Disjoint Set Union for Trees

The Disjoint Set Union (DSU) for Trees can be used to solve a lot of problems. It is effective in solving problems of the type: How many vertices in the subtree of a vertex V are following a particular property? For an organization that follows a Hierarchical employee structure, the administration might want to know the number of employees working under Manager ‘M’ who belongs to city ‘C’? Conventionally such sort of problems is solved by considering every individual working under ‘M’ and then seeing how many of them belongs to city ‘C’, but when the number of such queries are high then following the conventional approach is not at all feasible as for each query in worst case it takes O(N) time where N is the number of employees in hierarchical structure and therefore for Q such queries time complexity will be O(N*Q). Such problems can be effectively solved using DSU for trees with time complexity O(Q +N* Log
2
(N)). DSU for Tree is based on the idea of effectively merging the Disjoint Sets. Disjoint Set Union for trees uses the same structure as used in graphs. For graphs, the structure has two methods: unite and getParent. Similarly, for trees the same two methods are applicable but their definition gets tweaked as per the applicability. To validate the performance some data sets are generated and solution involving DSU for trees and two other brute force approaches were executed and runtime of each approach suggests that DSU for Tree takes approximately (Log(N)/N) times the runtime of brute force approaches.