Minimum Absolute Difference in BST

To find the minimum absolute difference between the values of any two different nodes in a Binary Search Tree (BST), we can utilize an in-order traversal. Since the BST follows the property that the left subtree has smaller values, and the right subtree has larger values, an in-order traversal will provide the elements in sorted order.

By traversing the tree in this manner and comparing adjacent values, we can find the minimum absolute difference. Here’s a Python solution that implements this approach:

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class Solution:
    def getMinimumDifference(self, root: Optional[TreeNode]) -> int:
        def in_order_traversal(node):
            if not node:
                return []
            return in_order_traversal(node.left) + [node.val] + in_order_traversal(node.right)

        sorted_values = in_order_traversal(root)
        min_difference = float('inf')

        for i in range(1, len(sorted_values)):
            difference = sorted_values[i] - sorted_values[i - 1]
            min_difference = min(min_difference, difference)

        return min_difference

Explanation:

  1. We define an inner function in_order_traversal to recursively traverse the tree in in-order (left-root-right) and return the values as a list.
  2. We call this function with the root of the tree to get the sorted values.
  3. Then we iterate through these sorted values and find the minimum difference between adjacent elements.
  4. We return the minimum difference found.

This code ensures that we find the minimum absolute difference between the values of any two different nodes in the given BST.

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# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, x):
#         self.val = x
#         self.left = None
#         self.right = None

class Solution:
    def getMinimumDifference(self, root: TreeNode) -> int:
        output = []
        self.inorder(root,output)
        mini_diff = float('inf')
        for i in range(1,len(output)):
            mini_diff = min(mini_diff,output[i]-output[i-1])
        return mini_diff

    def inorder(self,root,output):
        if root == None:
            return 
        else:
            self.inorder(root.left, output)
            output.append(root.val)
            self.inorder(root.right, output)

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# Definition for a binary tree node.
# class TreeNode
#     attr_accessor :val, :left, :right
#     def initialize(val = 0, left = nil, right = nil)
#         @val = val
#         @left = left
#         @right = right
#     end
# end
def inorder(node)
  if node.nil?
      return 
  end
  
  inorder(node.left)
  
  unless @prev.nil?
      @min = [node.val - @prev, @min].min.abs
  end
    @prev = node.val
    
  inorder(node.right)
end

# @param {TreeNode} root
# @return {Integer}
def get_minimum_difference(root)
  @min = Float::INFINITY
  inorder(root)
  @min  
end

Problem Classification

This problem belongs to Binary Search Trees (BST). A BST is a special type of binary tree where each node has a value, and the value of all nodes in the left subtree is less than the value of the root, and the value of all nodes in the right subtree is greater than the root.

What Components:

  1. A Binary Search Tree (BST) - It’s given as the input for the problem. We can assume that it’s properly constructed following the BST properties.
  2. Minimum absolute difference - The output of the problem. It’s the smallest absolute difference between the values of any two different nodes in the tree.

This is an algorithmic problem, which requires knowledge of the BST data structure and possibly the in-order traversal technique that allows for visiting nodes in ascending order. The goal is to find the pair of nodes in the BST such that the absolute difference between their values is minimized.

Language Agnostic Coding Drills

1. Dissect the code and identify each distinct concept it contains.

The code contains several distinct concepts:

a) Binary Trees: A binary tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child.

b) Tree Traversal: Specifically, in-order tree traversal. In this traversal method, the left subtree is visited first, then the root and later the right subtree.

c) Class Definition: The code defines a Python class, TreeNode, that represents a node in the binary tree.

d) Recursion: The inorder function calls itself to navigate through the tree, demonstrating the concept of recursion.

e) Lists: Lists are used to store the values of the tree nodes and to keep track of the minimum difference.

f) Loops: Loops are used to iterate through the list of node values.

g) Finding minimum: The concept of finding the minimum of a set of values is used when determining the minimum difference between node values.

2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty.

Ordered from simplest to most complex:

  1. Lists - Basic data structure used to store multiple items in a single variable.

  2. Loops - Essential for many tasks in programming. Used here to iterate over a list.

  3. Finding minimum - Requires understanding of comparison and updating a variable conditionally.

  4. Class Definition - Understanding how to create classes is crucial for object-oriented programming.

  5. Binary Trees - Requires understanding of nodes and the relationship between them.

  6. Recursion - A more advanced concept, requiring understanding of the call stack and how functions can call themselves.

  7. Tree Traversal - Specifically in-order traversal, which is a more complex application of recursion.

3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution.

The general approach to this problem is to leverage the properties of the binary search tree (BST), where in-order traversal visits nodes in ascending order. This means the difference between each pair of subsequent nodes visited during the in-order traversal gives us all the differences between nodes. We can then find the minimum of these differences.

Here are the steps:

a) Perform in-order traversal of the BST, saving the value of each visited node into a list. In Python, this would involve recursive calls.

b) Iterate through the list, calculating the difference between each pair of subsequent node values.

c) Keep track of the smallest difference found. This involves initializing a variable to a very large value (or infinity) and then updating it whenever a smaller difference is found.

d) Return the smallest difference.

Targeted Drills in Python

1. Python code for each identified concept

a) Lists

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# Create a list
my_list = [1, 2, 3, 4, 5]
# Add to a list
my_list.append(6)

b) Loops

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# Loop over a list
for i in my_list:
    print(i)

c) Finding minimum

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# Find minimum of a list
min_value = min(my_list)

d) Class Definition

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# Define a class
class MyClass:
    def __init__(self, x):
        self.x = x

e) Binary Trees

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# Define a binary tree node class
class TreeNode:
    def __init__(self, x):
        self.val = x
        self.left = None
        self.right = None

f) Recursion

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# Recursive function to calculate factorial
def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n-1)

g) Tree Traversal

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# In-order traversal of a binary tree
def inorder(root):
    if root:
        inorder(root.left)
        print(root.val)
        inorder(root.right)

2. Problem-specific concepts

The problem-specific concept is the usage of in-order tree traversal to solve this problem. Because of the properties of a Binary Search Tree (BST), an in-order traversal visits the nodes in ascending order. We leverage this property to find the minimum absolute difference between the values of any two different nodes in the tree.

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# In-order traversal of a binary tree and storing the values in a list
def inorder(root, output):
    if root:
        inorder(root.left, output)
        output.append(root.val)
        inorder(root.right, output)

3. Integrating the concepts to solve the problem

a) We start by defining the binary tree structure with the TreeNode class.

b) We perform an in-order traversal on the given tree using recursion, and store the values of each visited node in a list.

c) We then loop through the list, calculating the difference between each pair of subsequent node values.

d) We keep track of the smallest difference found. We initialize a variable to a very large value (or infinity) and then update it whenever a smaller difference is found.

e) We finally return the smallest difference as the result. This involves all the Python concepts learned from the basic to the complex, starting from lists, loops, finding the minimum, defining classes, understanding binary trees, applying recursion, and traversing trees.