Fenwick Tree 307

对于一个数组 arr[1...n]，Fenwick Tree 可以支持两种操作：

1. update(i, delta)：将 arr[i] += delta

2. query(i)：返回前缀和 arr[1] + arr[2] + ... + arr[i]

它的时间复杂度为：

• 构建：O(n)

• 查询：O(log n)

• 更新：O(log n)

# 注意，这里实现的时候，默认下表从1开始，因此实际使用时传入下标要+1
class FenwickTree:
    def __init__(self, n):
        self._sums = [0 for _ in range(n + 1)]
        
    def update(self, i, delta):
        while i < len(self._sums):
            self._sums[i] += delta
            i += i & -i
    
    def query(self, i):
        s = 0
        while i > 0:
            s += self._sums[i]
            i -= i & -i
        return s

 

307. Range Sum Query - Mutable

Given an integer array nums, handle multiple queries of the following types:

1. Update the value of an element in nums.
2. Calculate the sum of the elements of nums between indices left and right inclusive where left <= right.

Implement the NumArray class:

• NumArray(int[] nums) Initializes the object with the integer array nums.
• void update(int index, int val) Updates the value of nums[index] to be val.
• int sumRange(int left, int right) Returns the sum of the elements of nums between indices left and right inclusive (i.e. nums[left] + nums[left + 1] + ... + nums[right]).

Example 1:

Input
["NumArray", "sumRange", "update", "sumRange"]
[[[1, 3, 5]], [0, 2], [1, 2], [0, 2]]
Output
[null, 9, null, 8]

Explanation
NumArray numArray = new NumArray([1, 3, 5]);
numArray.sumRange(0, 2); // return 1 + 3 + 5 = 9
numArray.update(1, 2);   // nums = [1, 2, 5]
numArray.sumRange(0, 2); // return 1 + 2 + 5 = 8

Constraints:

• 1 <= nums.length <= 3 * 104
• -100 <= nums[i] <= 100
• 0 <= index < nums.length
• -100 <= val <= 100
• 0 <= left <= right < nums.length
• At most 3 * 104 calls will be made to update and sumRange.

class NumArray:
    def __init__(self, nums: List[int]):
        self.tree = FenwickTree(len(nums))
        self.nums = nums
        for i in range(len(nums)):
            self.tree.update(i + 1, nums[i])

    def update(self, index: int, val: int) -> None:
        self.tree.update(index + 1, val - self.nums[index])
        self.nums[index] = val

    def sumRange(self, left: int, right: int) -> int:
        return self.tree.query(right + 1) - self.tree.query(left)

# 注意，这里实现的时候，默认下表从1开始，因此实际使用时传入下标要+1
class FenwickTree:
    def __init__(self, n):
        self._sums = [0 for _ in range(n + 1)]
        
    def update(self, i, delta):
        while i < len(self._sums):
            self._sums[i] += delta
            i += i & -i
    
    def query(self, i):
        s = 0
        while i > 0:
            s += self._sums[i]
            i -= i & -i
        return s

# Your NumArray object will be instantiated and called as such:
# obj = NumArray(nums)
# obj.update(index,val)
# param_2 = obj.sumRange(left,right)