[Leetcode]307. Range Sum Query - Mutable
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这是Leetcode第307题,给一个数组,然后求指定下标之间的数之和,已知数组中的值可以更新,并且更新和求和操作会被频繁调用。
这是一道线段树的基础题,线段树是一种二叉搜索树。它将一段区间划分为若干单位区间,每一个节点都储存着一个区间。它功能强大,支持区间求和,区间最大值,区间修改,单点修改等操作。
线段树的思想和分治思想很相像。线段树的每一个节点都储存着一段区间[L…R]的信息,其中叶子节点L=R。它的大致思想是:将一段大区间平均地划分成2个小区间,每一个小区间都再平均分成2个更小区间……以此类推,直到每一个区间的L等于R(这样这个区间仅包含一个节点的信息,无法被划分)。通过对这些区间进行修改、查询,来实现对大区间的修改、查询。
这样一来,每一次修改、查询的时间复杂度都只为\(O(log_2n)\)
注意,可以用线段树维护的问题必须满足区间加法,否则是不可能将大问题划分成子问题来解决的。
通过这道题,也可以写一个线段树的模板,如下:
#Segment tree node
class Node(object):
def __init__(self, start, end):
self.start = start
self.end = end
self.total = 0
self.left = None
self.right = None
class NumArray(object):
def __init__(self, nums):
"""
initialize your data structure here.
:type nums: List[int]
"""
#helper function to create the tree from input array
def createTree(nums, l, r):
#base case
if l > r:
return None
#leaf node
if l == r:
n = Node(l, r)
n.total = nums[l]
return n
mid = (l + r) // 2
root = Node(l, r)
#recursively build the Segment tree
root.left = createTree(nums, l, mid)
root.right = createTree(nums, mid+1, r)
#Total stores the sum of all leaves under root
#i.e. those elements lying between (start, end)
root.total = root.left.total + root.right.total
return root
self.root = createTree(nums, 0, len(nums)-1)
def update(self, i, val):
"""
:type i: int
:type val: int
:rtype: int
"""
#Helper function to update a value
def updateVal(root, i, val):
#Base case. The actual value will be updated in a leaf.
#The total is then propogated upwards
if root.start == root.end:
root.total = val
return val
mid = (root.start + root.end) // 2
#If the index is less than the mid, that leaf must be in the left subtree
if i <= mid:
updateVal(root.left, i, val)
#Otherwise, the right subtree
else:
updateVal(root.right, i, val)
#Propogate the changes after recursive call returns
root.total = root.left.total + root.right.total
return root.total
return updateVal(self.root, i, val)
def sumRange(self, i, j):
"""
sum of elements nums[i..j], inclusive.
:type i: int
:type j: int
:rtype: int
"""
#Helper function to calculate range sum
def rangeSum(root, i, j):
#If the range exactly matches the root, we already have the sum
if root.start == i and root.end == j:
return root.total
mid = (root.start + root.end) // 2
#If end of the range is less than the mid, the entire interval lies
#in the left subtree
if j <= mid:
return rangeSum(root.left, i, j)
#If start of the interval is greater than mid, the entire inteval lies
#in the right subtree
elif i >= mid + 1:
return rangeSum(root.right, i, j)
#Otherwise, the interval is split. So we calculate the sum recursively,
#by splitting the interval
else:
return rangeSum(root.left, i, mid) + rangeSum(root.right, mid+1, j)
return rangeSum(self.root, i, j)
参考:
线段树详解
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