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Plan trips to destination cities so as to visit a maximal number of other unvisited cities en route.
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Task description

A country network consisting of N cities and N − 1 roads connecting them is given. Cities are labeled with distinct integers within the range [0..(N − 1)]. Roads connect cities in such a way that each distinct pair of cities is connected either by a direct road or through a path consisting of direct roads. There is exactly one way to reach any city from any other city.

Starting out from city K, you have to plan a series of daily trips. Each day you want to visit a previously unvisited city in such a way that, on a route to that city, you will also pass through a maximal number of other unvisited cities (which will then be considered to have been visited). We say that the destination city is our daily travel target.

In the case of a tie, you should choose the city with the minimal label. The trips cease when every city has been visited at least once.

For example, consider K = 2 and the following network consisting of seven cities and six roads:

You start in city 2. From here you make the following trips:

  • day 1 − from city 2 to city 0 (cities 1 and 0 become visited),
  • day 2 − from city 0 to city 6 (cities 4 and 6 become visited),
  • day 3 − from city 6 to city 3 (city 3 becomes visited),
  • day 4 − from city 3 to city 5 (city 5 becomes visited).

The goal is to find the sequence of travel targets. In the above example we have the following travel targets: (2, 0, 6, 3, 5).

Write a function:

vector<int> solution(int K, vector<int> &T);

that, given a non-empty array T consisting of N integers describing a network of N cities and N − 1 roads, returns the sequence of travel targets.

Array T describes a network of cities as follows:

  • if T[P] = Q and P ≠ Q, then there is a direct road between cities P and Q.

For example, given the following array T consisting of seven elements (this array describes the network shown above) and K = 2:

T[0] = 1 T[1] = 2 T[2] = 3 T[3] = 3 T[4] = 2 T[5] = 1 T[6] = 4

the function should return a sequence [2, 0, 6, 3, 5], as explained above.

Write an efficient algorithm for the following assumptions:

  • N is an integer within the range [1..90,000];
  • each element of array T is an integer within the range [0..(N−1)];
  • there is exactly one (possibly indirect) connection between any two distinct roads.
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Programming language used C++
Total time used 60 minutes
Effective time used 60 minutes
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