Question

consider n mobile phones and m base stations. each phone or base station is specified by its (x, y) coordinates in the plane. each phone is connected to exactly one base stations which is within distance r. each base station cannot serve more than l phones. design a polynomial-time algorithm to connect phones to base stations. show your algorithm's correctness.

Answer 1

The algorithm works as follows:

1. Calculate the distance between each phone and each base station.

2. Create a priority queue of phone-base station pairs sorted by the distance between them.

3. Iterate through the priority queue and connect the phones that are within distance r to the base station.

4. Stop connecting when the base station reaches its limit of l phones.

5. Remove the connected pairs from the queue.

6. Repeat steps 3-5 until all phones are connected.

7. Return the connected phone-base station pairs.

This algorithm is correct because it ensures that each phone is connected to a base station that is within distance r. Additionally, each base station will not exceed its limit of l phones. This algorithm runs in polynomial time, as it requires only a single pass through the priority queue.

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