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Position speed cameras so as to minimize the lengths of unmonitored paths.
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Task description

Recently, more and more illegal street races have been spotted at night in the city, and they have become a serious threat to public safety. Therefore, the Police Chief has decided to deploy speed cameras on the streets to collect evidence.

There are N+1 intersections in the city, connected by N roads. Every road has the same length of 1. A street race may take place between any two different intersections by using the roads connecting them. Limited by their budget, the police are able to deploy at most K speed cameras on these N roads. These K speed cameras should be installed such that the length of any possible street race route not covered by speed cameras should be as short as possible.

You are given a map of the city in the form of two arrays, A and B of length N, and an integer K:

  • For each J (0 ≤ J < N) there is a road connecting intersections A[J] and B[J].

The Police Chief would like to know the minimum length of the longest path out of surveillance after placing at most K speed cameras.

Write a function:

class Solution { public int solution(int[] A, int[] B, int K); }

that, given arrays A and B of N integers and integer K, returns the minimum length of the longest path unmonitored by speed cameras after placing at most K speed cameras.

For example, given K = 2 and the following arrays:

A[0] = 5 B[0] = 1 A[1] = 1 B[1] = 0 A[2] = 0 B[2] = 7 A[3] = 2 B[3] = 4 A[4] = 7 B[4] = 2 A[5] = 0 B[5] = 6 A[6] = 6 B[6] = 8 A[7] = 6 B[7] = 3 A[8] = 1 B[8] = 9

the function should return 2. Two speed cameras can be installed on the roads between intersections 1 and 0 and between intersections 0 and 7. (Another solution would be to install speed cameras between intersections 0 and 7 and between intersections 0 and 6.) By installing speed cameras according the first plan, one of the longest paths without a speed camera starts at intersection 8, passes through intersection 6 and ends at intersection 3, which consists of two roads. (Other longest paths are composed of intersections 5, 1, 9 and 7, 2, 4).

Write an efficient algorithm for the following assumptions:

  • N is an integer within the range [1..50,000];
  • each element of arrays A, B is an integer within the range [0..N];
  • K is an integer within the range [0..N];
  • the distance between any two intersections is not greater than 900.
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Programming language used Java
Total time used 2 minutes
Effective time used 2 minutes
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