A10835 | Mahmoud and a xor trip

Mahmoud and Ehab live in a country with $n$ cities numbered from $1$ to $n$ and connected by $n-1$ undirected roads. It's guaranteed that you can reach any city from any other using these roads. Each city has a number $a_{i}$ attached to it.

We define the distance from city $x$ to city $y$ as the xor of numbers attached to the cities on the path from $x$ to $y$ (including both $x$ and $y$ ). In other words if values attached to the cities on the path from $x$ to $y$ form an array $p$ of length $l$ then the distance between them is , where  is bitwise xor operation.

Mahmoud and Ehab want to choose two cities and make a journey from one to another. The index of the start city is always less than or equal to the index of the finish city (they may start and finish in the same city and in this case the distance equals the number attached to that city). They can't determine the two cities so they try every city as a start and every city with greater index as a finish. They want to know the total distance between all pairs of cities.

The first line contains integer $n$ ( $1<=n<=10^{5}$ ) — the number of cities in Mahmoud and Ehab's country.

Then the second line contains $n$ integers $a_{1},a_{2},...,a_{n}$ ( $0<=a_{i}<=10^{6}$ ) which represent the numbers attached to the cities. Integer $a_{i}$ is attached to the city $i$ .

Each of the next $n-1$ lines contains two integers $u$ and $v$ ( $1<=u,v<=n$ , $u≠v$ ), denoting that there is an undirected road between cities $u$ and $v$ . It's guaranteed that you can reach any city from any other using these roads.

Output one number denoting the total distance between all pairs of cities.