Question

1498. Number of Subsequences That Satisfy the Given Sum Condition

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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 <= 105
• 1 <= nums[i] <= 106
• 1 <= target <= 106

Solutions

Solution 1

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

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; ++i) {
      if (nums[i] * 2L > target) {
        break;
      }
      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;
  }
}

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; ++i) {
      if (nums[i] * 2L > target) {
        break;
      }
      int j = upper_bound(nums.begin() + i + 1, nums.end(), target - nums[i]) - nums.begin() - 1;
      ans = (ans + f[j - i]) % mod;
    }
    return ans;
  }
};

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
}