3243. Shortest Distance After Road Addition Queries I - Detailed Explanation

Problem Statement

You are given an integer n representing the number of cities (labeled from 0 to n–1), a list roads where each element is an array [u, v, w] representing an undirected road between cities u and v with weight w, two fixed integers start and target indicating the cities between which you want the shortest distance, and a list queries where each query is an array [u, v, w'] representing a road that could be temporarily added between cities u and v with weight w'. For each query, determine the minimum distance from start to target after adding the new road for that query only. If no valid path exists, return –1.

Examples

Example Input 1

• n = 5
• roads = [[0,1,2], [1,2,3], [2,3,4], [3,4,5]]
• start = 0, target = 4
• queries = [[0,4,1]]

Example Output 1

• [1]

Explanation: Without the new road, the only path from 0 to 4 is 0 → 1 → 2 → 3 → 4 with a total cost of 2+3+4+5 = 14. With the added road [0,4,1], a direct path 0 → 4 is available with a cost of 1, which is optimal.

Example Input 2

• n = 4
• roads = [[0,1,1], [1,2,1], [2,3,1]]
• start = 0, target = 3
• queries = [[0,2,1]]

Example Output 2

• [2]

Explanation: The original distance from 0 to 3 is 3. Adding the road [0,2,1] creates a new path 0 → 2 → 3 with a cost of 1+1 = 2.

Example Input 3

• n = 3
• roads = [[0,1,5]]
• start = 0, target = 2
• queries = [[1,2,2], [0,2,10]]

Example Output 3

• [7, 10]

Explanation: For the first query, adding [1,2,2] gives the path 0 → 1 → 2 with a cost of 5+2 = 7. For the second query, adding [0,2,10] provides a direct path from 0 to 2 with a cost of 10.

Constraints

• 1 ≤ n ≤ 100
• 1 ≤ roads.length, queries.length ≤ 10⁴
• 0 ≤ u, v < n
• 1 ≤ w, w' ≤ 10⁵
• The road network is undirected.
• The graph may have disconnected components.

Hints for the Solution

Hint 1: Precompute the shortest paths from start to every city and from target to every city using Dijkstra’s algorithm. This precomputation allows you to quickly evaluate the impact of adding any new road.
Hint 2: For each query [u, v, w'], consider both directions:

• Option 1: Use the precomputed distance from start to u, then the new road from u to v, and finally the distance from v to target.
• Option 2: Use the precomputed distance from start to v, then the new road from v to u, and finally the distance from u to target.
  Compare these with the original shortest distance from start to target and choose the minimum.

Brute Force Approach

Idea: For every query, temporarily add the road to the graph and run a shortest path algorithm (e.g., Dijkstra) from start to target.
Steps:

• For each query, insert the road into the graph.
• Run Dijkstra’s algorithm to compute the shortest path from start to target.
• Remove the temporary road after processing the query.
  Drawbacks:
• Inefficient for a large number of queries and a nontrivial graph.
  Time Complexity: O(q × (m log n)), where q is the number of queries and m is the number of roads.

Optimal Approach (Precomputation)

Idea: Precompute the shortest distances from start and target to all nodes and then use these values to quickly evaluate each query.

Steps:

1. Precomputation:
   • Run Dijkstra’s algorithm from start to compute dist_from_start[] for all nodes.
   • Run Dijkstra’s algorithm from target to compute dist_from_target[] for all nodes.
2. Query Evaluation:
   • For each query [u, v, w'], calculate:
     • candidate1 = dist_from_start[u] + w' + dist_from_target[v]
     • candidate2 = dist_from_start[v] + w' + dist_from_target[u]
   • Compare these candidates with the original distance from start to target.
   • Return the minimum valid distance, or –1 if no path exists.
     Time Complexity:

• Precomputation: Two Dijkstra runs take O(m log n) each.
• Query Evaluation: O(1) per query.
• Overall: O(m log n + q).

Step-by-Step Walkthrough with Visual Example

Consider Example Input 1 with cities labeled 0 through 4 and the following graph:

   (2)  
0 ----- 1  
        |  
       (3)  
        |  
        2  
        |  
       (4)  
        |  
        3  
        |  
       (5)  
        |  
        4

1. Precomputation:

   • From start (0):
     • dist_from_start[0] = 0,
     • dist_from_start[1] = 2,
     • dist_from_start[2] = 5,
     • dist_from_start[3] = 9,
     • dist_from_start[4] = 14.
   • From target (4):
     • dist_from_target[4] = 0,
     • dist_from_target[3] = 5,
     • dist_from_target[2] = 9,
     • dist_from_target[1] = 12,
     • dist_from_target[0] = 14.

2. Processing the Query [0,4,1]:

   • Option 1: 0 → 0 (0) + new road (1) + 4 → 4 (0) = 1.
   • Option 2: 0 → 4 (14) + new road (1) + 0 → 0 (0) = 15.
   • Compare with the original path cost of 14.
   • The minimum is 1.

Common Mistakes

• Ignoring Both Directions: Always check both (u to v and v to u) since the graph is undirected.
• Handling Unreachable Nodes: Ensure that if any segment of the precomputed path is unreachable (represented as infinity), that candidate is disregarded.
• Forgetting the Original Distance: Do not overlook the possibility that the original path might already be optimal.

Edge Cases and Alternative Variations

Disconnected Graph:

• If there is no original path from start to target, the new road might be the only viable option. If both precomputed values are infinite, return –1.

Directed Graph Variation:

• For directed graphs, run Dijkstra on the original graph and on its reverse, adjusting the candidate computations accordingly.

Permanent Road Addition Variation:

• If queries permanently add a road, you must update the graph and recompute the shortest paths dynamically using appropriate algorithms for dynamic graphs.

Python Code Implementation