java code

There are N cities numbered from $1$ to N$.$

You are given connections, where each connections$[$i$]=[$city$1,$ city$2,$ cost$]$ represents the cost to connect city$1$ and city$2$together. $($A connection is bidirectional: connecting city$1$ and city$2$ is the same as connecting city$2$ and city$1.)$

Return the minimum cost so that for every pair of cities, there exists a path of connections $($possibly of length $1)$ that connects those two cities together. The cost is the sum of the connection costs used. If the task is impossible, return $-1.$

Example $1$:

Input: N $=3,$ connections $=[[1,2,5],[1,3,6],[2,3,1]]$

Output: $6$

Explanation:

Choosing any $2$ edges will connect all cities so we choose the minimum $2.$

Example $2$:

Input: N $=4,$ connections $=[[1,2,3],[3,4,4]]$

Output: $-1$

Explanation:

There is no way to connect all cities even if all edges are used.

Note:

$1<=$ N $<=10000$

$1<=$ connections.length $<=10000$

$1<=$ connections$[$i$][0],$ connections$[$i$][1]<=$ N

$0<=$ connections$[$i$][2]<=10^5$

connections$[$i$][0]!=$ connections$[$i$][1]$

Solution:

Try to connect cities with minimum cost, then find small cost edge first, if two cities connected by the edge do no have same ancestor, then union them.

When number of unions equal to $1,$ all cities are connected.

Time Complexity: O$($mlogm $+$ mlogN$).$ sort takes O$($mlogm$).$ find takes O$($logN$).$ With path compression and unino by weight, amatorize O$(1).$

Space: O$($N$).$