Minimize Direction Violations in a Directed Road Network
Company: Google
Role: Software Engineer
Category: Coding & Algorithms
Difficulty: medium
Interview Round: Onsite
# Minimize Direction Violations in a Directed Road Network
There are `n` nodes numbered from `0` through `n - 1`. Every input edge `[u, v]` has an original direction from `u` to `v`, but you may physically traverse it in either direction.
- Traversing an edge in its original direction costs `0`.
- Traversing it against its original direction costs `1`.
Given `start` and `end`, return the minimum total cost of any route from `start` to `end`. Return `-1` if the two nodes are disconnected even when edges may be traversed both ways.
## Function Signature
```python
def minimum_direction_violations(
n: int,
edges: list[list[int]],
start: int,
end: int,
) -> int:
...
```
## Constraints
- `1 <= n <= 200_000`
- `0 <= len(edges) <= 300_000`
- `0 <= u, v < n` and `u != v`
- Multiple directed edges between the same pair of nodes are allowed.
- `0 <= start, end < n`
## Examples
```text
Input: n = 5, edges = [[0, 1], [2, 1], [2, 3], [4, 3]], start = 0, end = 4
Output: 2
```
```text
Input: n = 4, edges = [[0, 1], [1, 2]], start = 0, end = 3
Output: -1
```
```text
Input: n = 3, edges = [[0, 1], [1, 2]], start = 2, end = 0
Output: 2
```
Quick Answer: Find a route through a directed road network that minimizes travel against the original edge directions. Model forward and reverse traversal costs correctly, handle parallel edges and disconnected nodes, and choose a shortest-path approach that scales to hundreds of thousands of vertices and edges.