[Data Structure] Tree - relative

Segment Tree

First, try to build the segment tree.

lintcode

suggest code: Currently recursion recommended. (For coding exercise, u can just start with "Interval minimum number" below.)

"""
Definition of SegmentTreeNode:
class SegmentTreeNode:
    def __init__(self, start, end):
        self.start, self.end = start, end
        self.left, self.right = None, None
"""


class Solution:
    """
    @param: start: start value.
    @param: end: end value.
    @return: The root of Segment Tree.
    """
    def build(self, start, end):
        # write your code here
        if start > end:
            return
        root = SegmentTreeNode(start, end)
        if start == end:
            return root
        mid = (end - start) / 2 + start
        root.left = self.build(start, mid)
        root.right = self.build(mid + 1, end)
        return root

Edge case:
- Trees are building by pointers, so u must consider the null pointers!!!
- if start > end

Try to know about segment tree.
- wiki
- Exactly as its name, segment tree is used to store segments.
- A segment tree is a tree for storing intervals, or segments. It allows querying which of the stored segments contains a given pionts.
- It is, in principle, a static structure; that is, it's a structure that cannot be modified once it's built.
- Complexity:
  - A segment tree of a set I of n intervals uses O(nlogn) storage and can be built in O(nlogn) time.
  - Segements trees support searching for all the intervals that contain a query point in O(logn + k), k being the number of retrieved intervals or segments.

Try to use segment tree for simple query...

Recursion recommended too. But to careful to consider all the cases, see below:

"""
Definition of SegmentTreeNode:
class SegmentTreeNode:
    def __init__(self, start, end, max):
        self.start, self.end, self.max = start, end, max
        self.left, self.right = None, None
"""


class Solution:
    """
    @param: root: The root of segment tree.
    @param: start: start value.
    @param: end: end value.
    @return: The maximum number in the interval [start, end]
    """
    def query(self, root, start, end):
        # write your code here
        if start > end:
            return
        if (start <= root.start and end >= root.end):
            return root.max
        mid = (root.end - root.start) / 2 + root.start
        if end <= mid:
            return self.query(root.left, start, end)
        if start > mid:
            return self.query(root.right, start, end)
        return max(self.query(root.left, start, mid), self.query(root.right, mid + 1, end))

Then try to modify the tree...
Please be familiar with this way of recursion: return current val for each recursion.

"""
Definition of SegmentTreeNode:
class SegmentTreeNode:
    def __init__(self, start, end, max):
        self.start, self.end, self.max = start, end, max
        self.left, self.right = None, None
"""


class Solution:
    """
    @param: root: The root of segment tree.
    @param: index: index.
    @param: value: value
    @return: 
    """
    def modify(self, root, index, value):
        # write your code here
        if not root or index < root.start or index > root.end:
            return
        self.helper(root, index, value)
    
    def helper(self, root, index, value):
        if root.start == root.end:
            if root.start == index:
                root.max = value
            return root.max
        mid = (root.end - root.start) / 2 + root.start
        if index <= mid:
            val = self.helper(root.left, index, value)
            if root.right:
                val = max(val, root.right.max)
            root.max = val
        else:
            val = self.helper(root.right, index, value)
            if root.left:
                val = max(val, root.left.max)
            root.max = val
        return root.max

Now, try some usages.

Interval minimum number: lintcode

This is the most representative usage of segment: do interval aggregative operation.

"""
Definition of Interval.
class Interval(object):
    def __init__(self, start, end):
        self.start = start
        self.end = end
"""
class SegTreeNode:
    def __init__(self, start, end, val):
        self.start, self.end, self.val = start, end, val
        self.left, self.right = None, None


class Solution:
    """
    @param: A: An integer array
    @param: queries: An query list
    @return: The result list
    """
    def intervalMinNumber(self, A, queries):
        # write your code here
        if not A or not queries:
            return []
        # 1. construct a segment tree
        root = self.construct_st(A, 0, len(A) - 1)
        # 2. search for each query
        res = []
        for query in queries:
            res.append(self.query_st(root, query.start, query.end))
        return res
    
    def construct_st(self, nums, start, end):
        if start == end:
            return SegTreeNode(start, end, nums[start])
        mid = (end - start) / 2 + start
        left = self.construct_st(nums, start, mid)
        right = self.construct_st(nums, mid + 1, end)
        root = SegTreeNode(start, end, min(left.val, right.val))
        root.left = left
        root.right = right
        return root
    
    def query_st(self, root, start, end):
        if start <= root.start and end >= root.end:
            return root.val
        root_mid = (root.end - root.start) / 2 + root.start
        if end <= root_mid:
            return self.query_st(root.left, start, end)
        if start > root_mid:
            return self.query_st(root.right, start, end)
        return min(self.query_st(root.left, start, root_mid), self.query_st(root.right, root_mid + 1, end))

For complicated implementation, try Interval Sum II.

Now, try some usage that not that obvious segment tree.

Count of Smaller Number.
A useful thing to notice is that the value of from this array is value from 0 to 10000. So we can use a segment tree to denote [start, end, val] and val denotes how many numbers from the array is between [start, end].

class SegTreeNode:
    def __init__(self, start, end, val):
        self.start, self.end, self.val = start, end, val
        self.left, self.right = None, None

class Solution:
    """
    @param: A: An integer array
    @param: queries: The query list
    @return: The number of element in the array that are smaller that the given integer
    """
    def countOfSmallerNumber(self, A, queries):
        # write your code here
        if not A:
            return [0 for i in range(len(queries))]
        
        # 1. construct a segment tree
        dict_A = {}
        for val in A:
            dict_A[val] = dict_A.get(val, 0) + 1
        root = self.con_st(0, 10000, dict_A)
        # 2. search
        res = []
        for query in queries:
            res.append(self.query_st(root, query))
        return res
    
    def con_st(self, start, end, dict_A):
        if start > end:
            return
        if start == end:
            return SegTreeNode(start, end, dict_A.get(start, 0))
        mid = (end - start) / 2 + start
        left = self.con_st(start, mid, dict_A)
        right = self.con_st(mid + 1, end, dict_A)
        root = SegTreeNode(start, end, left.val + right.val)
        root.left, root.right = left, right
        return root
    
    def query_st(self, root, query):
        if not root:
            return 0
        if query > root.end:
            return root.val
        if query <= root.start:
            return 0
        mid = (root.end - root.start) / 2 + root.start
        if query > mid:
            return root.left.val + self.query_st(root.right, query)
        if query <= mid:
            return self.query_st(root.left, query)

Then, try this: Count of smaller number before itself.

Obivously, we can use binary search to solve this just almost the same as 'Count of smaller number'. But for this problem, it will cost O(N^2) time, for the O(N) time for insert into a list.

But still, I tried binary search, as below:

class Solution:
    """
    @param: A: an integer array
    @return: A list of integers includes the index of the first number and the index of the last number
    """

    def countOfSmallerNumberII(self, A):
        # write your code here
        if not A:
            return []
        # binary search, O(NlogN) time(O(N^2) for insert takes O(N) each time), O(N) space
        sorted_subarr = []
        res = []
        for val in A:
            ind = self.binary_search(sorted_subarr, val)
            res.append(ind)
        return res

    def binary_search(self, sorted_subarr, val):
        if not sorted_subarr:
            sorted_subarr.append(val)
            return 0
        # 1. find the right-most number who is smaller than val
        left, right = 0, len(sorted_subarr) - 1
        while left < right - 1:
            mid = (right - left) / 2 + left
            if sorted_subarr[mid] < val:
                left = mid
            else:
                right = mid
        # 2. insert val into sorted_subarr
        if sorted_subarr[right] < val:
            sorted_subarr.insert(right + 1, val)
            return right + 1
        elif sorted_subarr[left] < val:
            sorted_subarr.insert(left + 1, val)
            return left + 1
        else:
            sorted_subarr.insert(left, val)
            return left

And a better choice is using segment tree. For each val in A, we search segment tree then insert it into tree.

class SegTreeNode:
    def __init__(self, start, end, val):
        self.val, self.start, self.end = val, start, end
        self.left, self.right = None, None


class Solution:
    """
    @param: A: an integer array
    @return: A list of integers includes the index of the first number and the index of the last number
    """

    def countOfSmallerNumberII(self, A):
        # write your code here
        if not A:
            return []
        # 1. build segment tree, using O(10000log10000)
        root = self.build_st(0, max(A))
        # 2. search for target value then update segment tree
        res = []
        for val in A:
            res.append(self.search_st(root, val))
            self.update_st(root, val)
        return res

    def build_st(self, start, end):
        root = SegTreeNode(start, end, 0)
        if start == end:
            return root
        mid = (end - start) / 2 + start
        root.left = self.build_st(start, mid)
        root.right = self.build_st(mid + 1, end)
        return root

    def search_st(self, root, val):
        if val > root.end:
            return root.val
        if val <= root.start:
            return 0
        mid = (root.end - root.start) / 2 + root.start
        if val <= mid:
            return self.search_st(root.left, val)
        return root.left.val + self.search_st(root.right, val)

    def update_st(self, root, val):
        root.val += 1
        if root.start == root.end:
            return
        mid = (root.end - root.start) / 2 + root.start
        if val <= mid:
            self.update_st(root.left, val)
        else:   
            self.update_st(root.right, val)

Summary of segment tree

线段树的每个节点表示一个区间，子节点分别表示父节点的左右半区间。
线段树的构建、查询和更新都可以使用递归。构建是O(NlogN) time, 查询和更新时O(logN) time，其中n是root的end - start。
常见应用：求区间的最大最小值、区间的sum等。

Binary Search Tree

The definition of bst is very easy: x.left < x, x.right > x

Range search in bst: Search Range in BST.
BST(tree)惯用套路：recursion

class Solution:
    """
    @param: root: param root: The root of the binary search tree
    @param: k1: An integer
    @param: k2: An integer
    @return: return: Return all keys that k1<=key<=k2 in ascending order
    """
    def searchRange(self, root, k1, k2):
        # write your code here
        if not root or k1 > k2:
            return []
        results = []
        self.helper(root, k1, k2, results)
        return results
    
    def helper(self, root, start, end, results):
        if not root or start > end:
            return
        if root.val < start:
            self.helper(root.right, start, end, results)
        elif root.val > end:
            self.helper(root.left, start, end, results)
        else:
            self.helper(root.left, start, root.val, results)
            results.append(root.val)
            self.helper(root.right, root.val, end, results)

发表于 2017-09-29 10:28 橘子不是唯一的水果阅读(240) 评论(0) 编辑收藏举报

会员力量，点亮园子希望

刷新页面返回顶部

[Data Structure] Tree - relative

Segment Tree

Summary of segment tree

Binary Search Tree

公告

导航