范围更新查询与二进制数组中的 1 异或。

给定一个大小为N的二进制数组arr[] 。任务是回答Q查询，这些查询可以是以下任何一种类型：
Type 1 – 1 lr ：对从 l 到 r 的所有数组元素执行按位异或运算，值为 1。
类型 2 – 2 lr ：返回子数组 [l, r] 中值为 1 的两个元素之间的最小距离。
类型 3 – 3 lr ：返回子数组 [l, r] 中两个值为 1 的元素之间的最大距离。
类型 4 – 4 lr ：返回子数组 [l, r] 中值为 0 的两个元素之间的最小距离。
类型 5 – 5 lr ：返回子数组 [l, r] 中值为 0 的两个元素之间的最大距离。
例子：

Input : arr[] = {1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0}, q=5 
Output : 2 2 3 2
Explanation :
query 1 : Type 2, l=3, r=7 
Range 3 to 7 contains { 1, 0, 1, 0, 1 }. 
So, the minimum distance between two elements with value 1 is 2.
query 2 : Type 3, l=2, r=5 
Range 2 to 5 contains { 0, 1, 0, 1 }. 
So, the maximum distance between two elements with value 1 is 2.
query 3 : Type 1, l=1, r=4 
After update array becomes {1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0}
query 4 : Type 4, l=3, r=7 
Range 3 to 7 in updated array contains { 0, 1, 1, 0, 1 }. 
So, the minimum distance between two elements with value 0 is 3.
query 5 : Type 5, l=4, r=9 
Range 4 to 9 contains { 1, 1, 0, 1, 0, 1 }. 
So, the maximum distance between two elements with value 0 is 2.
 

方法：
我们将创建一个段树并使用带有延迟传播的范围更新来解决这个问题。

1. 线段树中的每个节点的索引为最左边的 1 和最右边的 1、最左边的 0 和最右边的 0 以及包含子数组 {l, r} 中值为 1 的任何元素之间的最大和最小距离的整数作为子数组 {l, r} 中任何值为 0 的元素之间的最大和最小距离。
2. 现在，在这个线段树中，我们可以合并左右节点，如下所示：

CPP

// l1 = leftmost index of 1, l0 = leftmost index of 0.
// r1 = rightmost index of 1, r0 = rightmost index of 0.
// max1 = maximum distance between two 1’s.
// max0 = maximum distance between two 0’s.
// min1 = minimum distance between two 1’s.
// min0 = minimum distance between two 0’s.
node Merge(node left, node right)
{
    node cur;
     
    if left.l0 is valid
        cur.l0 = left.l0
    else
        cur.l0 = r.l0
    // We will do this for all values
    // i.e. cur.r0, cur.l1, cur.r1, cur.l0
     
    // To find the min and max difference between two 1's and 0's
    // we will take min/max value of left side, right side and
    // difference between rightmost index of 1/0 in right node
    // and leftmost index of 1/0 in left node respectively.
         
     cur.min0 = minimum of left.min0 and right.min0
  
     if left.r0 is valid and right.l0 is valid
        cur.min0 = minimum of cur.min0 and (right.l0 - left.r0)
    // We will do this for all max/min values
    // i.e. cur.min0, cur.min1, cur.max1, cur.max0
         
    return cur;
}

2. 为了处理范围更新查询，我们将使用延迟传播。更新查询要求我们对从 l 到 r 范围内的所有元素与 1 进行异或运算，根据观察，我们知道：

0 xor 1 = 1
       1 xor 1 = 0

1. 因此，我们可以观察到，在这次更新之后，所有的 0 将变为 1，所有的 1 将变为 0。因此，在我们的线段树节点中，0 和 1 的所有对应值也将被交换，即

l0 and l1 will get swapped
       r0 and r1 will get swapped
       min0 and min1 will get swapped
       max0 and max1 will get swapped

2. 然后，最后为了找到任务 2、3、4 和 5 的答案，我们只需要调用给定范围 {l, r} 的查询函数，我为了找到任务 1 的答案，我们需要调用范围更新函数.

下面是上述方法的实现：

CPP

// C++ program for the given problem
#include 
using namespace std;
 
int lazy[100001];
 
// Class for each node
// in the segment tree
class node {
public:
    int l1, r1, l0, r0;
    int min0, max0, min1, max1;
 
    node()
    {
        l1 = r1 = l0 = r0 = -1;
 
        max1 = max0 = INT_MIN;
        min1 = min0 = INT_MAX;
    }
 
} seg[100001];
 
// A utility function for
// merging two nodes
node MergeUtil(node l, node r)
{
    node x;
 
    x.l0 = (l.l0 != -1) ? l.l0 : r.l0;
    x.r0 = (r.r0 != -1) ? r.r0 : l.r0;
 
    x.l1 = (l.l1 != -1) ? l.l1 : r.l1;
    x.r1 = (r.r1 != -1) ? r.r1 : l.r1;
 
    x.min0 = min(l.min0, r.min0);
    if (l.r0 != -1 && r.l0 != -1)
        x.min0 = min(x.min0, r.l0 - l.r0);
 
    x.min1 = min(l.min1, r.min1);
    if (l.r1 != -1 && r.l1 != -1)
        x.min1 = min(x.min1, r.l1 - l.r1);
 
    x.max0 = max(l.max0, r.max0);
    if (l.l0 != -1 && r.r0 != -1)
        x.max0 = max(x.max0, r.r0 - l.l0);
 
    x.max1 = max(l.max1, r.max1);
    if (l.l1 != -1 && r.r1 != -1)
        x.max1 = max(x.max1, r.r1 - l.l1);
 
    return x;
}
 
// utility function
// for updating a node
node UpdateUtil(node x)
{
    swap(x.l0, x.l1);
    swap(x.r0, x.r1);
    swap(x.min1, x.min0);
    swap(x.max0, x.max1);
 
    return x;
}
 
// A recursive function that constructs
// Segment Tree for given string
void Build(int qs, int qe, int ind, int arr[])
{
    // If start is equal to end then
    // insert the array element
    if (qs == qe) {
        if (arr[qs] == 1) {
            seg[ind].l1 = seg[ind].r1 = qs;
        }
        else {
            seg[ind].l0 = seg[ind].r0 = qs;
        }
 
        lazy[ind] = 0;
        return;
    }
    int mid = (qs + qe) >> 1;
 
    // Build the segment tree
    // for range qs to mid
    Build(qs, mid, ind << 1, arr);
 
    // Build the segment tree
    // for range mid+1 to qe
    Build(mid + 1, qe, ind << 1 | 1, arr);
 
    // merge the two child nodes
    // to obtain the parent node
    seg[ind] = MergeUtil(
        seg[ind << 1],
        seg[ind << 1 | 1]);
}
 
// Query in a range qs to qe
node Query(int qs, int qe,
           int ns, int ne, int ind)
{
    if (lazy[ind] != 0) {
        seg[ind] = UpdateUtil(seg[ind]);
        if (ns != ne) {
            lazy[ind * 2] ^= lazy[ind];
            lazy[ind * 2 + 1] ^= lazy[ind];
        }
        lazy[ind] = 0;
    }
 
    node x;
 
    // If the range lies in this segment
    if (qs <= ns && qe >= ne)
        return seg[ind];
 
    // If the range is out of the bounds
    // of this segment
    if (ne < qs || ns > qe || ns > ne)
        return x;
 
    // Else query for the right and left
    // child node of this subtree
    // and merge them
    int mid = (ns + ne) >> 1;
 
    node l = Query(qs, qe, ns,
                   mid, ind << 1);
    node r = Query(qs, qe,
                   mid + 1, ne,
                   ind << 1 | 1);
 
    x = MergeUtil(l, r);
    return x;
}
 
// range update using lazy prpagation
void RangeUpdate(int us, int ue,
                 int ns, int ne, int ind)
{
    if (lazy[ind] != 0) {
        seg[ind] = UpdateUtil(seg[ind]);
        if (ns != ne) {
            lazy[ind * 2] ^= lazy[ind];
            lazy[ind * 2 + 1] ^= lazy[ind];
        }
        lazy[ind] = 0;
    }
 
    // If the range is out of the bounds
    // of this segment
    if (ns > ne || ns > ue || ne < us)
        return;
 
    // If the range lies in this segment
    if (ns >= us && ne <= ue) {
        seg[ind] = UpdateUtil(seg[ind]);
        if (ns != ne) {
            lazy[ind * 2] ^= 1;
            lazy[ind * 2 + 1] ^= 1;
        }
        return;
    }
 
    // Else query for the right and left
    // child node of this subtree
    // and merge them
    int mid = (ns + ne) >> 1;
    RangeUpdate(us, ue, ns, mid, ind << 1);
    RangeUpdate(us, ue, mid + 1, ne, ind << 1 | 1);
 
    node l = seg[ind << 1], r = seg[ind << 1 | 1];
    seg[ind] = MergeUtil(l, r);
}
 
// Driver code
int main()
{
 
    int arr[] = { 1, 1, 0,
                  1, 0, 1,
                  0, 1, 0,
                  1, 0, 1,
                  1, 0 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    // Build the segment tree
    Build(0, n - 1, 1, arr);
 
    // Query of Type 2 in the range 3 to 7
    node ans = Query(3, 7, 0, n - 1, 1);
    cout << ans.min1 << "\n";
 
    // Query of Type 3 in the range 2 to 5
    ans = Query(2, 5, 0, n - 1, 1);
    cout << ans.max1 << "\n";
 
    // Query of Type 1 in the range 1 to 4
    RangeUpdate(1, 4, 0, n - 1, 1);
 
    // Query of Type 4 in the range 3 to 7
    ans = Query(3, 7, 0, n - 1, 1);
    cout << ans.min0 << "\n";
 
    // Query of Type 5 in the range 4 to 9
    ans = Query(4, 9, 0, n - 1, 1);
    cout << ans.max0 << "\n";
 
    return 0;
}

蟒蛇3

# Python program for the given problem
from sys import maxsize
from typing import List
INT_MAX = maxsize
INT_MIN = -maxsize
lazy = [0 for _ in range(100001)]
 
# Class for each node
# in the segment tree
class node:
    def __init__(self) -> None:
        self.l1 = self.r1 = self.l0 = self.r0 = -1
        self.max0 = self.max1 = INT_MIN
        self.min0 = self.min1 = INT_MAX
 
seg = [node() for _ in range(100001)]
 
# A utility function for
# merging two nodes
def MergeUtil(l: node, r: node) -> node:
    x = node()
 
    x.l0 = l.l0 if (l.l0 != -1) else r.l0
    x.r0 = r.r0 if (r.r0 != -1) else l.r0
 
    x.l1 = l.l1 if (l.l1 != -1) else r.l1
    x.r1 = r.r1 if (r.r1 != -1) else l.r1
 
    x.min0 = min(l.min0, r.min0)
    if (l.r0 != -1 and r.l0 != -1):
        x.min0 = min(x.min0, r.l0 - l.r0)
 
    x.min1 = min(l.min1, r.min1)
    if (l.r1 != -1 and r.l1 != -1):
        x.min1 = min(x.min1, r.l1 - l.r1)
 
    x.max0 = max(l.max0, r.max0)
    if (l.l0 != -1 and r.r0 != -1):
        x.max0 = max(x.max0, r.r0 - l.l0)
 
    x.max1 = max(l.max1, r.max1)
    if (l.l1 != -1 and r.r1 != -1):
        x.max1 = max(x.max1, r.r1 - l.l1)
 
    return x
 
# utility function
# for updating a node
def UpdateUtil(x: node) -> node:
    x.l0, x.l1 = x.l1, x.l0
    x.r0, x.r1 = x.r1, x.r0
    x.min1, x.min0 = x.min0, x.min1
    x.max0, x.max1 = x.max1, x.max0
 
    return x
 
# A recursive function that constructs
# Segment Tree for given string
def Build(qs: int, qe: int, ind: int, arr: List[int]) -> None:
 
  # If start is equal to end then
    # insert the array element
    if (qs == qe):
        if (arr[qs] == 1):
            seg[ind].l1 = seg[ind].r1 = qs
        else:
            seg[ind].l0 = seg[ind].r0 = qs
 
        lazy[ind] = 0
        return
 
    mid = (qs + qe) >> 1
 
    # Build the segment tree
    # for range qs to mid
    Build(qs, mid, ind << 1, arr)
 
    # Build the segment tree
    # for range mid+1 to qe
    Build(mid + 1, qe, ind << 1 | 1, arr)
 
    # merge the two child nodes
    # to obtain the parent node
    seg[ind] = MergeUtil(seg[ind << 1], seg[ind << 1 | 1])
 
# Query in a range qs to qe
def Query(qs: int, qe: int, ns: int, ne: int, ind: int) -> node:
    if (lazy[ind] != 0):
        seg[ind] = UpdateUtil(seg[ind])
        if (ns != ne):
            lazy[ind * 2] ^= lazy[ind]
            lazy[ind * 2 + 1] ^= lazy[ind]
        lazy[ind] = 0
    x = node()
 
    # If the range lies in this segment
    if (qs <= ns and qe >= ne):
        return seg[ind]
 
    # If the range is out of the bounds
    # of this segment
    if (ne < qs or ns > qe or ns > ne):
        return x
 
    # Else query for the right and left
    # child node of this subtree
    # and merge them
    mid = (ns + ne) >> 1
    l = Query(qs, qe, ns, mid, ind << 1)
    r = Query(qs, qe, mid + 1, ne, ind << 1 | 1)
    x = MergeUtil(l, r)
    return x
 
# range update using lazy prpagation
def RangeUpdate(us: int, ue: int, ns: int, ne: int, ind: int) -> None:
    if (lazy[ind] != 0):
        seg[ind] = UpdateUtil(seg[ind])
        if (ns != ne):
            lazy[ind * 2] ^= lazy[ind]
            lazy[ind * 2 + 1] ^= lazy[ind]
        lazy[ind] = 0
 
    # If the range is out of the bounds
    # of this segment
    if (ns > ne or ns > ue or ne < us):
        return
 
    # If the range lies in this segment
    if (ns >= us and ne <= ue):
        seg[ind] = UpdateUtil(seg[ind])
        if (ns != ne):
            lazy[ind * 2] ^= 1
            lazy[ind * 2 + 1] ^= 1
        return
 
    # Else query for the right and left
    # child node of this subtree
    # and merge them
    mid = (ns + ne) >> 1
    RangeUpdate(us, ue, ns, mid, ind << 1)
    RangeUpdate(us, ue, mid + 1, ne, ind << 1 | 1)
    l = seg[ind << 1]
    r = seg[ind << 1 | 1]
    seg[ind] = MergeUtil(l, r)
 
# Driver code
if __name__ == "__main__":
    arr = [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0]
    n = len(arr)
 
    # Build the segment tree
    Build(0, n - 1, 1, arr)
 
    # Query of Type 2 in the range 3 to 7
    ans = Query(3, 7, 0, n - 1, 1)
    print(ans.min1)
 
    # Query of Type 3 in the range 2 to 5
    ans = Query(2, 5, 0, n - 1, 1)
    print(ans.max1)
 
    # Query of Type 1 in the range 1 to 4
    RangeUpdate(1, 4, 0, n - 1, 1)
 
    # Query of Type 4 in the range 3 to 7
    ans = Query(3, 7, 0, n - 1, 1)
    print(ans.min0)
 
    # Query of Type 5 in the range 4 to 9
    ans = Query(4, 9, 0, n - 1, 1)
    print(ans.max0)
 
# This code is contributed by sanjeev2552

输出：

2
2
3
2

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