Remove PQ in Kruskal README and add properties of MST

Author: junnengsoo

docs/assets/images/MSTProperty3.png

@@ -6,6 +6,23 @@ Minimum Spanning Tree (MST) algorithms are used to find the minimum spanning tre
 spanning tree of a graph is a connected, acyclic subgraph that includes all the vertices of the original graph. An MST 
 is a spanning tree with the minimum possible total edge weight.
 
+### 4 Properties of MST
+1. An MST should not have any cycles
+2. If you cut an MST at any single edge, the two pieces will also be MSTs
+3. **Cycle Property:** For every cycle, the maximum weight edge is not in the MST
+
+![MST Property 3](../../../../../docs/assets/images/MSTProperty3.png)
+
+Image Source: CS2040S 22/23 Sem 2 Lecture Slides
+
+4. **Cut Property:** For every partition of the nodes, the minimum weight edge across the cut is in the MST
+
+![MST Property 4](../../../../../docs/assets/images/MSTProperty4.png)
+   
+Image Source: CS2040S 22/23 Sem 2 Lecture Slides
+
+Note that the other edges across the partition may or may not be in the MST.
+
 ## Prim's Algorithm and Kruskal's Algorithm
 
 We will discuss more implementation-specific details and complexity analysis in the respective folders. In short,
@@ -18,15 +35,6 @@ edges by weight and adding the edge with the minimum weight that does not form a
 
 ## Notes
 
-### Difference in use of Priority Queue in Prim's and Kruskal's Algorithm
-Prim's Algorithm uses a priority queue to keep track of the minimum weight edge that connects the current tree to an
-unexplored node, which could possibly be updated each time a node is popped from the queue.
-
-Kruskal's Algorithm uses a priority queue to sort all the edges by weight and the elements will not be updated at any
-point in time.
-
-See the individual READMEs for more details.
-
 ### Difference between Minimum Spanning Tree and Shortest Path
 It is important to note that a Minimum Spanning Tree of a graph does not represent the shortest path between all the
 nodes. See below for an example:

docs/assets/images/MSTProperty4.png

@@ -7,13 +7,10 @@ to the MST, provided it does not form a cycle with the already included edges. T
 included in the MST.
 
 ## Implementation Details
-Similar to Prim's Algorithm, Kruskal's Algorithm uses a priority queue (binary heap). However, instead of comparing
-the minimum edge weight to each vertex, all the weights of the individual edges are compared instead. Note that we do
-not need any decrease key operations as all edges are considered independently and will not be updated at any point in
-time.
+Kruskal's Algorithm uses a simple `ArrayList` to sort the edges by weight. 
 
-A [disjoint set](/dataStructures/disjointSet/weightedUnion) data structure is used to keep track of the connectivity of
-vertices and detect cycles.
+A [`DisjointSet`](/dataStructures/disjointSet/weightedUnion) data structure is also used to keep track of the
+connectivity of vertices and detect cycles.
 
 ## Complexity Analysis
 

src/main/java/algorithms/minimumSpanningTree/README.md

@@ -8,18 +8,22 @@ node, and the unexplored node to the current tree, until all nodes are included
 
 ### Implementation Details
 
-A priority queue (binary heap) is utilised to keep track of the minimum weight edge that connects the current tree to an
-unexplored node. In an ideal scenario, the minimum weight edge to each node in the priority queue should be updated each
+A `PriorityQueue` (binary heap) is utilised to keep track of the minimum weight edge that connects the current tree to
+an unexplored node. In an ideal scenario, the minimum weight edge to each node in the priority queue should be updated each
 time a lighter edge is found to maintain a single unique node in the priority queue. This means that a decrease key
-operation is required. However, we know that the decrease key operation of a binary heap implementation of a priority
-queue will take O(V) time, which will result in a larger time complexity for the entire algorithm compared to using only
-O(log V) operations for each edge.
+operation is required. 
+
+**Decrease Key Operation:** 
 
-Hence, in our implementation, to avoid the use of a decrease key operation, we will simply insert duplicate nodes with
+However, we know that the decrease key operation of a binary heap implementation of a priority
+queue will take O(V) time, which will result in a larger time complexity for the entire algorithm compared to using only
+O(log V) operations for each edge. Hence, in our implementation, to avoid the use of a decrease key operation, we will simply insert duplicate nodes with
 their new minimum weight edge, which will take O(log E) = O(log V) given an upper bound of E = V^2, into the queue,
 while leaving the old node in the queue. Additionally, we will track if a node has already been added into the MST to
 avoid adding duplicate nodes.
 
+**Priority Queue Implementation:**
+
 Note that a priority queue is an abstract data type that can be implemented using different data structures. In this
 implementation, the default Java `PriorityQueue` is used, which is a binary heap. By implementing the priority queue
 with an AVL tree, a decrease key operation that has a time complexity of O(log V) can also be achieved.