Given $n$ cities, some are directly connected, and others are not. You are given an $n \times n$ matrix called cities, where cities[i][j] $= 1$ means city $i^{th}$ and city $j^{th}$ are directly connected, and cities[i][j] $= 0$ means they are not directly connected.

Note: If city $A$ is directly connected to city $B$ and city $B$ is directly connected to city $C$, then city $A$ is indirectly connected to city $C$.

The task is to return the total number of groups of connected cities directly or indirectly. These groups are referred to as provincesA province is a group of directly or indirectly connected cities with no other cities outside of the group..

Constraints:

• $1\leq$ $n$ $\leq200$

• $n ==$ cities.length

• $n ==$ cities[i].length, where $0\leq$ $i$ $\leq n$

• cities[i][j] is either $0$ or $1$.

• cities[i][i] $== 1$

• cities[i][j] $==$ cities[j][i]

This solution uses the union find approach to find the number of provinces formed by the given cities. The union find approach groups cities that are connected directly or indirectly by using the Find operation to check which group a city belongs to and the Union operation to merge the roots of connected cities. Each root in the graph would act as a city, and a cluster of connected roots would represent a province, and a city with no connections would also be represented as a province.

Here’s the step-by-step implementation of the solution:

• Calculate the number of cities n from the input matrix and initialize an object of the UnionFind class, which helps manage and track groups of connected cities.

• Use two loops to go through each unique pair of cities (i, j) where i < j. This ensures that the same pair is not checked twice.

  • For each pair, check if the cities i and j are directly connected by looking at the value of cities[i][j].

    • If cities[i][j] is $1$, cities i and j are directly connected.

      • Use the find operation to find the subsets both cities belong to.

      • If the cities belong to two different subsets, run the union operation on the cities i and j to merge their separate subsets into one subset. Also, decrease the count variable by $1$, which tracks the number of provinces in the UnionFind class, as merging the two subsets would reduce the total number of provinces.

      • If the two cities belong to the same subset, skip them because they have already been connected.

    • Otherwise, move to the next city if cities[i][j] is $0$ because it means cities i and j are not connected.

• After going through all pairs of cities, return the count, containing the number of provinces.

Let’s look at the following illustration to get a better understanding of the solution: