Dynamic connectivity is a data structure that dynamically maintains the information about thee connected components of graph. In simple words suppose there is a graph G(V, E) in which no. of vertices V is constant but no. of edges E is variable. There are three ways in which we can change the number of edges
- Incremental Connectivity : Edges are only added to the graph.
- Decremental Connectivity : Edges are only deleted from the graph.
- 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.
- An edge is added to the graph.
- Information about two nodes x and y whether they are in the same connected components or not.
Input : V = 7 Number of operations = 11 1 0 1 2 0 1 2 1 2 1 0 2 2 0 2 2 2 3 2 3 4 1 0 5 2 4 5 2 5 6 1 2 6 Note: 7 represents number of nodes, 11 represents number of queries. There are two types of queries Type 1 : 1 x y in this if the node x and y are connected print Yes else No Type 2 : 2 x y in this add an edge between node x and y Output : No Yes No Yes Explanation : Initially there are no edges so node 0 and 1 will be disconnected so answer will be No Node 0 and 2 will be connected through node 1 so answer will be Yes similarly for other queries we can find whether two nodes are connected or not
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.
C++ implementation is given below here we are using union by rank and path compression
No Yes No Yes
The amortized time complexity is O(alpha(n)) per operation where alpha is inverse ackermann function which is nearly constant.
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