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ambitious

For each node in a tree find the sum of distances to all other nodes.

Programming language:

A computer network consisting of N routers and N−1 links connecting them is given. Routers are labeled with distinct integers within the range [0..(N−1)]. Links connect routers in such a way that each distinct pair of routers is connected either by a direct link or through a path consisting of direct links. There is exactly one way to reach any router from another and the number of direct links that must be traversed is called the *distance* between these two routers. For example, consider the following network consisting of ten routers and nine links:

Routers 2 and 4 are connected directly, so the distance between them is 1. Routers 4 and 7 are connected through a path consisting of direct links 4−0, 0−9 and 9−7; hence the distance between them is 3.

The location of a router in the network determines how quickly a packet dispatched by that router can reach other routers. The *peripherality* of a router is the average distance to all other routers on the network. For example, the peripherality of router 4 in the network shown above is 2.11, because:

distance to 0: 1 distance to 1: 3 distance to 2: 1 distance to 3: 3 distance to 5: 1 distance to 6: 3 distance to 7: 3 distance to 8: 2 distance to 9: 2 average: 19/9 = 2.11

The peripherality of router 0 is 1.66 and no other router has lower peripherality.

Write a function

class Solution { public int solution(int[] T); }

that, given a non-empty array T consisting of N integers describing a network of N routers and N−1 links, returns the label of the router that has minimum peripherality. If there is more than one router that has minimum peripherality, the function should return the lowest label.

Array T describes a network of routers as follows:

- if T[P] = Q and P ≠ Q, then there is a direct link between routers P and Q.

For example, given the following array T consisting of ten elements:

the function should return 0, because this array describes the network shown above and router 0 has minimum peripherality.

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [1..100,000];
- 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 routers.

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