给定一个数组“ arr”，其所有数字都初始化为“ 0”，则可以通过以下方式随时更新该数组：
数组中任何子数组的所有元素都可以翻转，即’1’=>’0’或’0’=>’1’。
任务是从任何传入查询[l，r]的数组中查找1的数目，其中“ l”和“ r”是给定数组中的有效索引。
例子：
Input: n = 7
arr = {0, 0, 0, 0, 0, 0, 0}
Update from index '2' to '5', {0, 0, 1, 1, 1, 1, 0}
Query from index '1' to '6'
Update from index '1' to '3', {0, 1, 0, 0, 1, 1, 0}
Query from index '1' to '4'
Output:
4
2
Input: n = 5
arr = {0, 0, 0, 0, 0}
Update from index '0' to '2', {1, 1, 1, 0, 0}
Query from index '2' to '4'
Update from index '2' to '4', {1, 1, 0, 1, 1}
Query from index '0' to '3'
Output:
1
3方法：可以使用段树解决此问题，您可以在此处阅读有关段树的更多信息。
- 在所有节点上用值“ 0”初始化段树。
- 在任何更新中，都使用更新函数仅将更新的值存储在该索引处，以便可以在O(Log n)时间执行查询，这被称为“延迟传播方法”。
- 在构建段树时，请使用合并函数，例如：
- 如果左右子数组中存在待处理的更新，则分别按leftSub.end – leftSub.start +1 – leftSub.count和rightSub.end – rightSub.start +1 – rightSub.count递增计数。
- 否则，添加leftSub.count和rightSub.count。
下面是上述方法的实现：
C++
// C++ implementation of the approach
#include
// Structure which contains
// all the necessary functions
// to solve a segment tree
struct SegmentTemplate {
int start, end;
int count = 0;
bool pendingUpdate = false;
// The function which
// contains merge logic
void merge(SegmentTemplate& leftSub,
SegmentTemplate& rightSub)
{
// If the left subarray
// contains some pending update
if (leftSub.pendingUpdate)
count += leftSub.end - leftSub.start +
1 - leftSub.count;
else
count += leftSub.count;
// If the right subarray
// contains some pending update
if (rightSub.pendingUpdate)
count += rightSub.end - rightSub.start +
1 - rightSub.count;
else
count += rightSub.count;
}
// To return the count
int query()
{
return count;
}
// If the segment tree
// has pending update
bool hasPendingUpdate()
{
return pendingUpdate;
}
// Apply pending update
void applyPendingUpdate()
{
count = end - start + 1 - count;
pendingUpdate = false;
}
// For a node to
// add pending update
void addUpdate()
{
pendingUpdate = !pendingUpdate;
}
};
// The class that performs
// all the segment tree functions
class SegmentTree {
SegmentTemplate* treeBuild;
int N;
public:
SegmentTree(int N)
{
this->N = N;
treeBuild = new SegmentTemplate[getSegmentTreeSize(N)];
buildTree(1, 0, N - 1);
}
// For the query
int query(int start, int end)
{
// Stores the result
SegmentTemplate result
= query(1, start, end);
return result.query();
}
// Updates at the
// specific location
void update(int start, int end)
{
update(1, start, end);
}
private:
void buildTree(int stIndex, int start, int end)
{
// Defining index in treeBuild
treeBuild[stIndex].start
= start,
treeBuild[stIndex].end = end;
if (start == end) {
return;
}
// Dividing the array into two
// parts for assigning the indices
// and building segment tree
int mid = (start + end) / 2,
leftChildIndex = 2 * stIndex,
rightChildIndex = leftChildIndex + 1;
buildTree(leftChildIndex, start, mid);
buildTree(rightChildIndex, mid + 1, end);
// Combine the array
treeBuild[stIndex]
.merge(treeBuild[leftChildIndex],
treeBuild[rightChildIndex]);
}
// Gets the segment tree size
int getSegmentTreeSize(int N)
{
int size = 1;
for (; size < N; size <<= 1)
;
return size << 1;
}
// Function for getting
// to know the queries
SegmentTemplate query(int stIndex, int start, int end)
{
// When reaches a particular index
if (treeBuild[stIndex].start == start
&& treeBuild[stIndex].end == end) {
SegmentTemplate result = treeBuild[stIndex];
if (result.hasPendingUpdate())
result.applyPendingUpdate();
return result;
}
// Dividing the array into two
// parts for assigning the indices
// and querying segment trees
int mid = (treeBuild[stIndex].start
+ treeBuild[stIndex].end)
/ 2,
leftChildIndex = stIndex * 2,
rightChildIndex = leftChildIndex + 1;
SegmentTemplate result;
if (start > mid)
result = query(rightChildIndex, start, end);
else if (end <= mid)
result = query(leftChildIndex, start, end);
else {
SegmentTemplate leftResult = query(leftChildIndex, start, mid),
rightResult = query(rightChildIndex, mid + 1, end);
result.start = leftResult.start;
result.end = rightResult.end;
result.merge(leftResult, rightResult);
}
// If tree has update,
// apply the pending update
if (treeBuild[stIndex].hasPendingUpdate()) {
result.addUpdate();
result.applyPendingUpdate();
}
return result;
}
// Function for updating the tree
void update(int stIndex, int start, int end)
{
// Adds the update in O(1)
// time to the node
if (treeBuild[stIndex].start == start
&& treeBuild[stIndex].end == end) {
treeBuild[stIndex].addUpdate();
return;
}
// Dividing the array into two
// parts for assigning the indices
// and querying segment trees
int mid = (treeBuild[stIndex].start
+ treeBuild[stIndex].end)
/ 2,
leftChildIndex = stIndex * 2,
rightChildIndex = leftChildIndex + 1;
if (start > mid)
update(rightChildIndex, start, end);
else if (end <= mid)
update(leftChildIndex, start, end);
else {
update(leftChildIndex, start, mid);
update(rightChildIndex, mid + 1, end);
}
// Calling the build function
// for building the segment tree
treeBuild[stIndex]
.merge(treeBuild[leftChildIndex],
treeBuild[rightChildIndex]);
}
};
// Driver function
int main()
{
int N = 7;
SegmentTree st(N);
// Updating the segment tree
st.update(2, 5);
// Executing the query
printf("%d\n", st.query(1, 6));
// Updating the segment tree
st.update(1, 3);
// Executing the query
printf("%d\n", st.query(1, 6));
return 0;
} Python3
# Python3 implementation of the approach
# Structure which contains
# all the necessary functions
# to solve a segment tree
class SegmentTemplate:
def __init__(self) -> None:
self.start = 0
self.end = 0
self.count = 0
self.pendingUpdate = False
# The function which
# contains merge logic
def merge(self, leftSub, rightSub) -> None:
# If the left subarray
# contains some pending update
if (leftSub.pendingUpdate):
self.count += (leftSub.end - leftSub.start +
1 - leftSub.count)
else:
self.count += leftSub.count
# If the right subarray
# contains some pending update
if (rightSub.pendingUpdate):
self.count += (rightSub.end - rightSub.start +
1 - rightSub.count)
else:
self.count += rightSub.count
# To return the count
def query(self) -> int:
return self.count
# If the segment tree
# has pending update
def hasPendingUpdate(self) -> bool:
return self.pendingUpdate
# Apply pending update
def applyPendingUpdate(self) -> None:
self.count = self.end - self.start + 1 - self.count
self.pendingUpdate = False
# For a node to
# add pending update
def addUpdate(self) -> None:
self.pendingUpdate = not self.pendingUpdate
# The class that performs
# all the segment tree functions
class SegmentTree:
def __init__(self, N) -> None:
self.treeBuild = [SegmentTemplate() for _ in range(
self._getSegmentTreeSize(N))]
self.N = N
self._buildTree(1, 0, N - 1)
# For the query
def query(self, start: int, end: int) -> int:
# Stores the result
result = self._query(1, start, end)
return result.query()
# Updates at the
# specific location
def update(self, start: int, end: int) -> None:
self._update(1, start, end)
def _buildTree(self, stIndex: int, start: int, end: int) -> None:
# Defining index in treeBuild
self.treeBuild[stIndex].start = start
self.treeBuild[stIndex].end = end
if (start == end):
return
# Dividing the array into two
# parts for assigning the indices
# and building segment tree
mid = (start + end) // 2
leftChildIndex = 2 * stIndex
rightChildIndex = leftChildIndex + 1
self._buildTree(leftChildIndex, start, mid)
self._buildTree(rightChildIndex, mid + 1, end)
# Combine the array
self.treeBuild[stIndex].merge(self.treeBuild[leftChildIndex],
self.treeBuild[rightChildIndex])
# Gets the segment tree size
def _getSegmentTreeSize(self, N: int) -> int:
size = 1
while size < N:
size <<= 1
return size << 1
# Function for getting
# to know the queries
def _query(self, stIndex: int,
start: int, end: int) -> SegmentTemplate:
# When reaches a particular index
if (self.treeBuild[stIndex].start == start and
self.treeBuild[stIndex].end == end):
result = self.treeBuild[stIndex]
if (result.hasPendingUpdate()):
result.applyPendingUpdate()
return result
# Dividing the array into two
# parts for assigning the indices
# and querying segment trees
mid = (self.treeBuild[stIndex].start +
self.treeBuild[stIndex].end) // 2
leftChildIndex = stIndex * 2
rightChildIndex = leftChildIndex + 1
result = SegmentTemplate()
if (start > mid):
result = self._query(rightChildIndex, start, end)
elif (end <= mid):
result = self._query(leftChildIndex, start, end)
else:
leftResult = self._query(leftChildIndex, start, mid)
rightResult = self._query(rightChildIndex, mid + 1, end)
result.start = leftResult.start
result.end = rightResult.end
result.merge(leftResult, rightResult)
# If tree has update,
# apply the pending update
if (self.treeBuild[stIndex].hasPendingUpdate()):
result.addUpdate()
result.applyPendingUpdate()
return result
# Function for updating the tree
def _update(self, stIndex: int, start: int, end: int) -> None:
# Adds the update in O(1)
# time to the node
if (self.treeBuild[stIndex].start == start and
self.treeBuild[stIndex].end == end):
self.treeBuild[stIndex].addUpdate()
return
# Dividing the array into two
# parts for assigning the indices
# and querying segment trees
mid = (self.treeBuild[stIndex].start +
self.treeBuild[stIndex].end) // 2
leftChildIndex = stIndex * 2
rightChildIndex = leftChildIndex + 1
if (start > mid):
self._update(rightChildIndex, start, end)
elif (end <= mid):
self._update(leftChildIndex, start, end)
else:
self._update(leftChildIndex, start, mid)
self._update(rightChildIndex, mid + 1, end)
# Calling the build function
# for building the segment tree
self.treeBuild[stIndex].merge(self.treeBuild[leftChildIndex],
self.treeBuild[rightChildIndex])
# Driver code
if __name__ == "__main__":
N = 7
st = SegmentTree(N)
# Updating the segment tree
st.update(2, 5)
# Executing the query
print("{}".format(st.query(1, 6)))
# Updating the segment tree
st.update(1, 3)
# Executing the query
print("{}".format(st.query(1, 6)))
# This code is contributed by sanjeev2552输出：
4
3