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第 37 场双周赛 Q4
设计
线段树
数学

1622. 奇妙序列

English Version

题目描述

请你实现三个 API append，addAll 和 multAll 来实现奇妙序列。

请实现 Fancy 类 ：

  • Fancy() 初始化一个空序列对象。
  • void append(val) 将整数 val 添加在序列末尾。
  • void addAll(inc) 将所有序列中的现有数值都增加 inc 。
  • void multAll(m) 将序列中的所有现有数值都乘以整数 m 。
  • int getIndex(idx) 得到下标为 idx 处的数值（下标从 0 开始），并将结果对 109 + 7 取余。如果下标大于等于序列的长度，请返回 -1 。

 

示例：

输入：
["Fancy", "append", "addAll", "append", "multAll", "getIndex", "addAll", "append", "multAll", "getIndex", "getIndex", "getIndex"]
[[], [2], [3], [7], [2], [0], [3], [10], [2], [0], [1], [2]]
输出：
[null, null, null, null, null, 10, null, null, null, 26, 34, 20]

解释：
Fancy fancy = new Fancy();
fancy.append(2);   // 奇妙序列：[2]
fancy.addAll(3);   // 奇妙序列：[2+3] -> [5]
fancy.append(7);   // 奇妙序列：[5, 7]
fancy.multAll(2);  // 奇妙序列：[5*2, 7*2] -> [10, 14]
fancy.getIndex(0); // 返回 10
fancy.addAll(3);   // 奇妙序列：[10+3, 14+3] -> [13, 17]
fancy.append(10);  // 奇妙序列：[13, 17, 10]
fancy.multAll(2);  // 奇妙序列：[13*2, 17*2, 10*2] -> [26, 34, 20]
fancy.getIndex(0); // 返回 26
fancy.getIndex(1); // 返回 34
fancy.getIndex(2); // 返回 20

 

提示：

  • 1 <= val, inc, m <= 100
  • 0 <= idx <= 105
  • 总共最多会有 105 次对 append，addAll，multAll 和 getIndex 的调用。

解法

方法一：线段树

线段树将整个区间分割为多个不连续的子区间，子区间的数量不超过 log(width)。更新某个元素的值，只需要更新 log(width) 个区间，并且这些区间都包含在一个包含该元素的大区间内。区间修改时，需要使用懒标记保证效率。

  • 线段树的每个节点代表一个区间；
  • 线段树具有唯一的根节点，代表的区间是整个统计范围，如 [1, N]；
  • 线段树的每个叶子节点代表一个长度为 1 的元区间 [x, x]；
  • 对于每个内部节点 [l, r]，它的左儿子是 [l, mid]，右儿子是 [mid + 1, r], 其中 mid = ⌊(l + r) / 2⌋ (即向下取整)。

Python3

MOD = int(1e9 + 7)


class Node:
    def __init__(self, l, r):
        self.left = None
        self.right = None
        self.l = l
        self.r = r
        self.mid = (l + r) >> 1
        self.v = 0
        self.add = 0
        self.mul = 1


class SegmentTree:
    def __init__(self):
        self.root = Node(1, int(1e5 + 1))

    def modifyAdd(self, l, r, inc, node=None):
        if l > r:
            return
        if node is None:
            node = self.root
        if node.l >= l and node.r <= r:
            node.v = (node.v + (node.r - node.l + 1) * inc) % MOD
            node.add += inc
            return
        self.pushdown(node)
        if l <= node.mid:
            self.modifyAdd(l, r, inc, node.left)
        if r > node.mid:
            self.modifyAdd(l, r, inc, node.right)
        self.pushup(node)

    def modifyMul(self, l, r, m, node=None):
        if l > r:
            return
        if node is None:
            node = self.root
        if node.l >= l and node.r <= r:
            node.v = (node.v * m) % MOD
            node.add = (node.add * m) % MOD
            node.mul = (node.mul * m) % MOD
            return
        self.pushdown(node)
        if l <= node.mid:
            self.modifyMul(l, r, m, node.left)
        if r > node.mid:
            self.modifyMul(l, r, m, node.right)
        self.pushup(node)

    def query(self, l, r, node=None):
        if l > r:
            return 0
        if node is None:
            node = self.root
        if node.l >= l and node.r <= r:
            return node.v
        self.pushdown(node)
        v = 0
        if l <= node.mid:
            v = (v + self.query(l, r, node.left)) % MOD
        if r > node.mid:
            v = (v + self.query(l, r, node.right)) % MOD
        return v

    def pushup(self, node):
        node.v = (node.left.v + node.right.v) % MOD

    def pushdown(self, node):
        if node.left is None:
            node.left = Node(node.l, node.mid)
        if node.right is None:
            node.right = Node(node.mid + 1, node.r)
        left, right = node.left, node.right
        if node.add != 0 or node.mul != 1:
            left.v = (left.v * node.mul + (left.r - left.l + 1) * node.add) % MOD
            right.v = (right.v * node.mul + (right.r - right.l + 1) * node.add) % MOD
            left.add = (left.add * node.mul + node.add) % MOD
            right.add = (right.add * node.mul + node.add) % MOD
            left.mul = (left.mul * node.mul) % MOD
            right.mul = (right.mul * node.mul) % MOD
            node.add = 0
            node.mul = 1


class Fancy:
    def __init__(self):
        self.n = 0
        self.tree = SegmentTree()

    def append(self, val: int) -> None:
        self.n += 1
        self.tree.modifyAdd(self.n, self.n, val)

    def addAll(self, inc: int) -> None:
        self.tree.modifyAdd(1, self.n, inc)

    def multAll(self, m: int) -> None:
        self.tree.modifyMul(1, self.n, m)

    def getIndex(self, idx: int) -> int:
        return -1 if idx >= self.n else self.tree.query(idx + 1, idx + 1)


# Your Fancy object will be instantiated and called as such:
# obj = Fancy()
# obj.append(val)
# obj.addAll(inc)
# obj.multAll(m)
# param_4 = obj.getIndex(idx)

Java

class Node {
    Node left;
    Node right;
    int l;
    int r;
    int mid;
    long v;
    long add;
    long mul = 1;

    public Node(int l, int r) {
        this.l = l;
        this.r = r;
        this.mid = (l + r) >> 1;
    }
}

class SegmentTree {
    private Node root = new Node(1, (int) 1e5 + 1);
    private static final int MOD = (int) 1e9 + 7;

    public SegmentTree() {
    }

    public void modifyAdd(int l, int r, int inc) {
        modifyAdd(l, r, inc, root);
    }

    public void modifyAdd(int l, int r, int inc, Node node) {
        if (l > r) {
            return;
        }
        if (node.l >= l && node.r <= r) {
            node.v = (node.v + (node.r - node.l + 1) * inc) % MOD;
            node.add = (node.add + inc) % MOD;
            return;
        }
        pushdown(node);
        if (l <= node.mid) {
            modifyAdd(l, r, inc, node.left);
        }
        if (r > node.mid) {
            modifyAdd(l, r, inc, node.right);
        }
        pushup(node);
    }

    public void modifyMul(int l, int r, int m) {
        modifyMul(l, r, m, root);
    }

    public void modifyMul(int l, int r, int m, Node node) {
        if (l > r) {
            return;
        }
        if (node.l >= l && node.r <= r) {
            node.v = (node.v * m) % MOD;
            node.add = (node.add * m) % MOD;
            node.mul = (node.mul * m) % MOD;
            return;
        }
        pushdown(node);
        if (l <= node.mid) {
            modifyMul(l, r, m, node.left);
        }
        if (r > node.mid) {
            modifyMul(l, r, m, node.right);
        }
        pushup(node);
    }

    public int query(int l, int r) {
        return query(l, r, root);
    }

    public int query(int l, int r, Node node) {
        if (l > r) {
            return 0;
        }
        if (node.l >= l && node.r <= r) {
            return (int) node.v;
        }
        pushdown(node);
        int v = 0;
        if (l <= node.mid) {
            v = (v + query(l, r, node.left)) % MOD;
        }
        if (r > node.mid) {
            v = (v + query(l, r, node.right)) % MOD;
        }
        return v;
    }

    public void pushup(Node node) {
        node.v = (node.left.v + node.right.v) % MOD;
    }

    public void pushdown(Node node) {
        if (node.left == null) {
            node.left = new Node(node.l, node.mid);
        }
        if (node.right == null) {
            node.right = new Node(node.mid + 1, node.r);
        }
        if (node.add != 0 || node.mul != 1) {
            Node left = node.left, right = node.right;
            left.v = (left.v * node.mul + (left.r - left.l + 1) * node.add) % MOD;
            right.v = (right.v * node.mul + (right.r - right.l + 1) * node.add) % MOD;
            left.add = (left.add * node.mul + node.add) % MOD;
            right.add = (right.add * node.mul + node.add) % MOD;
            left.mul = (left.mul * node.mul) % MOD;
            right.mul = (right.mul * node.mul) % MOD;
            node.add = 0;
            node.mul = 1;
        }
    }
}

class Fancy {
    private int n;
    private SegmentTree tree = new SegmentTree();

    public Fancy() {
    }

    public void append(int val) {
        ++n;
        tree.modifyAdd(n, n, val);
    }

    public void addAll(int inc) {
        tree.modifyAdd(1, n, inc);
    }

    public void multAll(int m) {
        tree.modifyMul(1, n, m);
    }

    public int getIndex(int idx) {
        return idx >= n ? -1 : tree.query(idx + 1, idx + 1);
    }
}

/**
 * Your Fancy object will be instantiated and called as such:
 * Fancy obj = new Fancy();
 * obj.append(val);
 * obj.addAll(inc);
 * obj.multAll(m);
 * int param_4 = obj.getIndex(idx);
 */

C++

const int MOD = 1e9 + 7;

class Node {
public:
    Node* left;
    Node* right;
    int l;
    int r;
    int mid;
    long long v;
    long long add;
    long long mul;

    Node(int l, int r) {
        this->l = l;
        this->r = r;
        this->mid = (l + r) >> 1;
        this->left = this->right = nullptr;
        v = add = 0;
        mul = 1;
    }
};

class SegmentTree {
private:
    Node* root;

public:
    SegmentTree() {
        root = new Node(1, 1e5 + 1);
    }

    void modifyAdd(int l, int r, int inc) {
        modifyAdd(l, r, inc, root);
    }

    void modifyAdd(int l, int r, int inc, Node* node) {
        if (l > r) return;
        if (node->l >= l && node->r <= r) {
            node->v = (node->v + (node->r - node->l + 1) * inc) % MOD;
            node->add = (node->add + inc) % MOD;
            return;
        }
        pushdown(node);
        if (l <= node->mid) modifyAdd(l, r, inc, node->left);
        if (r > node->mid) modifyAdd(l, r, inc, node->right);
        pushup(node);
    }

    void modifyMul(int l, int r, int m) {
        modifyMul(l, r, m, root);
    }

    void modifyMul(int l, int r, int m, Node* node) {
        if (l > r) return;
        if (node->l >= l && node->r <= r) {
            node->v = (node->v * m) % MOD;
            node->add = (node->add * m) % MOD;
            node->mul = (node->mul * m) % MOD;
            return;
        }
        pushdown(node);
        if (l <= node->mid) modifyMul(l, r, m, node->left);
        if (r > node->mid) modifyMul(l, r, m, node->right);
        pushup(node);
    }

    int query(int l, int r) {
        return query(l, r, root);
    }

    int query(int l, int r, Node* node) {
        if (l > r) return 0;
        if (node->l >= l && node->r <= r) return node->v;
        pushdown(node);
        int v = 0;
        if (l <= node->mid) v = (v + query(l, r, node->left)) % MOD;
        if (r > node->mid) v = (v + query(l, r, node->right)) % MOD;
        return v;
    }

    void pushup(Node* node) {
        node->v = (node->left->v + node->right->v) % MOD;
    }

    void pushdown(Node* node) {
        if (!node->left) node->left = new Node(node->l, node->mid);
        if (!node->right) node->right = new Node(node->mid + 1, node->r);
        if (node->add || node->mul != 1) {
            long add = node->add, mul = node->mul;
            Node* left = node->left;
            Node* right = node->right;
            left->v = (left->v * mul + (left->r - left->l + 1) * add) % MOD;
            right->v = (right->v * mul + (right->r - right->l + 1) * add) % MOD;
            left->add = (left->add * mul + add) % MOD;
            right->add = (right->add * mul + add) % MOD;
            left->mul = (left->mul * mul) % MOD;
            right->mul = (right->mul * mul) % MOD;
            node->add = 0;
            node->mul = 1;
        }
    }
};

class Fancy {
public:
    int n;
    SegmentTree* tree;

    Fancy() {
        n = 0;
        tree = new SegmentTree();
    }

    void append(int val) {
        ++n;
        tree->modifyAdd(n, n, val);
    }

    void addAll(int inc) {
        tree->modifyAdd(1, n, inc);
    }

    void multAll(int m) {
        tree->modifyMul(1, n, m);
    }

    int getIndex(int idx) {
        return idx >= n ? -1 : tree->query(idx + 1, idx + 1);
    }
};

/**
 * Your Fancy object will be instantiated and called as such:
 * Fancy* obj = new Fancy();
 * obj->append(val);
 * obj->addAll(inc);
 * obj->multAll(m);
 * int param_4 = obj->getIndex(idx);
 */