Merge pull request #220 from Jatin86400/patch-2

updated the readme.md file

Author: prateekiiest

Competitive Coding/Graphs/Detecting cycles/detecting-cycles_readme.md

@@ -1,11 +1,11 @@
-Representation
+# Disjoint-set 
 
 A disjoint-set forest consists of a number of elements each of which stores an id, a parent pointer, and, in efficient algorithms, a value called the "rank".
 The parent pointers of elements are arranged to form one or more trees, each representing a set. If an element's parent pointer points to no other element, then the element is the root of a tree and is the representative member of its set. A set may consist of only a single element. However, if the element has a parent, the element is part of whatever set is identified by following the chain of parents upwards until a representative element (one without a parent) is reached at the root of the tree.
 Forests can be represented compactly in memory as arrays in which parents are indicated by their array index.
 
-MAKE SET
-MakeSet[edit]
+# MAKE SET
+
 The MakeSet operation makes a new set by creating a new element with a unique id, a rank of 0, and a parent pointer to itself. The parent pointer to itself indicates that the element is the representative member of its own set.
 The MakeSet operation has O(1) time complexity.
 Pseudocode:
@@ -16,17 +16,22 @@ function MakeSet(x)
      x.parent := x
      x.rank   := 0
 
-FIND
+![alt text](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRkmVdlJLBBEPsQUbG3igzyitb6PDbqCuVbVtQzc009uyLX65U-)
+
+# FIND
 Find(x) follows the chain of parent pointers from x upwards through the tree until an element is reached whose parent is itself. This element is the root of the tree and is the representative member of the set to which x belongs, and may be x itself.
 Path compression, is a way of flattening the structure of the tree whenever Find is used on it. Since each element visited on the way to a root is part of the same set, all of these visited elements can be reattached directly to the root. The resulting tree is much flatter, speeding up future operations not only on these elements, but also on those referencing them.
+
 Pseudocode:
 
 function Find(x)
    if x.parent != x
      x.parent := Find(x.parent)
    return x.parent
+   
+![alt text](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRptXJI8DhupTU5GNaF5qKZX94VPYB7GTiXkYMTadEGRuFMS5PKtg)
 
-UNION
+# UNION
 
 Union(x,y) uses Find to determine the roots of the trees x and y belong to. If the roots are distinct, the trees are combined by attaching the root of one to the root of the other. If this is done naively, such as by always making x a child of y, the height of the trees can grow as {\displaystyle O(n)} O(n). To prevent this union by rank is used.
 Union by rank always attaches the shorter tree to the root of the taller tree. Thus, the resulting tree is no taller than the originals unless they were of equal height, in which case the resulting tree is taller by one node.
@@ -49,4 +54,6 @@ function Union(x, y)
    else
      //Arbitrarily make one root the new parent
      yRoot.parent := xRoot    
-     xRoot.rank   := xRoot.rank + 1
+     xRoot.rank   := xRoot.rank + 1
+     
+![alt text](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcSZkZmlExRXCYS-k3TO9dSM0dH_Ql2WuiB-VTIGj6090KEI0Jmjjw)