1. Chapter 9: Graphs Spanning Trees Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Lydia Sinapova, Simpson College

2. 2 Spanning Trees  Spanning Trees in Unweighted Graphs  Minimal Spanning Trees in Weighted Graphs - Prim’s Algorithm  Minimal Spanning Trees in Weighted Graphs - Kruskal’s Algorithm  Animation Animation

3. 3 Definitions Spanning tree: a tree that contains all vertices in the graph. Number of nodes: |V| Number of edges: |V|-1

4. 4 Spanning trees for unweighted graphs - data structures A table (an array) T size = number of vertices, Tv = parent of vertex v Adjacency lists A queue of vertices to be processed

5. 5 Algorithm - initialization 1.Choose a vertex S and store it in a queue, set a counter = 0 (counts the processed nodes), and Ts = 0 (to indicate the root), Ti = -1,i  s (to indicate vertex not processed)

6. 6 Algorithm – basic loop 2. While queue not empty and counter < |V| -1 : Read a vertex V from the queue For all adjacent vertices U : If Tu = -1 (not processed) Tu  V counter  counter + 1 store U in the queue

7. 7 Algorithm – results and complexity Result: Root: S, such that Ts = 0 Edges in the tree: (Tv, V) Complexity: O(|E| + |V|) - we process all edges and all nodes

8. 8 Example 1 2 3 4 5 Table 0 // 1 is the root 1 // edge (1,2) 2 // edge (2,3) 1 // edge (1,4) 2// edge (2,5) Edge format: (Tv,V)

9. 9 Minimum Spanning Tree - Prim's algorithm Given: Weighted graph. Find a spanning tree with the minimal sum of the weights. Similar to shortest paths in a weighted graphs. Difference: we record the weight of the current edge, not the length of the path.

10. 10 Data Structures A table : rows = number of vertices, three columns: T(v,1) = weight of the edge from parent to vertex V T(v,2) = parent of vertex V T(v,3) = True, if vertex V is fixed in the tree, False otherwise

11. 11 Data Structures (cont) Adjacency lists A priority queue of vertices to be processed. Priority of each vertex is determined by the weight of edge that links the vertex to its parent. The priority may change if we change the parent, provided the vertex is not yet fixed in the tree.

12. 12 Prim’s Algorithm Initialization For all V, set first column to –1 (not included in the tree) second column to 0 (no parent) third column to False (not fixed in the tree) Select a vertex S, set T(s,1) = 0 (root: no parent) Store S in a priority queue with priority 0.

13. 13 While (queue not empty) loop 1. DeleteMin V from the queue, set T(v,3) = True 2. For all adjacent to V vertices W: If T(w,3)= True do nothing – the vertex is fixed If T(w,1) = -1 (i.e. vertex not included) T(w,1)  weight of edge (v,w) T(w,2)  v (the parent of w) insert w in the queue with priority = weight of (v,w)

14. 14 While (queue not empty) loop (cont) If T(w,1) > weight of (v,w) T(w,1)  weight of edge (v,w) T(w,2)  v (the parent of w) update priority of w in the queue remove, and insert with new priority = weight of (v,w)

15. 15 Results and Complexity At the end of the algorithm, the tree would be represented in the table with its edges {(v, T(v,2) ) | v = 1, 2, …|V|} Complexity: O(|E|log(|V|))

16. 16 Kruskal's Algorithm The algorithm works with : set of edges, tree forests, each vertex belongs to only one tree in the forest.

17. 17 The Algorithm 1. Build |V| trees of one vertex only - each vertex is a tree of its own. Store edges in priority queue 2. Choose an edge (u,v) with minimum weight if u and v belong to one and the same tree, do nothing if u and v belong to different trees, link the trees by the edge (u,v) 3. Perform step 2 until all trees are combined into one tree only

18. 18 Example 1 2 3 4 5 1 5 3 4 6 24 Edges in Priority Queue: (1,2) 1 (3,5) 2 (2,4) 3 (1,3) 4 (4,5) 4 (2,5) 5 (1,5) 6

19. 19 Example (cont) 12345 Forest of 5 trees Edges in Priority Queue: (1,2) 1 (3,5) 2 (2,4) 3 (1,3) 4 (4,5) 4 (2,5) 5 (1,5) 6 12 Process edge (1,2) Process edge (3,5) 12 3 1 1 2 5 345 4

20. 20 Example (cont) Edges in Priority Queue: (2,4) 3 (1,3) 4 (4,5) 4 (2,5) 5 (1,5) 6 Process edge (2,4) 12 1 4 3 Process edge (1,3) 2 5 1 2 4 3 1 3 4 All trees are combined in one, the algorithm stops 3 2 5

21. 21 Operations on Disjoint Sets Comparison: Are the vertices in the same tree Union: combine two disjoint sets to form one set - the new tree Implementation Union/find operations: the unions are represented as trees - nlogn complexity. The set of trees is implemented by an array.

22. 22 Complexity of Kruskal’s Algorithm O(|E|log(|V|)Explanation: The complexity is determined by the heap operations and the union/find operations Union/find operations on disjoint sets represented as trees: tree search operations, with complexity O(log|V|) DeleteMin in Binary heaps with N elements is O(logN), When performed for N elements, it is O(NlogN).

23. 23 Complexity (cont) Kruskal’s algorithm works with a binary heap that contains all edges: O(|E|log(|E|)) However, a complete graph has |E| = |V|*(|V|-1)/2, i.e. |E| = O(|V| 2 ) Thus for the binary heap we have O(|E|log(|E|)) = O(|E|log (|V| 2 ) = O(2|E|log (|V|) = O(|E|log(|V|))

24. 24 Complexity (cont) Since for each edge we perform DeleteMin operation and union/find operation, the overall complexity is: O(|E| (log(|V|) + log(|V|)) = O(|E|log(|V|))

25. 25 Discussion Sparse trees - Kruskal's algorithm : Guided by edges Dense trees - Prim's algorithm : The process is limited by the number of the processed vertices