线段树

概念

线段树是一种用于解决区间更新和区间查询问题的高效数据结构

  • 这里我把区间更新分成两种（upd函数）
    • 区间添加更新，给这个区间的子数组都添加上某个值
    • 区间覆盖更新，把这个区间的子数组都赋值为某个值
  • 而区间查询也大概分为两种（query函数）
    • 区间和查询，求这个区间的数组的元素值之和
    • 区间最大值查询，求这个区间的数组的元素值的最大值

单点更新可以用区间更新的模板，只要把左右两个坐标设置成一样的就可以了
但是单点更新可以用树状数组解决，优先用树状数组解决，更简单

数组无lazy线段树

模板：区间覆盖更新，区间和查询

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class SegmentTree {
    int[] nums;
    int len;
    int[] tree;
    public SegmentTree(int[] nums) {
        this.nums = nums;
        this.len = nums.length;
        tree = new int[4 * len];
        build(1, 0, len - 1);
    }
    void upd(int idx, int l, int r, int tl, int tr, int val) {
        if (tl <= l && r <= tr) {
            tree[idx] = val;
            if (l != r) { // 没有用lazy数组的情况，就需要给所有子区间更新
                int m = l + (r - l) / 2;
                upd(2 * idx, l, m, tl, tr, val);
                upd(2 * idx + 1, m + 1, r, tl, tr, val);
            }
        } else {
            int m = l + (r - l) / 2;
            if (tl <= m) upd(2 * idx, l, m, tl, tr, val);
            if (m + 1 <= tr) upd(2 * idx + 1, m + 1, r, tl, tr, val);
            tree[idx] = tree[2 * idx] + tree[2 * idx + 1];
        }
    }
    int query(int idx, int l, int r, int tl, int tr) {
        if (tl <= l && r <= tr) {
            return tree[idx];
        } else {
            int m = l + (r - l) / 2, res = 0;
            if (tl <= m) res += query(2 * idx, l, m, tl, tr);
            if (m + 1 <= tr) res += query(2 * idx + 1, m + 1, r, tl, tr);
            return res;
        }
    }
    void build(int idx, int l, int r) {
        if (l == r) {
            tree[idx] = nums[l];
        } else {
            int m = l + (r - l) / 2;
            build(2 * idx, l, m);
            build(2 * idx + 1, m + 1, r);
            tree[idx] = tree[2 * idx] + tree[2 * idx + 1];
        }
    }
}

数组lazy线段树

模板：区间添加更新，区间和查询

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class SegmentTree {
    int[] tree;
    int[] lazy;
    int[] nums;
    int n;
    public SegmentTree(int[] nums) {
        this.nums = nums;
        this.n = nums.length;
        this.tree = new int[4 * n];
        this.lazy = new int[4 * n];
        build(1, 0, n - 1);
    }
    void pushDown(int idx, int l, int r) {
        if (lazy[idx] != 0) {
            int m = l + (r - l) / 2;
            tree[2 * idx] = lazy[idx] * (m - l + 1);
            tree[2 * idx + 1] = lazy[idx] * (r - m);
            lazy[2 * idx] = lazy[idx];
            lazy[2 * idx + 1] = lazy[idx];
            lazy[idx] = 0;
        }
    }
    void pushUp(int idx) {
        tree[idx] = tree[2 * idx] + tree[2 * idx + 1];
    }
    void upd(int idx, int l, int r, int tl, int tr, int val) {
        if (tl <= l && r <= tr) {
            tree[idx] = val * (r - l + 1);
            lazy[idx] = val;
        } else {
            pushDown(idx, l, r);
            int m = l + (r - l) / 2;
            if (tl <= m) upd(2 * idx, l, m, tl, tr, val);
            if (m + 1 <= tr) upd(2 * idx + 1, m + 1, r, tl, tr, val);
            pushUp(idx);
        }
    }
    int query(int idx, int l, int r, int tl, int tr) {
        if (tl <= l && r <= tr) {
            return tree[idx];
        } else {
            pushDown(idx, l, r);
            int m = l + (r - l) / 2;
            int res = 0;
            if (tl <= m) res += query(2 * idx, l, m, tl, tr);
            if (m + 1 <= tr) res += query(2 * idx + 1, m + 1, r, tl, tr);
            return res;
        }
    }
    void build(int idx, int l, int r) {
        if (l == r) {
            tree[idx] = nums[l];
        } else {
            int m = l + (r - l) / 2;
            build(2 * idx, l, m);
            build(2 * idx + 1, m + 1, r);
            pushUp(idx);
        }
    }
}

模板：区间添加更新，区间和查询（未验证）

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class SegmentTree {
    int[] tree;
    int[] lazy;
    int[] nums;
    int n;
    public SegmentTree(int[] nums) {
        this.nums = nums;
        this.n = nums.length;
        this.tree = new int[4 * n];
        this.lazy = new int[4 * n];
        build(1, 0, n - 1);
    }
    void pushDown(int idx, int l, int r) {
        if (lazy[idx] != 0) {
            int m = l + (r - l) / 2;
            tree[2 * idx] += lazy[idx] * (m - l + 1);
            tree[2 * idx + 1] += lazy[idx] * (r - m);
            lazy[2 * idx] += lazy[idx];
            lazy[2 * idx + 1] += lazy[idx];
            lazy[idx] = 0;
        }
    }
    void pushUp(int idx) {
        tree[idx] = tree[2 * idx] + tree[2 * idx + 1];
    }
    void upd(int idx, int l, int r, int tl, int tr, int val) {
        if (tl <= l && r <= tr) {
            tree[idx] += val * (r - l + 1);
            lazy[idx] += val;
        } else {
            pushDown(idx, l, r);
            int m = l + (r - l) / 2;
            if (tl <= m) upd(2 * idx, l, m, tl, tr, val);
            if (m + 1 <= tr) upd(2 * idx + 1, m + 1, r, tl, tr, val);
            pushUp(idx);
        }
    }
    int query(int idx, int l, int r, int tl, int tr) {
        if (tl <= l && r <= tr) {
            return tree[idx];
        } else {
            pushDown(idx, l, r);
            int m = l + (r - l) / 2;
            int res = 0;
            if (tl <= m) res += query(2 * idx, l, m, tl, tr);
            if (m + 1 <= tr) res += query(2 * idx + 1, m + 1, r, tl, tr);
            return res;
        }
    }
    void build(int idx, int l, int r) {
        if (l == r) {
            tree[idx] = nums[l];
        } else {
            int m = l + (r - l) / 2;
            build(2 * idx, l, m);
            build(2 * idx + 1, m + 1, r);
            pushUp(idx);
        }
    }
}

指针lazy线段树

对于有些题目值域过大，我们无法直接使用空间大小固定为 4×n 的常规线段树，就必须采用指针（动态开点）的方式了

模板：区间覆盖更新，区间和查询（未验证）

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class SegmentTree {
    class Node {
        Node l, r;
        int val, lazy;
    }

    Node root = new Node();

    void pushDown(Node cur, int len) {
        if (cur.l == null) cur.l = new Node();
        if (cur.r == null) cur.r = new Node();
        if (cur.lazy != 0) {
            cur.l.val = cur.lazy * (len - len / 2);
            cur.r.val = cur.lazy * (len / 2);
            cur.l.lazy = cur.lazy;
            cur.r.lazy = cur.lazy;
            cur.lazy = 0;
        }
    }

    void pushUp(Node cur) {
        cur.val = cur.l.val + cur.r.val;
    }

    void update(Node cur, int l, int r, int tl, int tr, int val) {
        if (tl <= l && r <= tr) {
            cur.val = val * (r - l + 1);
            cur.lazy = val;
        } else {
            pushDown(cur, r - l + 1);
            int m = l + (r - l) / 2;
            if (tl <= m) update(cur.l, l, m, tl, tr, val);
            if (m + 1 <= tr) update(cur.r, m + 1, r, tl, tr, val);
            pushUp(cur);
        }
    }

    int query(Node cur, int l, int r, int tl, int tr) {
        if (tl <= l && r <= tr) {
            return cur.val;
        } else {
            pushDown(cur, r - l + 1);
            int m = l + (r - l) / 2;
            int res = 0;
            if (tl <= m) res += query(cur.l, l, m, tl, tr);
            if (m + 1 <= tr) res += query(cur.r, m + 1, r, tl, tr);
            return res;
        }
    }
}

模板：区间添加更新，区间和查询（未验证）

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class SegmentTree {
    class Node {
        Node l, r;
        int val, lazy;
    }
    Node root = new Node();
    void pushDown(Node cur, int l, int r) {
        if (cur.l == null) cur.l = new Node();
        if (cur.r == null) cur.r = new Node();
        int m = l + (r - l) / 2;
        cur.l.val += cur.lazy * (m - l + 1);
        cur.r.val += cur.lazy * (r - m);
        cur.l.lazy += cur.lazy;
        cur.r.lazy += cur.lazy;
        cur.lazy = 0;
    }
    void pushUp(Node cur) {
        cur.val = cur.l.val + cur.r.val;
    }
    void upd(Node cur, int l, int r, int tl, int tr, int val) {
        if (tl <= l && r <= tr) {
            cur.val += val * (r - l + 1);
            cur.lazy += val;
        } else {
            pushDown(cur, l, r);
            int mid = l + (r - l) / 2;
            if (tl <= mid) upd(cur.l, l, mid, tl, tr, val);
            if (mid < tr) upd(cur.r, mid + 1, r, tl, tr, val);
            pushUp(cur);
        }
    }
    int query(Node cur, int l, int r, int tl, int tr) {
        if (tl <= l && r <= tr) {
            return cur.val;
        } else {
            pushDown(cur, l, r);
            int mid = l + (r - l) / 2;
            int res = 0;
            if (tl <= mid) res += query(cur.l, l, mid, tl, tr);
            if (mid < tr) res += query(cur.r, mid + 1, r, tl, tr);
            pushUp(cur);
            return res;
        }
    }
    void build(Node cur, int l, int r, int[] nums) {
        if (l == r) {
            cur.val = nums[l];
        } else {
            int mid = l + (r - l) / 2;
            pushDown(cur, l, r);
            build(cur.l, l, mid, nums);
            build(cur.r, mid + 1, r, nums);
            pushUp(cur);
        }
    }
}