923. 3Sum With Multiplicity
Description
Given an integer array arr
, and an integer target
, return the number of tuples i, j, k
such that i < j < k
and arr[i] + arr[j] + arr[k] == target
.
As the answer can be very large, return it modulo 109 + 7
.
Example 1:
Input: arr = [1,1,2,2,3,3,4,4,5,5], target = 8 Output: 20 Explanation: Enumerating by the values (arr[i], arr[j], arr[k]): (1, 2, 5) occurs 8 times; (1, 3, 4) occurs 8 times; (2, 2, 4) occurs 2 times; (2, 3, 3) occurs 2 times.
Example 2:
Input: arr = [1,1,2,2,2,2], target = 5 Output: 12 Explanation: arr[i] = 1, arr[j] = arr[k] = 2 occurs 12 times: We choose one 1 from [1,1] in 2 ways, and two 2s from [2,2,2,2] in 6 ways.
Example 3:
Input: arr = [2,1,3], target = 6 Output: 1 Explanation: (1, 2, 3) occured one time in the array so we return 1.
Constraints:
3 <= arr.length <= 3000
0 <= arr[i] <= 100
0 <= target <= 300
Solutions
Solution 1: Counting + Enumeration
We can use a hash table or an array $cnt$ of length $101$ to count the occurrence of each element in the array $arr$.
Then, we enumerate each element $arr[j]$ in the array $arr$, first subtract one from $cnt[arr[j]]$, and then enumerate the elements $arr[i]$ before $arr[j]$, calculate $c = target - arr[i] - arr[j]$. If $c$ is in the range of $[0, 100]$, then the answer is increased by $cnt[c]$, and finally return the answer.
Note that the answer may exceed ${10}^9 + 7$, so take the modulus after each addition operation.
The time complexity is $O(n^2)$, where $n$ is the length of the array $arr$. The space complexity is $O(C)$, where $C$ is the maximum value of the elements in the array $arr$, in this problem $C = 100$.
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