Lecture 94:- Recursion - Level 4

At Level 4 of understanding recursion, let's explore even more advanced topics and complex problems involving recursion:

Advanced Divide and Conquer: The divide and conquer technique can be applied to more complex problems, such as finding the maximum subarray sum (Kadane's algorithm) or matrix multiplication (Strassen's algorithm).

Example - Maximum Subarray Sum using Divide and Conquer:

In this example, max_subarray_sum finds the maximum sum of a subarray using the divide and conquer technique.

Recursive Tree Traversals: Tree traversal algorithms (pre-order, in-order, post-order) can be implemented using recursion to visit nodes in a tree-based data structure.

Example - In-order Traversal:

In this example, in_order_traversal performs an in-order traversal of a binary tree.

Tail Call Optimization (TCO): Some programming languages and compilers optimize tail-recursive functions by reusing the current function's stack frame for the next recursive call, eliminating the need for additional stack space.

Mutually Recursive Functions with Tail Call Optimization: Tail call optimization can be applied to mutually recursive functions to optimize memory usage.

Recursive Sorting Algorithms: Recursive sorting algorithms like QuickSort and MergeSort use recursion to sort elements efficiently.

Expression Parsing and Evaluation: Recursive descent parsing is a top-down parsing technique used to parse expressions and evaluate them.

Recursion in Graph Algorithms: Graph algorithms like Depth-First Search (DFS) and Breadth-First Search (BFS) can be implemented using recursion.

Recursion at Level 4 delves into more advanced topics, such as advanced divide and conquer, recursive tree traversals, and recursion in graph algorithms. It also covers optimization techniques like tail call optimization and explores how recursion can be applied to more complex problems. Understanding these concepts and practicing their implementations will further enhance your problem-solving skills and algorithmic expertise.

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class TreeNode:    def __init__(self, val=0, left=None, right=None):        self.val = val        self.left = left        self.right = right def in_order_traversal(root):    if root is None:        return    in_order_traversal(root.left)    print(root.val)    in_order_traversal(root.right)

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def max_subarray_sum(arr, left, right):    if left == right:        return arr[left]    mid = (left + right) // 2    left_max = max_subarray_sum(arr, left, mid)    right_max = max_subarray_sum(arr, mid + 1, right)    cross_max = max_crossing_sum(arr, left, mid, right)    return max(left_max, right_max, cross_max) def max_crossing_sum(arr, left, mid, right):    left_sum = float('-inf')    sum = 0    for i in range(mid, left - 1, -1):        sum += arr[i]        left_sum = max(left_sum, sum)    right_sum = float('-inf')    sum = 0    for i in range(mid + 1, right + 1):        sum += arr[i]        right_sum = max(right_sum, sum)    return left_sum + right_sum