Dynamic Connectivity | Set 1 (Incremental)

Dynamic connectivity is a data structure that dynamically maintains the information about the connected components of graph. In simple words suppose there is a graph G(V, E) in which number of vertices V is constant but number of edges E is variable. There are three ways in which we can change the number of edges

1. Incremental Connectivity : Edges are only added to the graph.
2. Decremental Connectivity : Edges are only deleted from the graph.
3. Fully Dynamic Connectivity : Edges can both be deleted and added to the graph.

In this article only Incremental connectivity is discussed. There are mainly two operations that need to be handled. 

1. [1, u, v]: An edge between u and v is added to the graph.
2. [2, u, v]: Check if u and v are connected.

Example:

Input: V = 5, queries[][] = [[1, 0, 1], [2, 0, 1], [2, 1, 2], [1, 0, 2], [2, 1, 2], [1, 3, 4], [2, 2, 4], [1, 1, 3], [2, 2, 4]]
Output: true false true false true
Explanation: Initial state: Components = [[0], [1], [2], [3], [4]]

• Query [1, 0, 1]: Union(0,1) --> Components = [[0,1], [2], [3], [4]]
• Query [2, 0, 1]: Check connectivity --> Connected: true
• Query [2, 1, 2]: Check connectivity --> Connected: false
• Query [1, 0, 2]: Union(0,2) --> Components = [[0,1,2], [3], [4]]
• Query [2, 1, 2]: Check connectivity --> Connected: true
• Query [1, 3, 4]: Union(3,4) --> Components = [[0,1,2], [3,4]]
• Query [2, 2, 4]: Check connectivity --> Connected: false
• Query [1, 1, 3]: Union(1,3) --> Components = [[0,1,2,3,4]]
• Query [2, 2, 4]: Check connectivity --> Connected: true

Approach:

To solve the problems of incremental connectivity disjoint data structure is used. Here each connected component represents a set and if the two nodes belong to the same set it means that they are connected. 

Implementation is given below here we are using union by rank and path compression 

true false true false true 

Time Complexity: O(α(n)), Inverse Ackermann, nearly constant time, because of path compression and union by rank optimization.
Space Complexity: O(n), For parent and rank arrays as arrays store disjoint set info for n elements.