[leetcode / lintcode] Tree - relative

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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)
        

         

 








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