Minimum Spanning Tree What is a Minimum Spanning Tree. Constructing a Minimum Spanning Tree. What is a Minimum-Cost Spanning Tree. Prim’s Algorithm. –Animated Example. –Implementation. Kruskal’s algorithm. –Animated Example. –Implementation. Review Questions.

What is a Minimum Spanning Tree. Minimum Spanning Tree is applicable only to connected undirected graphs. The idea is to find another graph with the same number of vertices, but with minimum number of edges. Let G = (V,E) be connected undirected graph. A minimum spanning tree for G is a graph, T = (V’, E’) with the following properties: –V’ = V –T is connected –T is acyclic.

Example of Minimum Spanning Tree The following figure shows a graph G1 together with its three possible minimum spanning trees. abd cef abd cef The example shows that the minimal spanning tree for a graph is generally not unique. (a) (b)(c)(d)

Constructing Minimum Spanning Tree A spanning tree can be constructed using any of the traversal algorithms discussed, and taking note of the edges that are actually followed during the traversals. The spanning tree obtained using breadth-first traversal is called breadth-first spanning tree, while that obtained with depth-first traversal is called depth-first spanning tree. For example, for the graph in figure 1, figure (c) is a depth-first spanning tree, while figure (d) is a breadth-first spanning tree.

What is a Minimum-Cost Spanning Tree Minimum-Cost spanning tree is applicable only to edge-weighted connected undirected graphs. For an edge-weighted, connected, undirected graph, G, the total cost of G is the sum of the weights on all its edges. A minimum-cost spanning tree for G is a minimum spanning tree of G that has the least total cost.

Prim’s Algorithm Prim’s algorithm finds a minimum cost spanning tree by selecting edges from the graph one-by-one as follows: It starts with a tree, T, consisting of the starting vertex, x. Then, it finds and adds the shortest edge connecting T to the rest of the graph. The process is repeated until all vertices are added to T. Let T be a tree consisting of only the starting vertex x While (T has fewer than n vertices) { find the smallest edge connecting T to G-T add it to T }

Animated Example for Prim’s Algorithm.

Implementation of Prim’s Algorithm. Prims algorithn can be implememted similar to the Dijskra’s algorithm as shown below: 1 public static Graph primsAlgorithm(Graph g, int start) { 2 int n = g.getNumberOfVertices(); 3 Entry table[] = new Entry[n]; 4 for(int v = 0; v < n; v++) 5 table[v] = new Entry(); 6 table[start].distance = 0; 7 PriorityQueue queue=new BinaryHeap(g.getNumberOfEdges()); 8 queue.enqueue(new Association(new Int(0), 9 g.getVertex(start))); 10 while(!queue.isEmpty()) { 11 Association association=(Association)queue.dequeueMin(); 12 Vertex v0 = (Vertex)association.getValue(); 13 int n0 = v0.getNumber(); 14 if(!table[n0].known) { 15 table[n0].known = true; 16 Enumeration p = v0.getEmanatingEdges();

Implementation of Prim’s Algorithm Cont'd 17 while (p.hasMoreElements()) { 18 Edge edge = (Edge) p.nextElement(); 19 Vertex v1 = edge.getMate(v0); 20 int n1 = v1.getNumber(); 21 Int weight = (Int)edge.getWeight(); 22 int d = weight.intValue(); 23 if(!table[n1].known && table[n1].distance>d) { 24 table[n1].distance = d; 25 table[n1].predecessor = n0; 26 queue.enqueue(new Association(new Int(d), v1)); 27 } 28 } 29 } 30 } 31 Graph result = new GraphAsLists(n); 32 for(int v = 0; v < n; v++) 33 result.addVertex(v); 34 for(int v = 0; v < n; v++) 35 if( v != start) 36 result.addEdge(v, table[v].predecessor, 37 new Int(table[v].distance)); 38 return result;}

Kruskal's Algorithm. Kruskal’s algorithm also finds the minimum cost spanning tree of a graph by adding edges one-by-one. However, Kruskal’s algorithm forms a forest of trees, which are joined together incrementally to form the minimum cost spanning tree. sort the edges of G in increasing order of weights form a forest by taking each vertex of G to be a tree for each edge e in sorted list if the endpoints of e are in separate trees join the two tree to form a bigger tree by adding e return the resulting single tree

Animated Example for Kruskal’s Algorithm. Demonstrating Kruskal's algorithm. Priority Queue Current Edge ad ef ce de cd df ac ab bc

Implementation of Kruskal’s Algorithm. To implement Kruskal's algorithm, we need a data structure that maintains a partition of sets. It should have the following methods: –find(item) : returns the set containing the item. –join(s1, s2) : replace the sets, s1 and s2 with their union. We shall implement the partition data structure as a forest. That is, each set in the partition is represented as a tree S1 S2 S3S4

Implementation of Kruskal’s Algorithm Cont'd. The tree we shall use to represent a set in a partition is unlike any of the trees we have seen so far. The nodes have arbitrary degrees. There is no restriction in the ordering of the keys. Instead of pointing to children, each node of the tree points to its parent. This tree is represented as an array of objects, where Each object # i is stored at index I Each object points to its parent or (null if root) S1 S2 S3S4

Implementation of Kruskal’s Algorithm Cont'd. The ith array element holds the node that contains item i. Thus, to implement the find method, we start at index i of the array and follow the parent pointers until we reach a node whose parent is null and return it. To implemented the join method, we attach the smaller tree (the one with fewer keys) to the root of the larger one S1 S2 S3S4

Implementation of Kruskal’s Algorithm Cont'd. The following code shows the implementation of the Partition class: 1 public class Partition { 2 protected PartitionTree[] forest; 3 protected int count; 4 public class PartitionTree { 5 protected int item, count = 1; 6 protected PartitionTree parent; 7 public PartitionTree(int item) { 8 this.item = item; 9 parent = null; 10 } 11 } 12 public Partition(int size) 13 forest = new PartitionTree[size]; 14 for (int item = 0; item <size; item++) 15 forest[item] = new PartitionTree(item); 16 count = size; 17 } 18 //...

Implementation of Kruskal’s Algorithm Cont'd 18 public PartitionTree find(int item) { 19 PartitionTree ptr = forest[item]; 20 while (ptr.parent != null) { 21 ptr = ptr.parent; 22 } 23 return ptr; 24 } 25 public void join (PartitionTree t1, PartitionTree t2) { 26 if (!isPartitionTree(t1) || !isPartitionTree(t2)) 27 throw new IllegalArgumentException("bad argument"); 28 else if (t1.count > t2.count) { 29 t2.parent = t1; 30 t1.count+=t2.count; 31 } 32 else { 33 t1.parent = t2; 34 t2.count += t1.count; 35 } 36count } 38 //...

Implementation of Kruskal’s Algorithm Cont'd. We can use the Partition data structure to implement the Kruskal’s algorithm as follows: 1 public static Graph kruskalsAlgorithm(Graph g) { 2 int n = g.getNumberOfVertices(); 3 Graph result = new GraphAsLists(n); 4 for(int v = 0; v < n; v++) 5 result.addVertex(v); 6 PriorityQueue queue=new BinaryHeap(g.getNumberOfEdges()); 7 Enumeration p = g.getEdges(); 8 while(p.hasMoreElements()) { 9 Edge edge = (Edge) p.nextElement(); 10 Int weight = (Int)edge.getWeight(); 11 queue.enqueue(new Association(weight, edge)); 12 } 13 //...

Implementation of Kruskal’s Algorithm Cont'd 13 Partition partition = new Partition(n); 14 while (!queue.isEmpty() && partition.getCount() > 1) { 15 Association association = 16 (Association) queue.dequeueMin(); 17 Edge edge = (Edge)association.getValue(); 18 Int weight = (Int) edge.getWeight(); 19 int n0 = edge.getV0().getNumber(); 20 int n1 = edge.getV1().getNumber(); 21 Partition.PartitionTree t1 = partition.find(n0); 22 Partition.PartitionTree t2 = partition.find(n1); 23 if(t1 != t2) { 24 partition.join(t1, t2); 25 result.addEdge(n0, n1, weight); 26 } 27 } 28 return result; 29

Review Questions 1.Find the breadth-first spanning tree and depth-first spanning tree of the graph GA shown above. 2.For the graph GB shown above, trace the execution of Prim's algorithm as it finds the minimum-cost spanning tree of the graph starting from vertex a. 3.Repeat question 2 above using Kruskal's algorithm. 4.Complete the implementation of the Partition class by writing a method isPartitionTree(PartitionTree tree) that returns true if tree is a valid partition tree in the partition forest. GBGB