2657. Find the Prefix Common Array of Two Arrays
Description
You are given two 0-indexed integer permutations A and B of length n.
A prefix common array of A and B is an array C such that C[i] is equal to the count of numbers that are present at or before the index i in both A and B.
Return the prefix common array of A and B.
A sequence of n integers is called a permutation if it contains all integers from 1 to n exactly once.
Example 1:
Input: A = [1,3,2,4], B = [3,1,2,4] Output: [0,2,3,4] Explanation: At i = 0: no number is common, so C[0] = 0. At i = 1: 1 and 3 are common in A and B, so C[1] = 2. At i = 2: 1, 2, and 3 are common in A and B, so C[2] = 3. At i = 3: 1, 2, 3, and 4 are common in A and B, so C[3] = 4.
Example 2:
Input: A = [2,3,1], B = [3,1,2] Output: [0,1,3] Explanation: At i = 0: no number is common, so C[0] = 0. At i = 1: only 3 is common in A and B, so C[1] = 1. At i = 2: 1, 2, and 3 are common in A and B, so C[2] = 3.
Constraints:
1 <= A.length == B.length == n <= 501 <= A[i], B[i] <= nIt is guaranteed that A and B are both a permutation of n integers.
Solutions
Solution 1: Counting
We can use two arrays $cnt1$ and $cnt2$ to record the occurrence times of each element in arrays $A$ and $B$ respectively, and use an array $ans$ to record the answer.
Traverse arrays $A$ and $B$, increment the occurrence times of $A[i]$ in $cnt1$, and increment the occurrence times of $B[i]$ in $cnt2$. Then enumerate $j \in [1,n]$, calculate the minimum occurrence times of each element $j$ in $cnt1$ and $cnt2$, and accumulate them into $ans[i]$.
After the traversal, return the answer array $ans$.
The time complexity is $O(n^2)$, and the space complexity is $O(n)$. Here, $n$ is the length of arrays $A$ and $B$.
Python3
class Solution:
def findThePrefixCommonArray(self, A: List[int], B: List[int]) -> List[int]:
ans = []
cnt1 = Counter()
cnt2 = Counter()
for a, b in zip(A, B):
cnt1[a] += 1
cnt2[b] += 1
t = sum(min(v, cnt2[x]) for x, v in cnt1.items())
ans.append(t)
return ans
Java
class Solution {
public int[] findThePrefixCommonArray(int[] A, int[] B) {
int n = A.length;
int[] ans = new int[n];
int[] cnt1 = new int[n + 1];
int[] cnt2 = new int[n + 1];
for (int i = 0; i < n; ++i) {
++cnt1[A[i]];
++cnt2[B[i]];
for (int j = 1; j <= n; ++j) {
ans[i] += Math.min(cnt1[j], cnt2[j]);
}
}
return ans;
}
}
C++
class Solution {
public:
vector<int> findThePrefixCommonArray(vector<int>& A, vector<int>& B) {
int n = A.size();
vector<int> ans(n);
vector<int> cnt1(n + 1), cnt2(n + 1);
for (int i = 0; i < n; ++i) {
++cnt1[A[i]];
++cnt2[B[i]];
for (int j = 1; j <= n; ++j) {
ans[i] += min(cnt1[j], cnt2[j]);
}
}
return ans;
}
};
Go
func findThePrefixCommonArray(A []int, B []int) []int {
n := len(A)
cnt1 := make([]int, n+1)
cnt2 := make([]int, n+1)
ans := make([]int, n)
for i, a := range A {
b := B[i]
cnt1[a]++
cnt2[b]++
for j := 1; j <= n; j++ {
ans[i] += min(cnt1[j], cnt2[j])
}
}
return ans
}
TypeScript
function findThePrefixCommonArray(A: number[], B: number[]): number[] {
const n = A.length;
const cnt1: number[] = Array(n + 1).fill(0);
const cnt2: number[] = Array(n + 1).fill(0);
const ans: number[] = Array(n).fill(0);
for (let i = 0; i < n; ++i) {
++cnt1[A[i]];
++cnt2[B[i]];
for (let j = 1; j <= n; ++j) {
ans[i] += Math.min(cnt1[j], cnt2[j]);
}
}
return ans;
}
Solution 2: Bit Operation (XOR Operation)
We can use an array $vis$ of length $n+1$ to record the occurrence situation of each element in arrays $A$ and $B$, the initial value of array $vis$ is $1$. In addition, we use a variable $s$ to record the current number of common elements.
Next, we traverse arrays $A$ and $B$, update $vis[A[i]] = vis[A[i]] \oplus 1$, and update $vis[B[i]] = vis[B[i]] \oplus 1$, where $\oplus$ represents XOR operation.
If at the current position, the element $A[i]$ has appeared twice (i.e., it has appeared in both arrays $A$ and $B$), then the value of $vis[A[i]]$ will be $1$, and we increment $s$. Similarly, if the element $B[i]$ has appeared twice, then the value of $vis[B[i]]$ will be $1$, and we increment $s$. Then add the value of $s$ to the answer array $ans$.
After the traversal, return the answer array $ans$.
The time complexity is $O(n)$, and the space complexity is $O(n)$. Here, $n$ is the length of arrays $A$ and $B$.
Python3
class Solution:
def findThePrefixCommonArray(self, A: List[int], B: List[int]) -> List[int]:
ans = []
vis = [1] * (len(A) + 1)
s = 0
for a, b in zip(A, B):
vis[a] ^= 1
s += vis[a]
vis[b] ^= 1
s += vis[b]
ans.append(s)
return ans
Java
class Solution {
public int[] findThePrefixCommonArray(int[] A, int[] B) {
int n = A.length;
int[] ans = new int[n];
int[] vis = new int[n + 1];
Arrays.fill(vis, 1);
int s = 0;
for (int i = 0; i < n; ++i) {
vis[A[i]] ^= 1;
s += vis[A[i]];
vis[B[i]] ^= 1;
s += vis[B[i]];
ans[i] = s;
}
return ans;
}
}
C++
class Solution {
public:
vector<int> findThePrefixCommonArray(vector<int>& A, vector<int>& B) {
int n = A.size();
vector<int> ans;
vector<int> vis(n + 1, 1);
int s = 0;
for (int i = 0; i < n; ++i) {
vis[A[i]] ^= 1;
s += vis[A[i]];
vis[B[i]] ^= 1;
s += vis[B[i]];
ans.push_back(s);
}
return ans;
}
};
Go
func findThePrefixCommonArray(A []int, B []int) (ans []int) {
vis := make([]int, len(A)+1)
for i := range vis {
vis[i] = 1
}
s := 0
for i, a := range A {
b := B[i]
vis[a] ^= 1
s += vis[a]
vis[b] ^= 1
s += vis[b]
ans = append(ans, s)
}
return
}
TypeScript
function findThePrefixCommonArray(A: number[], B: number[]): number[] {
const n = A.length;
const vis: number[] = Array(n + 1).fill(1);
const ans: number[] = [];
let s = 0;
for (let i = 0; i < n; ++i) {
const [a, b] = [A[i], B[i]];
vis[a] ^= 1;
s += vis[a];
vis[b] ^= 1;
s += vis[b];
ans.push(s);
}
return ans;
}
Solution 3: Bit Manipulation (Space Optimization)
Since the elements of arrays $A$ and $B$ are in the range $[1, n]$ and do not exceed $50$, we can use an integer $x$ and an integer $y$ to represent the occurrence of each element in arrays $A$ and $B$, respectively. Specifically, we use the $i$-th bit of integer $x$ to indicate whether element $i$ has appeared in array $A$, and the $i$-th bit of integer $y$ to indicate whether element $i$ has appeared in array $B$.
The time complexity of this solution is $O(n)$, where $n$ is the length of arrays $A$ and $B$. The space complexity is $O(1)$.
Python3
class Solution:
def findThePrefixCommonArray(self, A: List[int], B: List[int]) -> List[int]:
ans = []
x = y = 0
for a, b in zip(A, B):
x |= 1 << a
y |= 1 << b
ans.append((x & y).bit_count())
return ans
Java
class Solution {
public int[] findThePrefixCommonArray(int[] A, int[] B) {
int n = A.length;
int[] ans = new int[n];
long x = 0, y = 0;
for (int i = 0; i < n; i++) {
x |= 1L << A[i];
y |= 1L << B[i];
ans[i] = Long.bitCount(x & y);
}
return ans;
}
}
C++
class Solution {
public:
vector<int> findThePrefixCommonArray(vector<int>& A, vector<int>& B) {
int n = A.size();
vector<int> ans(n);
long long x = 0, y = 0;
for (int i = 0; i < n; ++i) {
x |= (1LL << A[i]);
y |= (1LL << B[i]);
ans[i] = __builtin_popcountll(x & y);
}
return ans;
}
};
Go
func findThePrefixCommonArray(A []int, B []int) []int {
n := len(A)
ans := make([]int, n)
var x, y int
for i := 0; i < n; i++ {
x |= 1 << A[i]
y |= 1 << B[i]
ans[i] = bits.OnesCount(uint(x & y))
}
return ans
}
TypeScript
function findThePrefixCommonArray(A: number[], B: number[]): number[] {
const n = A.length;
const ans: number[] = [];
let [x, y] = [0n, 0n];
for (let i = 0; i < n; i++) {
x |= 1n << BigInt(A[i]);
y |= 1n << BigInt(B[i]);
ans.push(bitCount64(x & y));
}
return ans;
}
function bitCount64(i: bigint): number {
i = i - ((i >> 1n) & 0x5555555555555555n);
i = (i & 0x3333333333333333n) + ((i >> 2n) & 0x3333333333333333n);
i = (i + (i >> 4n)) & 0x0f0f0f0f0f0f0f0fn;
i = i + (i >> 8n);
i = i + (i >> 16n);
i = i + (i >> 32n);
return Number(i & 0x7fn);
}