Validate BFS order queries on a tree
Problem Description
You are provided with the structure of an an undirected tree. This tree consists of n distinct nodes, each identified by an arbitrary integer label. The tree's n-1 edges are defined by two equal-length arrays, u and v. For each index i, an undirected edge exists between node u[i] and node v[i].
You will then receive q queries. Each query consists of an array order of length n. This order array is a permutation of all n node labels, meaning each node label appears exactly once.
For each given query, your task is to determine whether the order array could represent a valid output sequence of a standard Breadth-First Search (BFS) traversal on the provided tree, adhering to the following specific rules:
- The BFS traversal must always start from the node specified by
order[0]. - A standard First-In, First-Out (FIFO) queue must be used for the BFS.
- When a node is dequeued, its unvisited neighbors can be enqueued into the queue in any arbitrary order. The specific order of enqueuing neighbors does not matter, as long as they are all enqueued before processing nodes from subsequent levels.
- The BFS output is defined as the sequence of nodes in the order they are dequeued.
Your solution should return a boolean value for each query, indicating whether the order is a valid BFS output.
Example
Given u = [1, 1, 4] and v = [3, 2, 3], which defines edges (1,3), (1,2), (4,3).
For order = [1,2,3,4], the answer is true. One valid BFS traversal starting at 1 could enqueue 2 then 3 (or 3 then 2). If 2 then 3 are enqueued, the dequeue order would be 1, 2, 3, 4 (assuming 4 is enqueued after 3 when 3 is dequeued).
Constraints
n(number of nodes) can be up to2 * 10^5.q(number of queries) can be large. The solution should aim for a time complexity significantly better thanO(q * n^2).
AI interview prep
Core dilemma
The central engineering conflict is efficiently validating if a given node permutation `order` *could* be a BFS output, given the flexibility of neighbor enqueue order, across a large number of queries (`q`) on a potentially large tree (`n`). This requires understanding the fundamental properties of BFS levels and how the 'any order' rule impacts the valid sequences within a level.
3 tradeoffs and 3 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Validate BFS order queries on a tree
Problem Description
You are provided with the structure of an an undirected tree. This tree consists of n distinct nodes, each identified by an arbitrary integer label. The tree's n-1 edges are defined by two equal-length arrays, u and v. For each index i, an undirected edge exists between node u[i] and node v[i].
You will then receive q queries. Each query consists of an array order of length n. This order array is a permutation of all n node labels, meaning each node label appears exactly once.
For each given query, your task is to determine whether the order array could represent a valid output sequence of a standard Breadth-First Search (BFS) traversal on the provided tree, adhering to the following specific rules:
- The BFS traversal must always start from the node specified by
order[0]. - A standard First-In, First-Out (FIFO) queue must be used for the BFS.
- When a node is dequeued, its unvisited neighbors can be enqueued into the queue in any arbitrary order. The specific order of enqueuing neighbors does not matter, as long as they are all enqueued before processing nodes from subsequent levels.
- The BFS output is defined as the sequence of nodes in the order they are dequeued.
Your solution should return a boolean value for each query, indicating whether the order is a valid BFS output.
Example
Given u = [1, 1, 4] and v = [3, 2, 3], which defines edges (1,3), (1,2), (4,3).
For order = [1,2,3,4], the answer is true. One valid BFS traversal starting at 1 could enqueue 2 then 3 (or 3 then 2). If 2 then 3 are enqueued, the dequeue order would be 1, 2, 3, 4 (assuming 4 is enqueued after 3 when 3 is dequeued).
Constraints
n(number of nodes) can be up to2 * 10^5.q(number of queries) can be large. The solution should aim for a time complexity significantly better thanO(q * n^2).