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2569. Handling Sum Queries After Update

Description

You are given two 0-indexed arrays nums1 and nums2 and a 2D array queries of queries. There are three types of queries:

  1. For a query of type 1, queries[i] = [1, l, r]. Flip the values from 0 to 1 and from 1 to 0 in nums1 from index l to index r. Both l and r are 0-indexed.
  2. For a query of type 2, queries[i] = [2, p, 0]. For every index 0 <= i < n, set nums2[i] = nums2[i] + nums1[i] * p.
  3. For a query of type 3, queries[i] = [3, 0, 0]. Find the sum of the elements in nums2.

Return an array containing all the answers to the third type queries.

 

Example 1:

Input: nums1 = [1,0,1], nums2 = [0,0,0], queries = [[1,1,1],[2,1,0],[3,0,0]]
Output: [3]
Explanation: After the first query nums1 becomes [1,1,1]. After the second query, nums2 becomes [1,1,1], so the answer to the third query is 3. Thus, [3] is returned.

Example 2:

Input: nums1 = [1], nums2 = [5], queries = [[2,0,0],[3,0,0]]
Output: [5]
Explanation: After the first query, nums2 remains [5], so the answer to the second query is 5. Thus, [5] is returned.

 

Constraints:

  • 1 <= nums1.length,nums2.length <= 105
  • nums1.length = nums2.length
  • 1 <= queries.length <= 105
  • queries[i].length = 3
  • 0 <= l <= r <= nums1.length - 1
  • 0 <= p <= 106
  • 0 <= nums1[i] <= 1
  • 0 <= nums2[i] <= 109

Solutions

Solution 1: Segment Tree

According to the problem description:

  • Operation \(1\) is to reverse all numbers in the index range \([l,..r]\) of array nums1, that is, change \(0\) to \(1\) and \(1\) to \(0\).
  • Operation \(3\) is to sum all numbers in array nums2.
  • Operation \(2\) is to add the sum of all numbers in array nums2 with \(p\) times the sum of all numbers in array nums1, that is, \(sum(nums2) = sum(nums2) + p * sum(nums1)\).

Therefore, we actually only need to maintain the segment sum of array nums1, which can be implemented through a segment tree.

We define each node of the segment tree as Node, each node contains the following attributes:

  • l: The left endpoint of the node, the index starts from \(1\).
  • r: The right endpoint of the node, the index starts from \(1\).
  • s: The segment sum of the node.
  • lazy: The lazy tag of the node.

The segment tree mainly has the following operations:

  • build(u, l, r): Build the segment tree.
  • pushdown(u): Propagate the lazy tag.
  • pushup(u): Update the information of the parent node with the information of the child nodes.
  • modify(u, l, r): Modify the segment sum. In this problem, it is to reverse each number in the segment, so the segment sum \(s = r - l + 1 - s\).
  • query(u, l, r): Query the segment sum.

First, calculate the sum of all numbers in array nums2, denoted as \(s\).

When executing operation \(1\), we only need to call modify(1, l + 1, r + 1).

When executing operation \(2\), we update \(s = s + p \times query(1, 1, n)\).

When executing operation \(3\), we just need to add \(s\) to the answer array.

The time complexity is \(O(n + m \times \log n)\), and the space complexity is \(O(n)\). Where \(n\) and \(m\) are the lengths of arrays nums1 and queries respectively.

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class Node:
    def __init__(self):
        self.l = self.r = 0
        self.s = self.lazy = 0


class SegmentTree:
    def __init__(self, nums):
        self.nums = nums
        n = len(nums)
        self.tr = [Node() for _ in range(n << 2)]
        self.build(1, 1, n)

    def build(self, u, l, r):
        self.tr[u].l, self.tr[u].r = l, r
        if l == r:
            self.tr[u].s = self.nums[l - 1]
            return
        mid = (l + r) >> 1
        self.build(u << 1, l, mid)
        self.build(u << 1 | 1, mid + 1, r)
        self.pushup(u)

    def modify(self, u, l, r):
        if self.tr[u].l >= l and self.tr[u].r <= r:
            self.tr[u].lazy ^= 1
            self.tr[u].s = self.tr[u].r - self.tr[u].l + 1 - self.tr[u].s
            return
        self.pushdown(u)
        mid = (self.tr[u].l + self.tr[u].r) >> 1
        if l <= mid:
            self.modify(u << 1, l, r)
        if r > mid:
            self.modify(u << 1 | 1, l, r)
        self.pushup(u)

    def query(self, u, l, r):
        if self.tr[u].l >= l and self.tr[u].r <= r:
            return self.tr[u].s
        self.pushdown(u)
        mid = (self.tr[u].l + self.tr[u].r) >> 1
        res = 0
        if l <= mid:
            res += self.query(u << 1, l, r)
        if r > mid:
            res += self.query(u << 1 | 1, l, r)
        return res

    def pushup(self, u):
        self.tr[u].s = self.tr[u << 1].s + self.tr[u << 1 | 1].s

    def pushdown(self, u):
        if self.tr[u].lazy:
            mid = (self.tr[u].l + self.tr[u].r) >> 1
            self.tr[u << 1].s = mid - self.tr[u].l + 1 - self.tr[u << 1].s
            self.tr[u << 1].lazy ^= 1
            self.tr[u << 1 | 1].s = self.tr[u].r - mid - self.tr[u << 1 | 1].s
            self.tr[u << 1 | 1].lazy ^= 1
            self.tr[u].lazy ^= 1


class Solution:
    def handleQuery(
        self, nums1: List[int], nums2: List[int], queries: List[List[int]]
    ) -> List[int]:
        tree = SegmentTree(nums1)
        s = sum(nums2)
        ans = []
        for op, a, b in queries:
            if op == 1:
                tree.modify(1, a + 1, b + 1)
            elif op == 2:
                s += a * tree.query(1, 1, len(nums1))
            else:
                ans.append(s)
        return ans

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class Node {
    int l, r;
    int s, lazy;
}

class SegmentTree {
    private Node[] tr;
    private int[] nums;

    public SegmentTree(int[] nums) {
        int n = nums.length;
        this.nums = nums;
        tr = new Node[n << 2];
        for (int i = 0; i < tr.length; ++i) {
            tr[i] = new Node();
        }
        build(1, 1, n);
    }

    private void build(int u, int l, int r) {
        tr[u].l = l;
        tr[u].r = r;
        if (l == r) {
            tr[u].s = nums[l - 1];
            return;
        }
        int mid = (l + r) >> 1;
        build(u << 1, l, mid);
        build(u << 1 | 1, mid + 1, r);
        pushup(u);
    }

    public void modify(int u, int l, int r) {
        if (tr[u].l >= l && tr[u].r <= r) {
            tr[u].lazy ^= 1;
            tr[u].s = tr[u].r - tr[u].l + 1 - tr[u].s;
            return;
        }
        pushdown(u);
        int mid = (tr[u].l + tr[u].r) >> 1;
        if (l <= mid) {
            modify(u << 1, l, r);
        }
        if (r > mid) {
            modify(u << 1 | 1, l, r);
        }
        pushup(u);
    }

    public int query(int u, int l, int r) {
        if (tr[u].l >= l && tr[u].r <= r) {
            return tr[u].s;
        }
        pushdown(u);
        int mid = (tr[u].l + tr[u].r) >> 1;
        int res = 0;
        if (l <= mid) {
            res += query(u << 1, l, r);
        }
        if (r > mid) {
            res += query(u << 1 | 1, l, r);
        }
        return res;
    }

    private void pushup(int u) {
        tr[u].s = tr[u << 1].s + tr[u << 1 | 1].s;
    }

    private void pushdown(int u) {
        if (tr[u].lazy == 1) {
            int mid = (tr[u].l + tr[u].r) >> 1;
            tr[u << 1].s = mid - tr[u].l + 1 - tr[u << 1].s;
            tr[u << 1].lazy ^= 1;
            tr[u << 1 | 1].s = tr[u].r - mid - tr[u << 1 | 1].s;
            tr[u << 1 | 1].lazy ^= 1;
            tr[u].lazy ^= 1;
        }
    }
}

class Solution {
    public long[] handleQuery(int[] nums1, int[] nums2, int[][] queries) {
        SegmentTree tree = new SegmentTree(nums1);
        long s = 0;
        for (int x : nums2) {
            s += x;
        }
        int m = 0;
        for (var q : queries) {
            if (q[0] == 3) {
                ++m;
            }
        }
        long[] ans = new long[m];
        int i = 0;
        for (var q : queries) {
            if (q[0] == 1) {
                tree.modify(1, q[1] + 1, q[2] + 1);
            } else if (q[0] == 2) {
                s += 1L * q[1] * tree.query(1, 1, nums2.length);
            } else {
                ans[i++] = s;
            }
        }
        return ans;
    }
}

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class Node {
public:
    int l = 0, r = 0;
    int s = 0, lazy = 0;
};

class SegmentTree {
public:
    SegmentTree(vector<int>& nums) {
        this->nums = nums;
        int n = nums.size();
        tr.resize(n << 2);
        for (int i = 0; i < tr.size(); ++i) {
            tr[i] = new Node();
        }
        build(1, 1, n);
    }

    void modify(int u, int l, int r) {
        if (tr[u]->l >= l && tr[u]->r <= r) {
            tr[u]->lazy ^= 1;
            tr[u]->s = tr[u]->r - tr[u]->l + 1 - tr[u]->s;
            return;
        }
        pushdown(u);
        int mid = (tr[u]->l + tr[u]->r) >> 1;
        if (l <= mid) {
            modify(u << 1, l, r);
        }
        if (r > mid) {
            modify(u << 1 | 1, l, r);
        }
        pushup(u);
    }

    int query(int u, int l, int r) {
        if (tr[u]->l >= l && tr[u]->r <= r) {
            return tr[u]->s;
        }
        pushdown(u);
        int mid = (tr[u]->l + tr[u]->r) >> 1;
        int res = 0;
        if (l <= mid) {
            res += query(u << 1, l, r);
        }
        if (r > mid) {
            res += query(u << 1 | 1, l, r);
        }
        return res;
    }

private:
    vector<Node*> tr;
    vector<int> nums;

    void build(int u, int l, int r) {
        tr[u]->l = l;
        tr[u]->r = r;
        if (l == r) {
            tr[u]->s = nums[l - 1];
            return;
        }
        int mid = (l + r) >> 1;
        build(u << 1, l, mid);
        build(u << 1 | 1, mid + 1, r);
        pushup(u);
    }

    void pushup(int u) {
        tr[u]->s = tr[u << 1]->s + tr[u << 1 | 1]->s;
    }

    void pushdown(int u) {
        if (tr[u]->lazy) {
            int mid = (tr[u]->l + tr[u]->r) >> 1;
            tr[u << 1]->s = mid - tr[u]->l + 1 - tr[u << 1]->s;
            tr[u << 1]->lazy ^= 1;
            tr[u << 1 | 1]->s = tr[u]->r - mid - tr[u << 1 | 1]->s;
            tr[u << 1 | 1]->lazy ^= 1;
            tr[u]->lazy ^= 1;
        }
    }
};

class Solution {
public:
    vector<long long> handleQuery(vector<int>& nums1, vector<int>& nums2, vector<vector<int>>& queries) {
        SegmentTree* tree = new SegmentTree(nums1);
        long long s = 0;
        for (int& x : nums2) {
            s += x;
        }
        vector<long long> ans;
        for (auto& q : queries) {
            if (q[0] == 1) {
                tree->modify(1, q[1] + 1, q[2] + 1);
            } else if (q[0] == 2) {
                s += 1LL * q[1] * tree->query(1, 1, nums1.size());
            } else {
                ans.push_back(s);
            }
        }
        return ans;
    }
};

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type node struct {
    l, r, s, lazy int
}

type segmentTree struct {
    nums []int
    tr   []*node
}

func newSegmentTree(nums []int) *segmentTree {
    n := len(nums)
    tr := make([]*node, n<<2)
    for i := range tr {
        tr[i] = &node{}
    }
    t := &segmentTree{nums, tr}
    t.build(1, 1, n)
    return t
}

func (t *segmentTree) build(u, l, r int) {
    t.tr[u].l, t.tr[u].r = l, r
    if l == r {
        t.tr[u].s = t.nums[l-1]
        return
    }
    mid := (l + r) >> 1
    t.build(u<<1, l, mid)
    t.build(u<<1|1, mid+1, r)
    t.pushup(u)
}

func (t *segmentTree) modify(u, l, r int) {
    if t.tr[u].l >= l && t.tr[u].r <= r {
        t.tr[u].lazy ^= 1
        t.tr[u].s = t.tr[u].r - t.tr[u].l + 1 - t.tr[u].s
        return
    }
    t.pushdown(u)
    mid := (t.tr[u].l + t.tr[u].r) >> 1
    if l <= mid {
        t.modify(u<<1, l, r)
    }
    if r > mid {
        t.modify(u<<1|1, l, r)
    }
    t.pushup(u)
}

func (t *segmentTree) query(u, l, r int) int {
    if t.tr[u].l >= l && t.tr[u].r <= r {
        return t.tr[u].s
    }
    t.pushdown(u)
    mid := (t.tr[u].l + t.tr[u].r) >> 1
    res := 0
    if l <= mid {
        res += t.query(u<<1, l, r)
    }
    if r > mid {
        res += t.query(u<<1|1, l, r)
    }
    return res
}

func (t *segmentTree) pushup(u int) {
    t.tr[u].s = t.tr[u<<1].s + t.tr[u<<1|1].s
}

func (t *segmentTree) pushdown(u int) {
    if t.tr[u].lazy == 1 {
        mid := (t.tr[u].l + t.tr[u].r) >> 1
        t.tr[u<<1].s = mid - t.tr[u].l + 1 - t.tr[u<<1].s
        t.tr[u<<1].lazy ^= 1
        t.tr[u<<1|1].s = t.tr[u].r - mid - t.tr[u<<1|1].s
        t.tr[u<<1|1].lazy ^= 1
        t.tr[u].lazy ^= 1
    }
}

func handleQuery(nums1 []int, nums2 []int, queries [][]int) (ans []int64) {
    tree := newSegmentTree(nums1)
    var s int64
    for _, x := range nums2 {
        s += int64(x)
    }
    for _, q := range queries {
        if q[0] == 1 {
            tree.modify(1, q[1]+1, q[2]+1)
        } else if q[0] == 2 {
            s += int64(q[1] * tree.query(1, 1, len(nums1)))
        } else {
            ans = append(ans, s)
        }
    }
    return
}

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