MST worksheet By Jim Xu

Transcription

1 Name: Name: Name: MST worksheet By Jim Xu Please work in groups of 2 or 3 to work the following problems. Use additional paper as needed, and staple the sheets together before turning them in. ONLY TURN IN 1 WORKSHEET/ANSWERS PER TEAM. 1. [10] Below is an example of a connected graph and a minimum spanning tree. Answer the questions below based on the graph. A. [3] The graph is a (weighted / unweighted) and (directed / undirected) graph. (Circle correct choice) B. [5] Using the Cut Property, briefly explain why the MST does not include the edge from McFane s Farm to Brewery? Call the edge from the Bakery to the Mayor s House (i, a) e and the edge from McFane s Farm to the Brewery (h, b) e. Assume that the subset of nodes A contains the Brewery node and the subset of nodes B contains McFane s Farm. Show that there is no cut where e can be used as the light edge to connect node sets A and B. (Hint: when you try different cuts, describe what nodes are in A and B. For example, if the cut crosses the edge (h, i) and e, then the nodes in A are {i,a b} and the node in B is {h}. In this case the light edge would be (h, i), NOT e.)

2 C. [2] For the minimum spanning tree problem, we proved that strategies do yield a spanning tree with minimum weight, based on which we explored the algorithms of Kruskal and Prim. 2. [5] Consider a graph with n nodes and k edges and a minimum spanning tree built based on it. The resulting MST will have nodes and edges. 3. [20] Experiment graph with algorithms. (Assume the starting point is C) A. [10] Perform Kruskal s MST Algorithm based on the graph. i. [1] Connecting edge is the first step of Kruskal s. ii. [4] Write down each edge and its weight you picked, in order, using Kruskal s algorithm. Stop adding edges when you have an MST.

3 iii. [5] Show how disjointed sets change as you apply Kruskal s algorithm. Based on your answer to the previous question, draw a picture of the following disjoint sets: (a) [1] Initially: (b) [2] After you have processed the first 2 edges you considered during Kruskal s algorithm: (c) [2] Just before you process the last edge that made the MST: B. [10] Perform Prim s MST Algorithm based on the graph (Starting Point is C) i. [1] Choosing edge is the first step of Prim.

4 ii. [5] Write down each node you picked in order (in the set S), and fill in the priority queue you maintained for each step of Prim s algorithm. (We added the information for the starting node C in both the priority queue and the set S.) Priority Queue: wt wt wt wt wt wt 0 { C A B F D E } S: {C, }

5 iii. [4] Jim thinks of two different ways to keep track of the minimum priority queue used by Prim s to select the light edge from the tree nodes to the remaining nodes. Consider these two operations of the minimum priority queue. Insert: To insert a new element to the priority queue, then re build it if necessary. ExtractMin: To find and extract the minimum element of the priority queue, then re build it if necessary. (a) [2] Using linked list data structure in python, write the analysis time complexity for the priority queue. Write whether either or both operations need to re build the queue. Insert: ExtractMin: (b) [2] Using heap in python, write the analysis time complexity for the priority queue. Write whether either or both operations need to re build the queue. Insert: ExtractMin:

6 4. [5] Discover the stability of MST. A. [2] If we add a new edge AF with weight 5 in original graph shown at the beginning of this problem, will the resulting MST change? B. [3] Now consider a new MST with n nodes and k edges (the weight of edges namely, w 1, w 2 w k ). Assume we add a new edge with weight w (k+1). Under what conditions on w (k+1) might the MST change? Briefly explain.

7 5. [10] (Assume the starting point is C) Following alphabetical order. Use DFS and BFS algorithms to answer the questions below. BFS DFS weight: weight: A. [7] Draw the BFS and DFS trees on the respective graphs above. (Remember to follow alphabetical node order, starting at node C.) B. [2] Write the weight of each resulting tree under the respective graph. C. [1] Are either of the search trees the same as the MST for this graph?