Prerequisites : Union Find (or Disjoint Set), Disjoint Set Data Structures (Java Implementation)
A disjoint-set data structure maintains a collection S = {S1, S2,…., Sk} of disjoint dynamic sets. We identify each set by a representative, which is some member of the set. In some applications, it doesn’t matter which member is used as the representative; we care only that if we ask for the representative of a dynamic set twice without modifying the set between the requests, we get the same answer both times. Other applications may require a prespecified rule for choosing the representative, such as choosing the smallest member in the set.
Example: Determining the connected components of an undirected graph. Below figure, shows a graph with four connected components.
Solution : One procedure X that follows uses the disjoint-set operations to compute the connected components of a graph. Once X has pre-processed the graph, the procedure Y answers queries about whether two vertices are in the same connected component. Below figure shows the collection of disjoint sets after processing each edge.
See here as the above example was discussed earlier.
Figure
(a) Linked-list representations of two sets. Set S1 contains members d, f, and g, with representative f, and set S2 contains members b, c, e, and h, with representative c.
Each object in the list contains a set member, a pointer to the next object in the list, and a pointer back to the set object. Each set object has pointers head and tail to the first and last objects, respectively.
b) The result of UNION(e, g), which appends the linked list containing e to the linked list containing g. The representative of the resulting set is f . The set object for e’s list, S2, is destroyed.
Above three figures are taken from the Cormen(CLRS) book. Above Figure shows a simple way to implement a disjoint-set data structure: each set is represented by its own linked list. The object for each set has attributes head, pointing to the 1st object in the list, and tail, pointing to the last object.
Each object in the list contains a set member, a pointer to the next object in the list, and a pointer back to the set object. Within each linked list, the objects may appear in any order. The representative is the set member in the 1st object in the list. To carry out MAKE-SET (x), we create a new linked list whose only object is x. For FIND-SET(x), we just follow the pointer from x back to its set object and then return the member in the object that head points to. For example, in the Figure, the call FIND-SET(g) would return f.
Algorithm:
Letting x denote an object, we wish to support the following operations:
- MAKE-SET(x) creates a new set whose only member (and thus representative) is x. Since the sets are disjoint, we require that x not already be in some other set.
- UNION (x, y) unites the dynamic sets that contain x and y, say Sx and Sy, into a new set that is the union of these two sets. We assume that the two sets are disjoint prior to the operation. The representative of the resulting set is any member of Sx U Sy, although many implementations of UNION specifically choose the representative of either Sx or Sy as the new representative. Since we require the sets in the collection to be disjoint, conceptually we destroy sets Sx and Sy, removing them from the collection S. In practice, we often absorb the elements of one of the sets into the other set.
- FIND-SET(x) returns a pointer to the representative of the (unique) set containing x.
Based on the above explanation, below are implementations:
// C++ program for implementation of disjoint// set data structure using linked list#include <bits/stdc++.h>using namespace std;
// to represent linked list which is a setstruct Item;
// to represent Node of linked list. Every// node has a pointer to representativestruct Node
{ int val;
Node *next;
Item *itemPtr;
};// A list has a pointer to head and tailstruct Item
{ Node *hd, *tl;
};// To represent union setclass ListSet
{private:
// Hash to store addresses of set representatives
// for given values. It is made global for ease of
// implementation. And second part of hash is actually
// address of Nodes. We typecast addresses to long
// before storing them.
unordered_map<int, Node *> nodeAddress;
public:
void makeset(int a);
Item* find(int key);
void Union(Item *i1, Item *i2);
};// To make a set with one object// with its representativevoid ListSet::makeset(int a)
{ // Create a new Set
Item *newSet = new Item;
// Create a new linked list node
// to store given key
newSet->hd = new Node;
// Initialize head and tail
newSet->tl = newSet->hd;
nodeAddress[a] = newSet->hd;
// Create a new set
newSet->hd->val = a;
newSet->hd->itemPtr = newSet;
newSet->hd->next = NULL;
}// To find representative address of a// keyItem *ListSet::find(int key)
{ Node *ptr = nodeAddress[key];
return (ptr->itemPtr);
}// union function for joining two subsets// of a universe. Merges set2 into set1// and deletes set1.void ListSet::Union(Item *set1, Item *set2)
{ Node *cur = set2->hd;
while (cur != 0)
{
cur->itemPtr = set1;
cur = cur->next;
}
// Join the tail of the set to head
// of the input set
(set1->tl)->next = set2->hd;
set1->tl = set2->tl;
delete set2;
}// Driver codeint main()
{ ListSet a;
a.makeset(13); //a new set is made with one object only
a.makeset(25);
a.makeset(45);
a.makeset(65);
cout << "find(13): " << a.find(13) << endl;
cout << "find(25): "
<< a.find(25) << endl;
cout << "find(65): "
<< a.find(65) << endl;
cout << "find(45): "
<< a.find(45) << endl << endl;
cout << "Union(find(65), find(45)) \n";
a.Union(a.find(65), a.find(45));
cout << "find(65]): "
<< a.find(65) << endl;
cout << "find(45]): "
<< a.find(45) << endl;
return 0;
} |
// Java program for implementation of disjoint// set data structure using linked listimport java.util.*;
// to represent Node of linked list. Every// node has a pointer to representativeclass Node {
int val;
Node next;
Item itemPtr;
}// A list has a pointer to head and tailclass Item {
Node hd, tl;
}// To represent union setpublic class ListSet {
// Hash to store addresses of set representatives
// for given values. It is made global for ease of
// implementation. And second part of hash is actually
// address of Nodes. We typecast addresses to long
// before storing them.
private Map<Integer, Node> nodeAddress
= new HashMap<>();
// To make a set with one object
// with its representative
public void makeset(int a)
{
// Create a new Set
Item newSet = new Item();
// Create a new linked list node
// to store given key
newSet.hd = new Node();
// Initialize head and tail
newSet.tl = newSet.hd;
nodeAddress.put(a, newSet.hd);
// Create a new set
newSet.hd.val = a;
newSet.hd.itemPtr = newSet;
newSet.hd.next = null;
}
// To find representative address of a
// key
public Item find(int key)
{
Node ptr = nodeAddress.get(key);
return ptr.itemPtr;
}
// union function for joining two subsets
// of a universe. Merges set2 into set1
// and deletes set1.
public void Union(Item set1, Item set2)
{
Node cur = set2.hd;
while (cur != null) {
cur.itemPtr = set1;
cur = cur.next;
}
// Join the tail of the set to head
// of the input set
set1.tl.next = set2.hd;
set1.tl = set2.tl;
set2 = null;
}
// Driver code
public static void main(String[] args)
{
ListSet a = new ListSet();
a.makeset(13);
a.makeset(25);
a.makeset(45);
a.makeset(65);
System.out.println("find(13): " + a.find(13).hashCode());
System.out.println("find(25): " + a.find(25).hashCode());
System.out.println("find(65): " + a.find(65).hashCode());
System.out.println("find(45): " + a.find(45).hashCode()
+ "\n");
System.out.println("Union(find(65), find(45))");
a.Union(a.find(65), a.find(45));
System.out.println("find(65): " + a.find(65).hashCode());
System.out.println("find(45): " + a.find(45).hashCode());
}
} |
from collections import defaultdict
class Node:
def __init__(self, val, item_ptr):
self.val = val
self.item_ptr = item_ptr
self.next = None
class Item:
def __init__(self, hd, tl):
self.hd = hd
self.tl = tl
class ListSet:
def __init__(self):
self.node_address = defaultdict(lambda: None)
def makeset(self, a):
new_set = Item(Node(a, None), None)
new_set.hd.item_ptr = new_set
new_set.tl = new_set.hd
self.node_address[a] = new_set.hd
def find(self, key):
node = self.node_address[key]
return node.item_ptr
def union(self, i1, i2):
cur = i2.hd
while cur:
cur.item_ptr = i1
cur = cur.next
i1.tl.next = i2.hd
i1.tl = i2.tl
del i2
def main():
a = ListSet()
a.makeset(13)
a.makeset(25)
a.makeset(45)
a.makeset(65)
print(f"find(13): {a.find(13)}")
print(f"find(25): {a.find(25)}")
print(f"find(65): {a.find(65)}")
print(f"find(45): {a.find(45)}")
print()
print("Union(find(65), find(45))")
a.union(a.find(65), a.find(45))
print(f"find(65): {a.find(65)}")
print(f"find(45): {a.find(45)}")
if __name__ == "__main__":
main()
|
// C# program for implementation of disjoint// set data structure using linked listusing System;
using System.Collections.Generic;
class Program {
// Driver Code
static void Main(string[] args)
{
ListSet a = new ListSet();
a.makeset(13);
a.makeset(25);
a.makeset(45);
a.makeset(65);
Console.WriteLine("find(13): "
+ a.find(13).GetHashCode());
Console.WriteLine("find(25): "
+ a.find(25).GetHashCode());
Console.WriteLine("find(65): "
+ a.find(65).GetHashCode());
Console.WriteLine(
"find(45): " + a.find(45).GetHashCode() + "\n");
Console.WriteLine("Union(find(65), find(45))");
a.Union(a.find(65), a.find(45));
Console.WriteLine("find(65): "
+ a.find(65).GetHashCode());
Console.WriteLine("find(45): "
+ a.find(45).GetHashCode());
}
}// to represent Node of linked list. Every// node has a pointer to representativeclass Node {
public int val;
public Node next;
public Item itemPtr;
}// A list has a pointer to head and tailclass Item {
public Node hd, tl;
}// To represent union setclass ListSet {
// Hash to store addresses of set representatives
// for given values. It is made global for ease of
// implementation. And second part of hash is actually
// address of Nodes. We typecast addresses to long
// before storing them.
private Dictionary<int, Node> nodeAddress
= new Dictionary<int, Node>();
// To make a set with one object
// with its representative
public void makeset(int a)
{
// Create a new Set
Item newSet = new Item();
// Create a new linked list node
// to store given key
newSet.hd = new Node();
// Initialize head and tail
newSet.tl = newSet.hd;
nodeAddress.Add(a, newSet.hd);
// Create a new set
newSet.hd.val = a;
newSet.hd.itemPtr = newSet;
newSet.hd.next = null;
}
// To find representative address of a
// key
public Item find(int key)
{
Node ptr = nodeAddress[key];
return ptr.itemPtr;
}
// union function for joining two subsets
// of a universe. Merges set2 into set1
// and deletes set1.
public void Union(Item set1, Item set2)
{
Node cur = set2.hd;
while (cur != null) {
cur.itemPtr = set1;
cur = cur.next;
}
// Join the tail of the set to head
// of the input set
set1.tl.next = set2.hd;
set1.tl = set2.tl;
set2 = null;
}
}// This code is contributed by Tapesh(tapeshdua420) |
// To represent linked list which is a setclass Item { constructor() {
this.hd = null;
this.tl = null;
}
}// To represent Node of linked list. Every// node has a pointer to representativeclass Node { constructor(val) {
this.val = val;
this.next = null;
this.itemPtr = null;
}
}// To represent union setclass ListSet { constructor() {
this.nodeAddress = new Map();
}
makeset(a) {
// Create a new Set
const newSet = new Item();
// Create a new linked list node
// to store the given key
newSet.hd = new Node(a);
// Initialize head and tail
newSet.tl = newSet.hd;
this.nodeAddress.set(a, newSet.hd);
// Create a new set
newSet.hd.itemPtr = newSet;
newSet.hd.next = null;
}
find(key) {
const ptr = this.nodeAddress.get(key);
return ptr ? ptr.itemPtr : null;
}
Union(set1, set2) {
let cur = set2.hd;
while (cur !== null) {
cur.itemPtr = set1;
cur = cur.next;
}
// Join the tail of the set to the head
// of the input set
set1.tl.next = set2.hd;
set1.tl = set2.tl;
set2.hd = null;
set2.tl = null;
}
}// Driver codeconst a = new ListSet();
a.makeset(13);a.makeset(25);a.makeset(45);a.makeset(65);console.log("find(13):", a.find(13));
console.log("find(25):", a.find(25));
console.log("find(65):", a.find(65));
console.log("find(45):", a.find(45));
console.log("\nUnion(find(65), find(45))");
a.Union(a.find(65), a.find(45));console.log("find(65):", a.find(65));
console.log("find(45):", a.find(45));
|
find(13): 0x1cf8c20 find(25): 0x1cf8ca0 find(65): 0x1cf8d70 find(45): 0x1cf8c80 Union(find(65), find(45)) find(65]): 0x1cf8d70 find(45]): 0x1cf8d70
Note: The node address will change every time, we run the program. Time complexities of MAKE-SET and FIND-SET are O(1). Time complexity for UNION is O(n).