3243. Shortest Distance After Road Addition Queries I

[mah-shamim]

Topics: Array, Breadth-First Search, Graph

You are given an integer n and a 2D integer array queries.

There are n cities numbered from 0 to n - 1. Initially, there is a unidirectional road from city i to city i + 1 for all 0 <= i < n - 1.

queries[i] = [ui, vi] represents the addition of a new unidirectional road from city ui to city vi. After each query, you need to find the length of the shortest path from city 0 to city n - 1.

Return an array answer where for each i in the range [0, queries.length - 1], answer[i] is the length of the shortest path from city 0 to city n - 1 after processing the first i + 1 queries.

Example 1:

• Input: n = 5, queries = [[2,4],[0,2],[0,4]]
• Output: [3,2,1]
• Explanation:
  
  After the addition of the road from 2 to 4, the length of the shortest path from 0 to 4 is 3.
  
  After the addition of the road from 0 to 2, the length of the shortest path from 0 to 4 is 2.
  
  After the addition of the road from 0 to 4, the length of the shortest path from 0 to 4 is 1.

Example 2:

• Input: n = 4, queries = [[0,3],[0,2]]
• Output: [1,1]
• Explanation:
  
  After the addition of the road from 0 to 3, the length of the shortest path from 0 to 3 is 1.
  
  After the addition of the road from 0 to 2, the length of the shortest path remains 1.

Constraints:

• 3 <= n <= 500
• 1 <= queries.length <= 500
• queries[i].length == 2
• 0 <= queries[i][0] < queries[i][1] < n
• 1 < queries[i][1] - queries[i][0]
• There are no repeated roads among the queries.

Hint:

1. Maintain the graph and use an efficient shortest path algorithm after each update.
2. We use BFS/Dijkstra for each query.

[mah-shamim]

We need to simulate adding roads between cities and calculate the shortest path from city 0 to city n - 1 after each road addition. Given the constraints and the nature of the problem, we can use Breadth-First Search (BFS) for unweighted graphs.

Approach:

1. Graph Representation:

   • We can represent the cities and roads using an adjacency list. Initially, each city i will have a road to city i + 1 for all 0 <= i < n - 1.
   • After each query, we add the road from u_i to v_i into the graph.

2. Shortest Path Calculation (BFS):

   • We can use BFS to calculate the shortest path from city 0 to city n - 1. BFS works well here because all roads have equal weight (each road has a length of 1).

3. Iterating over Queries:

   • For each query, we will add the new road to the graph and then use BFS to find the shortest path from city 0 to city n - 1. After processing each query, we store the result in the output array.

4. Efficiency:

   • Since we are using BFS after each query, and the graph size can be at most 500 cities with up to 500 queries, the time complexity for each BFS is O(n + m), where n is the number of cities and m is the number of roads. We need to perform BFS up to 500 times, so the solution will run efficiently within the problem's constraints.

Let's implement this solution in PHP: 3243. Shortest Distance After Road Addition Queries I

<?php
/**
 * @param Integer $n
 * @param Integer[][] $queries
 * @return Integer[]
 */
function shortestDistanceAfterQueries($n, $queries) {
    // Initialize adjacency list for the graph
    $graph = array_fill(0, $n, []);
    for ($i = 0; $i < $n - 1; $i++) {
        $graph[$i][] = $i + 1; // Add the default unidirectional road
    }

    $result = [];

    // Process each query
    foreach ($queries as $query) {
        list($u, $v) = $query;
        $graph[$u][] = $v; // Add the new road

        // Calculate the shortest path from 0 to n-1 after adding this road
        $result[] = bfs($graph, $n);
    }

    return $result;
}

/**
 * Function to find the shortest path using BFS
 *
 * @param $graph
 * @param $n
 * @return int
 */
function bfs($graph, $n) {
    $queue = [[0, 0]]; // [current node, distance from source]
    $visited = array_fill(0, $n, false);
    $visited[0] = true;

    while (!empty($queue)) {
        list($node, $dist) = array_shift($queue);

        if ($node == $n - 1) {
            return $dist; // Found the shortest path to city n-1
        }

        foreach ($graph[$node] as $neighbor) {
            if (!$visited[$neighbor]) {
                $visited[$neighbor] = true;
                $queue[] = [$neighbor, $dist + 1];
            }
        }
    }

    return -1; // If there's no path (shouldn't happen in this problem)
}

// Example 1
$n = 5;
$queries = [[2, 4], [0, 2], [0, 4]];
print_r(shortestDistanceAfterQueries($n, $queries));

// Example 2
$n = 4;
$queries = [[0, 3], [0, 2]];
print_r(shortestDistanceAfterQueries($n, $queries));
?>

Explanation:

1. Graph Initialization:

   • An adjacency list graph is used to represent the cities and roads.
   • Initially, roads exist only between consecutive cities (i to i+1).

2. BFS Function:

   • BFS is used to compute the shortest distance from city 0 to city n - 1. We maintain a queue for BFS and an array distances to store the minimum number of roads (edges) to reach each city.
   • Initially, the distance to city 0 is set to 0, and all other cities have a distance of infinity (PHP_INT_MAX).
   • As we process each city in the BFS queue, we update the distances for its neighboring cities and continue until all reachable cities are visited.

3. Query Processing:

   • For each query, the new road is added to the graph (u -> v).
   • BFS is called to calculate the shortest path from city 0 to city n-1 after the update.
   • The result of BFS is stored in the result array.

4. Output:

   • The result array contains the shortest path lengths after each query.

5. Time Complexity:

   • Each BFS takes O(n + m), where n is the number of cities and m is the number of roads. Since the number of queries is q, the overall time complexity is O(q * (n + m)), which should be efficient for the given constraints.

Example Walkthrough:

For the input n = 5 and queries = [[2, 4], [0, 2], [0, 4]]:

• Initially, the roads are [0 -> 1 -> 2 -> 3 -> 4].
• After the first query [2, 4], the roads are [0 -> 1 -> 2 -> 3 -> 4] and the shortest path from 0 to 4 is 3 (using the path 0 -> 1 -> 2 -> 4).
• After the second query [0, 2], the roads are [0 -> 2, 1 -> 2 -> 3 -> 4], and the shortest path from 0 to 4 is 2 (using the path 0 -> 2 -> 4).
• After the third query [0, 4], the roads are [0 -> 2, 1 -> 2 -> 3 -> 4], and the shortest path from 0 to 4 is 1 (direct road 0 -> 4).

Thus, the output is [3, 2, 1].

Final Thoughts:

• The solution uses BFS for each query to efficiently calculate the shortest path.
• The graph is dynamically updated as new roads are added in each query.
• The solution works well within the problem's constraints and efficiently handles up to 500 queries with up to 500 cities.

[kovatz]

Thanks for solving the problem "Shortest Distance After Road Addition Queries I"

[mah-shamim]

Thanks