给定一个数字数组，大小为N和Q。每个查询或范围可以由L(LeftIndex)和R(RightIndex)表示。查找在给定范围内出现偶数次的数字的XOR和。
先决条件：查询给定范围内不同数字的数量。 |用于范围查询的细分树
例子 ：
Input : arr[] = { 1, 2, 1, 3, 3, 2, 3 }
Q = 5
L = 3, R = 6
L = 3, R = 4
L = 0, R = 2
L = 0, R = 6
L = 0, R = 4
Output : 0
3
1
3
2
以上示例说明：
在查询1中，没有数字出现偶数次。
因此，XOR和为0。
在查询2中，{3}出现偶数次。 XOR和为3。
在查询3中，{1}出现偶数次。 XOR和为1。
在查询4中，{1，2}出现偶数次。 XOR-sum是1 xor 2 = 3。
在查询5中，{1，3}出现偶数次。 XOR-sum是1 xor 3 = 2。
段树或二进制索引树可用于有效解决此问题。
方法 ：
首先，很容易注意到查询的答案是查询范围内所有元素的XOR和与查询范围内不同元素的XOR和求和(因为将元素与自身结果的XOR求和空值)。使用前缀XOR-sums查找查询范围内所有数字的XOR-sum。
要查找范围内不同元素的XOR和：给定范围子数组中不同元素的数量。
现在，回到我们的主要问题，只需将BIT [i] = 1更改为BIT [i] = arr i并计算XOR-sum而不是sum。
以下是在CPP中使用二进制索引树的实现
// CPP Program to Find the XOR-sum
// of elements that appeared even
// number of times within a range
#include
using namespace std;
/* structure to store queries
L --> Left Bound of Query
R --> Right Bound of Query
idx --> Query Number */
struct que {
int L, R, idx;
};
// cmp function to sort queries
// according to R
bool cmp(que a, que b)
{
if (a.R != b.R)
return a.R < b.R;
else
return a.L < b.L;
}
/* N --> Number of elements present in
input array. BIT[0..N] --> Array that
represents Binary Indexed Tree*/
// Returns XOR-sum of arr[0..index]. This
// function assumes that the array is
// preprocessed and partial sums of array
// elements are stored in BIT[].
int getSum(int BIT[], int index)
{
// Iniialize result
int xorSum = 0;
// index in BITree[] is 1 more than
// the index in arr[]
index = index + 1;
// Traverse ancestors of BIT[index]
while (index > 0)
{
// Take XOR of current element
// of BIT to xorSum
xorSum ^= BIT[index];
// Move index to parent node
// in getSum View
index -= index & (-index);
}
return xorSum;
}
// Updates a node in Binary Index Tree
// (BIT) at given index in BIT. The
// given value 'val' is xored to BIT[i]
// and all of its ancestors in tree.
void updateBIT(int BIT[], int N,
int index, int val)
{
// index in BITree[] is 1 more than
// the index in arr[]
index = index + 1;
// Traverse all ancestors and
// take xor with 'val'
while (index <= N)
{
// Take xor with 'val' to
// current node of BIT
BIT[index] ^= val;
// Update index to that of
// parent in update View
index += index & (-index);
}
}
// Constructs and returns a Binary Indexed
// Tree for given array of size N.
int* constructBITree(int arr[], int N)
{
// Create and initialize BITree[] as 0
int* BIT = new int[N + 1];
for (int i = 1; i <= N; i++)
BIT[i] = 0;
return BIT;
}
// Function to answer the Queries
void answeringQueries(int arr[], int N,
que queries[], int Q, int BIT[])
{
// Creating an array to calculate
// prefix XOR sums
int* prefixXOR = new int[N + 1];
// map for coordinate compression
// as numbers can be very large but we
// have limited space
map mp;
for (int i = 0; i < N; i++) {
// If A[i] has not appeared yet
if (!mp[arr[i]])
mp[arr[i]] = i;
// calculate prefixXOR sums
if (i == 0)
prefixXOR[i] = arr[i];
else
prefixXOR[i] =
prefixXOR[i - 1] ^ arr[i];
}
// Creating an array to store the
// last occurrence of arr[i]
int lastOcc[1000001];
memset(lastOcc, -1, sizeof(lastOcc));
// sort the queries according to comparator
sort(queries, queries + Q, cmp);
// answer for each query
int res[Q];
// Query Counter
int j = 0;
for (int i = 0; i < Q; i++)
{
while (j <= queries[i].R)
{
// If last visit is not -1 update
// arr[j] to set null by taking
// xor with itself at the idx
// equal lastOcc[mp[arr[j]]]
if (lastOcc[mp[arr[j]]] != -1)
updateBIT(BIT, N,
lastOcc[mp[arr[j]]], arr[j]);
// Setting lastOcc[mp[arr[j]]] as j and
// updating the BIT array accordingly
updateBIT(BIT, N, j, arr[j]);
lastOcc[mp[arr[j]]] = j;
j++;
}
// get the XOR-sum of all elements within
// range using precomputed prefix XORsums
int allXOR = prefixXOR[queries[i].R] ^
prefixXOR[queries[i].L - 1];
// get the XOR-sum of distinct elements
// within range using BIT query function
int distinctXOR = getSum(BIT, queries[i].R) ^
getSum(BIT, queries[i].L - 1);
// store the final answer at the numbered query
res[queries[i].idx] = allXOR ^ distinctXOR;
}
// Output the result
for (int i = 0; i < Q; i++)
cout << res[i] << endl;
}
// Driver program to test above functions
int main()
{
int arr[] = { 1, 2, 1, 3, 3, 2, 3 };
int N = sizeof(arr) / sizeof(arr[0]);
int* BIT = constructBITree(arr, N);
// structure of array for queries
que queries[5];
// Intializing values (L, R, idx) to queries
queries[0].L = 3;
queries[0].R = 6, queries[0].idx = 0;
queries[1].L = 3;
queries[1].R = 4, queries[1].idx = 1;
queries[2].L = 0;
queries[2].R = 2, queries[2].idx = 2;
queries[3].L = 0;
queries[3].R = 6, queries[3].idx = 3;
queries[4].L = 0;
queries[4].R = 4, queries[4].idx = 4;
int Q = sizeof(queries) / sizeof(queries[0]);
// answer Queries
answeringQueries(arr, N, queries, Q, BIT);
return 0;
}
输出：
0
3
1
3
2
时间复杂度： O(Q * Log(N))，其中N是数组的大小，Q是查询的总数。