Count Valid Paths in a Tree

Count Valid Paths in a Tree

What is this problem about?

The Count Valid Paths in a Tree coding problem is an advanced graph problem that incorporates number theory. You are given a tree with $n$ nodes labeled from 1 to $n$. A path between two nodes $(u, v)$ is considered "valid" if there is exactly one prime number among the labels of the nodes on that path. The goal is to return the total number of such valid paths.

Why is this asked in interviews?

This "Hard" difficulty tree interview pattern is common at high-frequency trading firms like Gameskraft and top tech companies like Sprinklr. It tests a candidate's ability to combine multiple domains: Number Theory (Sieve of Eratosthenes), Graph Theory (Tree Traversal), and Dynamic Programming. It assesses how well you can optimize a path-counting problem using component decomposition.

Algorithmic pattern used

The optimal strategy uses a Modified DFS with Component Counting:

1. Sieve of Eratosthenes: Precompute all prime numbers up to $n$.
2. Component Decomposition: Remove all prime-labeled nodes from the tree. This breaks the tree into several connected components consisting only of non-prime nodes.
3. Path Logic: A valid path must contain exactly one prime. This means it must connect:
   • A prime node to a non-prime component.
   • Two non-prime components through a single prime node.
4. DFS: For each prime node, count the sizes of adjacent non-prime components to calculate the number of valid paths passing through or starting at that prime.

Example explanation

Consider a small tree: 1-2-3-4.

• Primes: 2, 3. Non-primes: 1, 4.
• Components of non-primes: {1} and {4}.
• Paths through prime 2:
  • (2, 1) - One prime.
  • (2, 3) - Two primes (Invalid).
  • (1, 2, 3) - Two primes (Invalid).
• Paths through prime 3:
  • (3, 4) - One prime.
  • (3, 2) - Two primes (Invalid).
  • (2, 3, 4) - Two primes (Invalid).
• Path (1, 2, 3, 4) - Two primes (Invalid). The algorithm efficiently counts the sizes of the "islands" of non-prime numbers surrounding each prime node.

Common mistakes candidates make

• Re-calculating Primes: Not using a sieve, leading to redundant primality checks.
• O(n^2) Path Check: Attempting to verify every path individually instead of using component sizes.
• Handling 1: Forgetting that the number 1 is NOT prime.

Interview preparation tip

When a tree problem asks about paths with "exactly one" of something, try removing those specific nodes and looking at the resulting "forest." This often simplifies the counting logic significantly.