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Minimize Direction Violations in a Directed Road Network

GoogleSoftware Engineer · Onsite
MediumSoftware EngineerOnsiteCoding & Algorithms

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Find the minimum number of edge reversals needed to travel from a source node to a target node in a directed graph. Tests graph modeling (0-1 weighted edges), edge cases like parallel edges and disconnected components, and efficient shortest-path traversal.

Problem Statement

You are given a directed road network represented by n intersections labeled from 0 to n - 1 and a list of directed roads roads, where each roads[i] = [u, v] represents a one-way street from intersection u to intersection v.

An emergency vehicle needs to travel from a starting intersection source to a destination intersection target. The vehicle can travel along any road in its designated direction at a cost of 0 direction violations. It may also travel against the designated direction (from v to u) at the cost of 1 direction violation.

Return the minimum number of direction violations required to travel from source to target. If target cannot be reached from source under any combination of forward and reverse traversals, return -1.

Function Signature

def min_direction_violations(n: int, roads: list[list[int]], source: int, target: int) -> int:
    pass

Input & Output Format

  • n: An integer representing the number of intersections.
  • roads: A list of pairs [u, v] where 0 <= u, v < n and u != v.
  • source: An integer representing the start intersection.
  • target: An integer representing the destination intersection.
  • Returns: An integer representing the minimum violations, or -1 if unreachable.

Constraints

  • 1 <= n <= 2 * 10^5
  • 0 <= len(roads) <= 3 * 10^5
  • `0 <= source, target 1: cost 0 (road exists 0 -> 1).
  • Travel 1 -> 2: cost 1 (traversing against original road 2 -> 1).
  • Travel 2 -> 3: cost 0 (road exists 2 -> 3). Total violations = 0 + 1 + 0 = 1.

### Example 2

```text
Input: n = 3, roads = [[0, 1]], source = 0, target = 2
Output: -1
Explanation: Node 2 is completely disconnected from the rest of the network.

Edge Cases to Consider

  • source == target: Result should be 0 immediately without needing any road traversals.
  • Multiple roads between the same pair of nodes (e.g., both [u, v] and [v, u] exist in roads): Traversal in either direction should cost 0.
  • Unconnected components: Handle properly when target cannot be reached.

Hints

Hint 1

Model the road network as a graph where each original road u -> v has a directed edge u -> v with weight 0 and a reverse edge v -> u with weight 1.

Hint 2

Since edge weights are strictly 0 and 1, standard Dijkstra's algorithm works, but a 0-1 BFS using a double-ended queue can achieve linear time.

Hint 3

Be careful when both u -> v and v -> u exist in the input: the traversal cost in both directions should be 0, not 1.

Follow-up questions

  • How would you reconstruct and return the actual sequence of vertices along the optimal path rather than just the minimum violation count?
  • If traveling against a one-way street had a variable penalty w > 0 depending on the street type, how would your approach and time complexity change?
  • How would you modify the solution if we wanted to find the minimum violations from a single source to ALL other vertices simultaneously in a distributed setting?