Distinct Gate Paths to LCA - Solution & Explanation

Problem Overview: You are given a tree where certain nodes represent gates. For pairs or groups of nodes, determine how many distinct paths reach their lowest common ancestor (LCA). The challenge is identifying unique gate-to-LCA routes efficiently without recomputing paths for every query.

Approach 1: Brute Force DFS per Query (O(n) per query time, O(n) space)

For every query, run a DFS from each gate node up toward the root and explicitly track the path until the LCA is found. Store visited nodes in a set to avoid double counting. This works because trees guarantee a single path between nodes, but repeatedly traversing the tree makes it expensive when queries are large. This approach is mainly useful for validating correctness or when the number of queries is very small. Traversal can be implemented using standard DFS on the tree.

Approach 2: Path Reconstruction Using Parent Pointers (O(h) per query time, O(n) space)

Precompute each node's parent and depth using a single DFS. For a query, walk both nodes upward until they meet at the LCA. While climbing, track which gates appear on the path using a hash set or frequency map. Because each step moves toward the root, the traversal cost depends on the tree height h. This reduces repeated full traversals but still becomes slow on skewed trees where h ≈ n. It demonstrates how path intersection naturally reveals the LCA.

Approach 3: Binary Lifting LCA with Prefix Path State (O((n + q) log n) time, O(n log n) space)

Preprocess the tree using binary lifting to answer LCA queries in O(log n). During the initial DFS, maintain prefix information along the root-to-node path—such as counts, bitmasks, or hash signatures representing which gates appear. For a query, compute the LCA using the binary lifting table. The distinct path contribution from each node can be derived using prefix values from the two nodes and subtracting the prefix of the LCA's parent. This transforms repeated path exploration into constant-time prefix arithmetic after the LCA lookup.

Recommended for interviews: The binary lifting + prefix state approach is what most interviewers expect for a hard tree problem. The brute-force DFS shows you understand path structure in trees, but the optimized solution proves you can combine preprocessing, prefix aggregation, and LCA queries to handle large inputs efficiently.