an array of integers nums and an integer target.
Return the number of non-empty subsequences of nums such that the sum of the minimum and maximum element on it is less or equal to target. Since the answer may be too large, return it modulo 109 + 7.
Example 1:
Input: nums = [3,5,6,7], target = 9
Output: 4
Explanation: There are 4 subsequences that satisfy the condition.
[3] -> Min value + max value <= target (3 + 3 <= 9)
[3,5] -> (3 + 5 <= 9)
[3,5,6] -> (3 + 6 <= 9)
[3,6] -> (3 + 6 <= 9)
Example 2:
Input: nums = [3,3,6,8], target = 10
Output: 6
Explanation: There are 6 subsequences that satisfy the condition. (nums can have repeated numbers).
[3] , [3] , [3,3], [3,6] , [3,6] , [3,3,6]
Example 3:
Input: nums = [2,3,3,4,6,7], target = 12
Output: 61
Explanation: There are 63 non-empty subsequences, two of them do not satisfy the condition ([6,7], [7]).
Number of valid subsequences (63 - 2 = 61).
Constraints:
1 <= nums.length <= 1051 <= nums[i] <= 1061 <= target <= 106
Solutions
Solution 1: Sorting + Binary Search
Since the problem is about subsequences and involves the sum of the minimum and maximum elements, we can first sort the array nums.
Then we enumerate the minimum element nums[i]. For each nums[i], we can find the maximum element nums[j] in nums[i + 1] to nums[n - 1] such that nums[i] + nums[j] ≤ target. The number of valid subsequences in this case is 2j - i, where 2j - i represents all possible subsequences from nums[i + 1] to nums[j]. We sum up the counts of all such subsequences.
The time complexity is O(n × log n), and the space complexity is O(n), where n is the length of the array nums.
PythonJavaC++GoTypeScriptRust
class Solution:
def numSubseq(self, nums: List[int], target: int) -> int:
mod = 10**9 + 7
nums.sort()
n = len(nums)
f = [1] + [0] * n
for i in range(1, n + 1):
f[i] = f[i - 1] * 2 % mod
ans = 0
for i, x in enumerate(nums):
if x * 2 > target:
break
j = bisect_right(nums, target - x, i + 1) - 1
ans = (ans + f[j - i]) % mod
return ans(code-box)
class Solution {
public int numSubseq(int[] nums, int target) {
Arrays.sort(nums);
final int mod = (int) 1e9 + 7;
int n = nums.length;
int[] f = new int[n + 1];
f[0] = 1;
for (int i = 1; i <= n; ++i) {
f[i] = (f[i - 1] * 2) % mod;
}
int ans = 0;
for (int i = 0; i < n && nums[i] * 2 <= target; ++i) {
int j = search(nums, target - nums[i], i + 1) - 1;
ans = (ans + f[j - i]) % mod;
}
return ans;
}
private int search(int[] nums, int x, int left) {
int right = nums.length;
while (left < right) {
int mid = (left + right) >> 1;
if (nums[mid] > x) {
right = mid;
} else {
left = mid + 1;
}
}
return left;
}
}(code-box)
class Solution {
public:
int numSubseq(vector<int>& nums, int target) {
sort(nums.begin(), nums.end());
const int mod = 1e9 + 7;
int n = nums.size();
int f[n + 1];
f[0] = 1;
for (int i = 1; i <= n; ++i) {
f[i] = (f[i - 1] * 2) % mod;
}
int ans = 0;
for (int i = 0; i < n && nums[i] * 2 <= target; ++i) {
int j = upper_bound(nums.begin() + i + 1, nums.end(), target - nums[i]) - nums.begin() - 1;
ans = (ans + f[j - i]) % mod;
}
return ans;
}
};(code-box)
func numSubseq(nums []int, target int) (ans int) {
sort.Ints(nums)
n := len(nums)
f := make([]int, n+1)
f[0] = 1
const mod int = 1e9 + 7
for i := 1; i <= n; i++ {
f[i] = f[i-1] * 2 % mod
}
for i, x := range nums {
if x*2 > target {
break
}
j := sort.SearchInts(nums[i+1:], target-x+1) + i
ans = (ans + f[j-i]) % mod
}
return
}(code-box)
function numSubseq(nums: number[], target: number): number {
nums.sort((a, b) => a - b);
const mod = 1e9 + 7;
const n = nums.length;
const f: number[] = Array(n + 1).fill(1);
for (let i = 1; i <= n; ++i) {
f[i] = (f[i - 1] * 2) % mod;
}
let ans = 0;
for (let i = 0; i < n && nums[i] * 2 <= target; ++i) {
const j = search(nums, target - nums[i], i + 1) - 1;
if (j >= i) {
ans = (ans + f[j - i]) % mod;
}
}
return ans;
}
function search(nums: number[], x: number, left: number): number {
let right = nums.length;
while (left < right) {
const mid = (left + right) >> 1;
if (nums[mid] > x) {
right = mid;
} else {
left = mid + 1;
}
}
return left;
}(code-box)
impl Solution {
pub fn num_subseq(mut nums: Vec<i32>, target: i32) -> i32 {
nums.sort();
const MOD: i32 = 1_000_000_007;
let n = nums.len();
let mut f = vec![1; n + 1];
for i in 1..=n {
f[i] = (f[i - 1] * 2) % MOD;
}
let mut ans = 0;
for i in 0..n {
if nums[i] * 2 > target {
break;
}
let mut l = i + 1;
let mut r = n;
while l < r {
let m = (l + r) / 2;
if nums[m] > target - nums[i] {
r = m;
} else {
l = m + 1;
}
}
let j = l - 1;
ans = (ans + f[j - i]) % MOD;
}
ans
}
}(code-box)
