Question

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)。

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)

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);
 */

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);
 */