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#!/bin/python3
import math
import os
// ... rest of solution available after purchaseExplanation
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To solve the Vaccination Drive problem, you need to figure out the shortest time it takes for all deficient centers (status 1) to receive vaccines from surplus centers (status 3) over a network of bidirectional roads. The roads connect various centers, and each traversal takes exactly one unit of time. The key challenge is to minimize the maximum delivery time across all deficient centers, considering that vaccines can be delivered in parallel from multiple surplus centers.
Start by understanding the problem constraints and the data structure involved. You have a graph where nodes represent vaccination centers and edges represent bidirectional roads connecting them. Each node has a status indicating if it is deficient, sufficient, or surplus. Your goal is to determine the shortest distance from any surplus center to each deficient center and find the maximum of these minimum distances.
Step 1: Represent the graph
Model the centers and roads as an undirected graph with nodes and edges. Each node is a vaccination center, and edges link centers with a traversal cost (time) of 1 unit. Since the roads are bidirectional, each edge allows movement in both directions.
Step 2: Identify surplus and deficient centers
Classify the nodes into three groups based on their status:
- Surplus centers (status 3), which serve as sources for vaccine delivery.
- Deficient centers (status 1), which require delivery.
- Sufficient centers (status 2), which do not affect delivery times but can be part of the path.
Step 3: Use multi-source shortest path search
Since vaccines can be shipped simultaneously from multiple surplus centers, the problem effectively becomes finding the shortest distance from any surplus center to every other center.
To solve this efficiently, perform a Breadth-First Search (BFS) starting simultaneously from all surplus centers. Instead of starting BFS from a single source, initialize the BFS queue with all surplus centers, each at distance zero.
This approach lets you propagate shortest distances outward from all surplus centers in parallel, calculating the minimum distance to every node.
Step 4: Traverse the graph with BFS
Perform BFS on the graph:
- Initialize a distance array or map to store the shortest distance to each center, starting with zero for all surplus centers and infinity for others.
- Visit nodes layer by layer, updating the distance to neighbors if the current path offers a shorter route.
- BFS guarantees that once a node is visited, its distance is the shortest from any surplus center.
Step 5: Extract distances for deficient centers
After completing BFS, look up the computed shortest distances for all deficient centers. These distances represent the minimum time needed for vaccines to reach each deficient center.
Step 6: Find the maximum delivery time
Since all deliveries can happen in parallel, the total minimum time to supply all deficient centers is the maximum shortest distance among them. This is because the slowest delivery determines when all centers are vaccinated.
Step 7: Handle unreachable deficient centers
If any deficient center is unreachable (distance remains infinity), it means no path exists from any surplus center to that node. In such cases, delivering vaccines to all deficient centers is impossible, and you may need to return an indicator (like -1) to denote failure.
Step 8: Complexity and efficiency
- Building the graph takes O(n + m), where n is the number of centers and m is the number of roads.
- BFS takes O(n + m) time, since each edge and node is processed once.
- Thus, the overall approach is efficient for large inputs.
Step 9: Why BFS and multi-source?
- BFS is ideal because the edges have uniform weight (1 unit time).
- Multi-source BFS simulates simultaneous deliveries from all surplus centers, reflecting the problem’s parallel nature.
- This avoids running BFS multiple times and is more efficient.
Summary:
1. Model centers and roads as a graph.
2. Identify surplus centers and initialize a BFS queue with them at zero distance.
3. Run BFS to find shortest distances to all centers.
4. Retrieve distances for deficient centers.
5. The answer is the maximum among these distances or -1 if any are unreachable.
This approach provides an optimal and clear method to determine the minimum time needed for vaccine delivery to all deficient centers given the problem’s constraints.
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