[Data Structure] Tree

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)