3607. Power Grid Maintenance

3607. Power Grid Maintenance

Difficulty: Medium

Topics: Array, Hash Table, Depth-First Search, Breadth-First Search, Union Find, Graph, Heap (Priority Queue), Ordered Set, Weekly Contest 457

You are given an integer c representing c power stations, each with a unique identifier id from 1 to c (1‑based indexing).

These stations are interconnected via n bidirectional cables, represented by a 2D array connections, where each element connections[i] = [uᵢ vᵢ] indicates a connection between station uᵢ and station vᵢ. Stations that are directly or indirectly connected form a power grid.

Initially, all stations are online (operational).

You are also given a 2D array queries, where each query is one of the following two types:

• [1, x]: A maintenance check is requested for station x. If station x is online, it resolves the check by itself. If station x is offline, the check is resolved by the operational station with the smallest id in the same power grid as x. If no operational station exists in that grid, return -1.

• [2, x]: Station x goes offline (i.e., it becomes non-operational).

Return an array of integers representing the results of each query of type [1, x] in the order they appear.

Note: The power grid preserves its structure; an offline (non‑operational) node remains part of its grid and taking it offline does not alter connectivity.

Example 1:

• Input: c = 5, connections = [[1,2],[2,3],[3,4],[4,5]], queries = [[1,3],[2,1],[1,1],[2,2],[1,2]]
• Output: [3,2,3]
• Explanation:
  • Initially, all stations {1, 2, 3, 4, 5} are online and form a single power grid.
  • Query [1,3]: Station 3 is online, so the maintenance check is resolved by station 3.
  • Query [2,1]: Station 1 goes offline. The remaining online stations are {2, 3, 4, 5}.
  • Query [1,1]: Station 1 is offline, so the check is resolved by the operational station with the smallest id among {2, 3, 4, 5}, which is station 2.
  • Query [2,2]: Station 2 goes offline. The remaining online stations are {3, 4, 5}.
  • Query [1,2]: Station 2 is offline, so the check is resolved by the operational station with the smallest id among {3, 4, 5}, which is station 3.

Example 2:

• Input: c = 3, connections = [], queries = [[1,1],[2,1],[1,1]]
• Output: [1,-1]
• Explanation:
  • There are no connections, so each station is its own-isolated grid.
  • Query [1,1]: Station 1 is online in its isolated grid, so the maintenance check is resolved by station 1.
  • Query [2,1]: Station 1 goes offline.
  • Query [1,1]: Station 1 is offline and there are no other stations in its grid, so the result is -1.

Constraints:

• 1 <= c <= 10⁵
• 0 <= n == connections.length <= min(10⁵, c * (c - 1) / 2)
• connections[i].length == 2
• 1 <= uᵢ, vᵢ <= c
• uᵢ != vᵢ
• 1 <= queries.length <= 2 * 10⁵
• queries[i].length == 2
• queries[i][0] is either 1 or 2.
• 1 <= queries[i][1] <= c

Hint:

1. Use DFS or BFS to assign each station a component ID
2. For each component, maintain a sorted set of online station IDs
3. For query [2, x], remove x from the set of its component
4. For query [1, x], if x is in its component’s set return x; otherwise if the set is non-empty return its smallest element; else return -1
5. Precompute all components and then handle each query in O(log n) time using the sorted sets

Solution:

We need to handle two types of queries on a power grid where stations can go offline, and I need to find the smallest operational station in the same connected component.

Approach

1. Union-Find for Connectivity: First, I'll use Union-Find to group stations into connected components based on the given connections.

2. Track Operational Status: Maintain an array to track which stations are operational (online).

3. Component Minimum Tracking: For each component, track the smallest operational station ID. When stations go offline, I need to efficiently update this minimum.

4. Query Processing:

   • For type 1 queries: If station is online, return itself; otherwise return the minimum operational station in its component
   • For type 2 queries: Mark station as offline and update the component's minimum if needed

Let's implement this solution in PHP: 3607. Power Grid Maintenance

<?php
/**
 * @param Integer[] $c
 * @param Integer[][] $connections
 * @param Integer[][] $queries
 * @return Integer[]
 */
function processQueries($c, $connections, $queries) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
print_r(processQueries(5, [[1,2],[2,3],[3,4],[4,5]], [[1,3],[2,1],[1,1],[2,2],[1,2]])); // Output: [3, 2, 3]
print_r(processQueries(3, [], [[1,1],[2,1],[1,1]]); // Output: [1,-1]
?>

Explanation:

1. Union-Find efficiently manages connectivity between stations
2. Min-heaps provide O(log k) access to the smallest operational station in each component
3. Lazy deletion from heaps avoids expensive removal operations
4. The solution handles both edge cases: when all stations in a component are offline, and when stations are isolated

Complexity

• Building components: O(c + n) where c is stations, n is connections
• Each query: O(log k) where k is component size (for heap operations)
• Total: O((c + n) + q log k) where q is number of queries

Space Complexity: O(c + n) for storing graph and component data

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