C-Plus-Plus/data_structures/dsu_path_compression.cpp

/**
 * @file
 * @brief [DSU (Disjoint sets)](https://en.wikipedia.org/wiki/Disjoint-set-data_structure)
 * @details
 * It is a very powerful data structure that keeps track of different
 * clusters(sets) of elements, these sets are disjoint(doesnot have a common element).
 * Disjoint sets uses cases : for finding connected components in a graph,
 * used in Kruskal's algorithm for finding Minimum Spanning tree.
 * Operations that can be performed:
 * 1) UnionSet(i,j): add(element i and j to the set)
 * 2) findSet(i): returns the representative of the set to which i belogngs to.
 * 3) get_max(i),get_min(i) : returns the maximum and minimum 
 * Below is the class-based approach which uses the heuristic of path compression.
 * Using path compression in findSet(i),we are able to get to the representative of i
 * in O(1) time.
 * @author [AayushVyasKIIT](https://github.com/AayushVyasKIIT)
 * @see dsu_union_rank.cpp
 */

#include <iostream>  /// for IO operations
#include <vector>    /// for std::vector
#include <cassert>  /// for assert

using std::cout;
using std::endl;
using std::vector;

/**
 * @brief Disjoint sets union data structure, class based representation.
 * @param n number of elements
 */
class dsu{
    private:
        vector<uint64_t> p; ///<keeps track of the parent of ith element
        vector<uint64_t> depth; ///<tracks the depth(rank) of i in the tree
        vector<uint64_t> setSize;///<size of each chunk(set)
        vector<uint64_t> maxElement;/// <maximum of each set to which i belongs to
        vector<uint64_t> minElement;/// <minimum of each set to which i belongs to
    public:
        /**
        * @brief contructor for initialising all data members.
        * @param n number of elements
        */
        explicit dsu(uint64_t n){
            p.assign(n,0);
            /// initially, all of them are their own parents
            for(uint64_t i=0;i<n;i++){
                p[i] = i;
            }
            /// initially all have depth are equals to zero
            depth.assign(n,0);
            maxElement.assign(n,0);
            minElement.assign(n,0);
            for(uint64_t i=0;i<n;i++){
                depth[i] = 0;
                maxElement[i] = i;
                minElement[i] = i;
            }
            setSize.assign(n,0);
            /// initially set size will be equals to one
            for(uint64_t i=0;i<n;i++){
                setSize[i]=1;
            }
        }

        /**
         * @brief Method to find the representative of the set to which i belongs to, T(n) = O(1)
         * @param i element of some set
         * @returns representative of the set to which i belongs to. 
         */
        uint64_t findSet(uint64_t i){
            /// using path compression
            if(p[i]==i){
                return i;
            }
            return (p[i] = findSet(p[i]));
        }
        /**
         * @brief Method that combines two disjoint sets to which i and j belongs to 
         * and make a single set having a common representative.
         * @param i element of some set
         * @param j element of some set
         * @returns void
        */
        void UnionSet(uint64_t i,uint64_t j){
            /// check if both belongs to the same set or not
            if(isSame(i,j)){
                return;
            }

            //we find the representative of the i and j
            uint64_t x = findSet(i);
            uint64_t y = findSet(j);

            //always keeping the min as x
            //shallow tree
            if(depth[x]>depth[y]){
                std::swap(x,y);
            }
            //making the shallower root's parent the deeper root
            p[x] = y;
            
            //if same depth then increase one's depth
            if(depth[x] == depth[y]){
                depth[y]++;
            }
            //total size of the resultant set.  
            setSize[y] += setSize[x];
            //changing the maximum elements
            maxElement[y] = std::max(maxElement[x],maxElement[y]);
            minElement[y] = std::min(minElement[x],minElement[y]);
        }
        /**
         * @brief A utility function which check whether i and j belongs to
         * same set or not
         * @param i element of some set
         * @param j element of some set
         * @returns `true` if element `i` and `j` ARE in the same set
         * @returns `false` if element `i` and `j` are NOT in same set
         */
        bool isSame(uint64_t i,uint64_t j){
            if(findSet(i) == findSet(j)){
                return true;
            }
            return false;
        }
        /**
         * @brief prints the minimum, maximum and size of the set to which i belongs to
         * @param i element of some set
         * @returns void
         */
        vector<uint64_t> get(uint64_t i){
            vector<uint64_t> ans;
            ans.push_back(get_min(i));
            ans.push_back(get_max(i));
            ans.push_back(size(i));
            return ans;

        }
        /**
         * @brief A utility function that returns the size of the set to which i belongs to
         * @param i element of some set
         * @returns size of the set to which i belongs to
         */
        uint64_t size(uint64_t i){
            return setSize[findSet(i)];
        }
        /**
         * @brief A utility function that returns the max element of the set to which i belongs to
         * @param i element of some set
         * @returns maximum of the set to which i belongs to
         */
        uint64_t get_max(uint64_t i){
            return maxElement[findSet(i)];
        }
        /**
         * @brief A utility function that returns the min element of the set to which i belongs to
         * @param i element of some set
         * @returns minimum of the set to which i belongs to 
         */
        uint64_t get_min(uint64_t i){
            return minElement[findSet(i)];
        }

};
/**
 * @brief Self-implementation Test case #1
 * @returns void
 */
static void test1() {
    /* the minimum, maximum and size of the set*/
    uint64_t n = 10;///< number of items
    dsu d(n+1);///< object of class disjoint sets
    //set 1
    d.UnionSet(1,2); //performs union operation on 1 and 2
    d.UnionSet(1,4); //performs union operation on 1 and 4
    vector<uint64_t> ans = {1,4,3};
    for(uint64_t i=0;i<ans.size();i++){
        assert(d.get(4).at(i) == ans[i]); //makes sure algorithm works fine
    }
    cout << "Test case# 1: passed"<<endl;
}
/**
 * @brief Self-implementation Test case #2
 * @returns void
 */
static void test2() {
    /* the minimum, maximum and size of the set */
    uint64_t n = 10;///< number of items
    dsu d(n+1);///< object of class disjoint sets
    //set 1
    d.UnionSet(3,5);
    d.UnionSet(5,6);
    d.UnionSet(5,7);
    vector<uint64_t> ans = {3,7,4};
    for(uint64_t i=0;i<ans.size();i++){
        assert(d.get(3).at(i) == ans[i]); //makes sure algorithm works fine
    }
    cout << "Test case# 2: passed"<<endl;
}


/**
 * @brief Main function
 * @returns 0 on exit
 * */
int main(){
    uint64_t n = 10;///< number of items
    dsu d(n+1);///< object of class disjoint sets

    test1(); //< test case# 1
    test2(); //< test case# 2
    
    return 0;
}