Question

308. 二维区域和检索 - 可变

English Version

题目描述

给你一个二维矩阵 matrix ，处理以下类型的多个查询:

1. 更新 matrix 中单元格的值。
2. 计算由 左上角 (row1, col1) 和 右下角 (row2, col2) 定义的 matrix 内矩阵元素的 和。

实现 NumMatrix 类：

• NumMatrix(int[][] matrix) 用整数矩阵 matrix 初始化对象。
• void update(int row, int col, int val) 更新 matrix[row][col] 的值到 val 。
• int sumRegion(int row1, int col1, int row2, int col2) 返回矩阵 matrix 中指定矩形区域元素的 和 ，该区域由 左上角 (row1, col1) 和 右下角 (row2, col2) 界定。

 

示例 1：

输入
["NumMatrix", "sumRegion", "update", "sumRegion"]
[[[[3, 0, 1, 4, 2], [5, 6, 3, 2, 1], [1, 2, 0, 1, 5], [4, 1, 0, 1, 7], [1, 0, 3, 0, 5]]], [2, 1, 4, 3], [3, 2, 2], [2, 1, 4, 3]]
输出
[null, 8, null, 10]

解释
NumMatrix numMatrix = new NumMatrix([[3, 0, 1, 4, 2], [5, 6, 3, 2, 1], [1, 2, 0, 1, 5], [4, 1, 0, 1, 7], [1, 0, 3, 0, 5]]);
numMatrix.sumRegion(2, 1, 4, 3); // 返回 8 (即, 左侧红色矩形的和)
numMatrix.update(3, 2, 2); // 矩阵从左图变为右图
numMatrix.sumRegion(2, 1, 4, 3); // 返回 10 (即，右侧红色矩形的和)

 

提示：

• m == matrix.length
• n == matrix[i].length
• 1 <= m, n <= 200
• -105 <= matrix[i][j] <= 105
• 0 <= row < m
• 0 <= col < n
• -105 <= val <= 105
• 0 <= row1 <= row2 < m
• 0 <= col1 <= col2 < n
• 最多调用104 次 sumRegion 和 update 方法

解法

方法一：树状数组

树状数组，也称作“二叉索引树”（Binary Indexed Tree）或 Fenwick 树。 它可以高效地实现如下两个操作：

1. 单点更新 update(x, delta)： 把序列 x 位置的数加上一个值 delta；
2. 前缀和查询 query(x)：查询序列 [1,...x] 区间的区间和，即位置 x 的前缀和。

这两个操作的时间复杂度均为 $O(\log n)$。

对于本题，可以构建二维树状数组。

class BinaryIndexedTree:
  def __init__(self, n):
    self.n = n
    self.c = [0] * (n + 1)

  @staticmethod
  def lowbit(x):
    return x & -x

  def update(self, x, delta):
    while x <= self.n:
      self.c[x] += delta
      x += BinaryIndexedTree.lowbit(x)

  def query(self, x):
    s = 0
    while x > 0:
      s += self.c[x]
      x -= BinaryIndexedTree.lowbit(x)
    return s


class NumMatrix:
  def __init__(self, matrix: List[List[int]]):
    self.trees = []
    n = len(matrix[0])
    for row in matrix:
      tree = BinaryIndexedTree(n)
      for j, v in enumerate(row):
        tree.update(j + 1, v)
      self.trees.append(tree)

  def update(self, row: int, col: int, val: int) -> None:
    tree = self.trees[row]
    prev = tree.query(col + 1) - tree.query(col)
    tree.update(col + 1, val - prev)

  def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
    return sum(
      tree.query(col2 + 1) - tree.query(col1)
      for tree in self.trees[row1 : row2 + 1]
    )


# Your NumMatrix object will be instantiated and called as such:
# obj = NumMatrix(matrix)
# obj.update(row,col,val)
# param_2 = obj.sumRegion(row1,col1,row2,col2)

class BinaryIndexedTree {
  private int n;
  private int[] c;

  public BinaryIndexedTree(int n) {
    this.n = n;
    c = new int[n + 1];
  }

  public void update(int x, int delta) {
    while (x <= n) {
      c[x] += delta;
      x += lowbit(x);
    }
  }

  public int query(int x) {
    int s = 0;
    while (x > 0) {
      s += c[x];
      x -= lowbit(x);
    }
    return s;
  }

  public static int lowbit(int x) {
    return x & -x;
  }
}

class NumMatrix {
  private BinaryIndexedTree[] trees;

  public NumMatrix(int[][] matrix) {
    int m = matrix.length;
    int n = matrix[0].length;
    trees = new BinaryIndexedTree[m];
    for (int i = 0; i < m; ++i) {
      BinaryIndexedTree tree = new BinaryIndexedTree(n);
      for (int j = 0; j < n; ++j) {
        tree.update(j + 1, matrix[i][j]);
      }
      trees[i] = tree;
    }
  }

  public void update(int row, int col, int val) {
    BinaryIndexedTree tree = trees[row];
    int prev = tree.query(col + 1) - tree.query(col);
    tree.update(col + 1, val - prev);
  }

  public int sumRegion(int row1, int col1, int row2, int col2) {
    int s = 0;
    for (int i = row1; i <= row2; ++i) {
      BinaryIndexedTree tree = trees[i];
      s += tree.query(col2 + 1) - tree.query(col1);
    }
    return s;
  }
}

/**
 * Your NumMatrix object will be instantiated and called as such:
 * NumMatrix obj = new NumMatrix(matrix);
 * obj.update(row,col,val);
 * int param_2 = obj.sumRegion(row1,col1,row2,col2);
 */

class BinaryIndexedTree {
public:
  int n;
  vector<int> c;

  BinaryIndexedTree(int _n)
    : n(_n)
    , c(_n + 1) {}

  void update(int x, int delta) {
    while (x <= n) {
      c[x] += delta;
      x += lowbit(x);
    }
  }

  int query(int x) {
    int s = 0;
    while (x > 0) {
      s += c[x];
      x -= lowbit(x);
    }
    return s;
  }

  int lowbit(int x) {
    return x & -x;
  }
};

class NumMatrix {
public:
  vector<BinaryIndexedTree*> trees;

  NumMatrix(vector<vector<int>>& matrix) {
    int m = matrix.size();
    int n = matrix[0].size();
    trees.resize(m);
    for (int i = 0; i < m; ++i) {
      BinaryIndexedTree* tree = new BinaryIndexedTree(n);
      for (int j = 0; j < n; ++j) tree->update(j + 1, matrix[i][j]);
      trees[i] = tree;
    }
  }

  void update(int row, int col, int val) {
    BinaryIndexedTree* tree = trees[row];
    int prev = tree->query(col + 1) - tree->query(col);
    tree->update(col + 1, val - prev);
  }

  int sumRegion(int row1, int col1, int row2, int col2) {
    int s = 0;
    for (int i = row1; i <= row2; ++i) {
      BinaryIndexedTree* tree = trees[i];
      s += tree->query(col2 + 1) - tree->query(col1);
    }
    return s;
  }
};

/**
 * Your NumMatrix object will be instantiated and called as such:
 * NumMatrix* obj = new NumMatrix(matrix);
 * obj->update(row,col,val);
 * int param_2 = obj->sumRegion(row1,col1,row2,col2);
 */

type BinaryIndexedTree struct {
  n int
  c []int
}

func newBinaryIndexedTree(n int) *BinaryIndexedTree {
  c := make([]int, n+1)
  return &BinaryIndexedTree{n, c}
}

func (this *BinaryIndexedTree) lowbit(x int) int {
  return x & -x
}

func (this *BinaryIndexedTree) update(x, delta int) {
  for x <= this.n {
    this.c[x] += delta
    x += this.lowbit(x)
  }
}

func (this *BinaryIndexedTree) query(x int) int {
  s := 0
  for x > 0 {
    s += this.c[x]
    x -= this.lowbit(x)
  }
  return s
}

type NumMatrix struct {
  trees []*BinaryIndexedTree
}

func Constructor(matrix [][]int) NumMatrix {
  n := len(matrix[0])
  var trees []*BinaryIndexedTree
  for _, row := range matrix {
    tree := newBinaryIndexedTree(n)
    for j, v := range row {
      tree.update(j+1, v)
    }
    trees = append(trees, tree)
  }
  return NumMatrix{trees}
}

func (this *NumMatrix) Update(row int, col int, val int) {
  tree := this.trees[row]
  prev := tree.query(col+1) - tree.query(col)
  tree.update(col+1, val-prev)
}

func (this *NumMatrix) SumRegion(row1 int, col1 int, row2 int, col2 int) int {
  s := 0
  for i := row1; i <= row2; i++ {
    tree := this.trees[i]
    s += tree.query(col2+1) - tree.query(col1)
  }
  return s
}

/**
 * Your NumMatrix object will be instantiated and called as such:
 * obj := Constructor(matrix);
 * obj.Update(row,col,val);
 * param_2 := obj.SumRegion(row1,col1,row2,col2);
 */

方法二：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。

• 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)。

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


class SegmentTree:
  def __init__(self, nums):
    n = len(nums)
    self.nums = nums
    self.tr = [Node() for _ in range(4 * n)]
    self.build(1, 1, n)

  def build(self, u, l, r):
    self.tr[u].l = l
    self.tr[u].r = r
    if l == r:
      self.tr[u].v = 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, x, v):
    if self.tr[u].l == x and self.tr[u].r == x:
      self.tr[u].v = v
      return
    mid = (self.tr[u].l + self.tr[u].r) >> 1
    if x <= mid:
      self.modify(u << 1, x, v)
    else:
      self.modify(u << 1 | 1, x, v)
    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].v
    mid = (self.tr[u].l + self.tr[u].r) >> 1
    v = 0
    if l <= mid:
      v += self.query(u << 1, l, r)
    if r > mid:
      v += self.query(u << 1 | 1, l, r)
    return v

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


class NumMatrix:
  def __init__(self, matrix: List[List[int]]):
    self.trees = [SegmentTree(row) for row in matrix]

  def update(self, row: int, col: int, val: int) -> None:
    tree = self.trees[row]
    tree.modify(1, col + 1, val)

  def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
    return sum(
      self.trees[row].query(1, col1 + 1, col2 + 1)
      for row in range(row1, row2 + 1)
    )


# Your NumMatrix object will be instantiated and called as such:
# obj = NumMatrix(matrix)
# obj.update(row,col,val)
# param_2 = obj.sumRegion(row1,col1,row2,col2)

class Node {
  int l;
  int r;
  int v;
}

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

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

  public void build(int u, int l, int r) {
    tr[u].l = l;
    tr[u].r = r;
    if (l == r) {
      tr[u].v = 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 x, int v) {
    if (tr[u].l == x && tr[u].r == x) {
      tr[u].v = v;
      return;
    }
    int mid = (tr[u].l + tr[u].r) >> 1;
    if (x <= mid) {
      modify(u << 1, x, v);
    } else {
      modify(u << 1 | 1, x, v);
    }
    pushup(u);
  }

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

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

class NumMatrix {
  private SegmentTree[] trees;

  public NumMatrix(int[][] matrix) {
    int m = matrix.length;
    trees = new SegmentTree[m];
    for (int i = 0; i < m; ++i) {
      trees[i] = new SegmentTree(matrix[i]);
    }
  }

  public void update(int row, int col, int val) {
    SegmentTree tree = trees[row];
    tree.modify(1, col + 1, val);
  }

  public int sumRegion(int row1, int col1, int row2, int col2) {
    int s = 0;
    for (int row = row1; row <= row2; ++row) {
      SegmentTree tree = trees[row];
      s += tree.query(1, col1 + 1, col2 + 1);
    }
    return s;
  }
}

/**
 * Your NumMatrix object will be instantiated and called as such:
 * NumMatrix obj = new NumMatrix(matrix);
 * obj.update(row,col,val);
 * int param_2 = obj.sumRegion(row1,col1,row2,col2);
 */

class Node {
public:
  int l;
  int r;
  int v;
};

class SegmentTree {
public:
  vector<Node*> tr;
  vector<int> nums;

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

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

  void modify(int u, int x, int v) {
    if (tr[u]->l == x && tr[u]->r == x) {
      tr[u]->v = v;
      return;
    }
    int mid = (tr[u]->l + tr[u]->r) >> 1;
    if (x <= mid)
      modify(u << 1, x, v);
    else
      modify(u << 1 | 1, x, v);
    pushup(u);
  }

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

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

class NumMatrix {
public:
  vector<SegmentTree*> trees;

  NumMatrix(vector<vector<int>>& matrix) {
    int m = matrix.size();
    trees.resize(m);
    for (int i = 0; i < m; ++i) trees[i] = new SegmentTree(matrix[i]);
  }

  void update(int row, int col, int val) {
    SegmentTree* tree = trees[row];
    tree->modify(1, col + 1, val);
  }

  int sumRegion(int row1, int col1, int row2, int col2) {
    int s = 0;
    for (int row = row1; row <= row2; ++row) s += trees[row]->query(1, col1 + 1, col2 + 1);
    return s;
  }
};

/**
 * Your NumMatrix object will be instantiated and called as such:
 * NumMatrix* obj = new NumMatrix(matrix);
 * obj->update(row,col,val);
 * int param_2 = obj->sumRegion(row1,col1,row2,col2);
 */