SDE - 1

Brute Force Method

Recursive Merge

Iterative Merge

In-Place Merge

Steps involved in solving the problem:

Create an empty adjacency list ‘T’ to represent the tree structure. For each edge in the tree, add the corresponding vertices to each other’s adjacency lists to represent the connections.
Initialize the variable ‘ans’ to n * (n – 1) * (n – 2) / 6, which represents the total number of tuples (i, j, k) with 1 ≤ i < j < k ≤ N.
Initialize a vector ‘sz’ of size ‘n’, where each element is initialized to 1. This vector will be used to store the size of each subtree rooted at every vertex.
Define a recursive function ‘dfs’ that takes two parameters ‘v’ (current vertex) and ‘pre’ (the parent vertex). The purpose of this function is to calculate the size of each subtree rooted at each vertex and update the ‘ans’ variable accordingly.
Inside the ‘dfs’ function, iterate through the adjacent vertices of ‘v’ (excluding the parent vertex ‘pre’). For each adjacent vertex ‘c’, recursively call the ‘dfs’ function with ‘c’ as the current vertex and ‘v’ as the parent vertex.
Calculate the ‘sum1‘ and ‘sum2‘ values for the current vertex ‘v’ based on the sizes of its subtrees and update the ‘sz’ vector accordingly.
Decrement ‘ans’ by (sum1 * sum1 – sum2) >> 1. This step removes the tuples (i, j, k) that form a simple path containing all three vertices i, j, and k in the tree.
Call the ‘dfs’ function with the starting vertex as 0 (or any other appropriate starting vertex) and the parent vertex as -1 (or any other value indicating that there is no parent).
Finally, print the ‘ans’ value, which represents the number of tuples (i, j, k) that satisfy the condition of not having a simple path containing all three vertices i, j, and k in the tree.

Tip 1: Clarify Requirements: Always ask clarifying questions to fully understand the problem and its constraints before designing.
Tip 2: Prioritize Scalability: Choose designs that can handle growing user bases, data volumes, and traffic patterns.
Tip 3: Explain Trade-offs: Clearly discuss the pros and cons of your design choices to showcase your understanding of flexibility and limitations.

Clarify Requirements: Always ask clarifying questions to fully understand the problem and constraints before designing.

Prioritize Scalability: Choose designs that can handle increasing user bases, data volumes, and traffic patterns.

Explain Trade-offs: Discuss the pros and cons of your design choices to demonstrate your understanding of flexibility and limitations.