Chapter 9 : Graphs Part II (Minimum Spanning Trees) CE 221 Data Structures and Algorithms Chapter 9 : Graphs Part II (Minimum Spanning Trees) Text: Read Weiss, §9.5 Izmir University of Economics 1

Izmir University of Economics Minimum Spanning Tree A minimum spanning tree of an undirected graph G is a tree formed from graph edges that connects all the vertices of G at a lowest cost. # of edges in a MST is |V|-1. If an edge e not in a MST T is added, acycle is created. The removal of an edge from a cycle reinstates the spanning tree property. As MST is created, if minimum cost edge not creating cycles is selected, then it’s a MST. So, greed works for MST problem. Izmir University of Economics 2

Izmir University of Economics Prim’s Algorithm Proceed in stages. In each stage, find a new vertex to add to the tree already formed by choosing an edge (u, v) such that the cost of (u, v) is the smallest among all the edges where u is in the tree and v is not. Prim’s algorithm for MSTs is essentially identical to Dijkstra’s for shortest paths. Update rule is even simpler: after a vertex is picked, for each unknown w adjacent to v, dw=min(dw, cv,w). Izmir University of Economics 3

Prim’s Example By Joe James Minimum Spanning Tree Prim’s Example By Joe James

Prim’s Algoritim for MST

Time Complexity of Prim’s Algoritim

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST

Prim’s Algoritim for MST Take away all unused edges, So the final MST will be obtained

Prim’s Algorithm – Example I Izmir University of Economics 18

Prim’s Algorithm – Example II Izmir University of Economics 19

Prim’s Algorithm – Example III Izmir University of Economics 20

Prim’s Algorithm – Example IV Izmir University of Economics 21

Prim’s Algorithm – Time Complexity The running time is O(|V|2) (|V| iterations in each of which a sequential scan is performed to find the minimum cost unknown edge) without heaps, which is optimal for dense graphs. O(|E|*log|V|) (an underlying binary heap with |V| elements is used, and |V| deleteMin (for picking min cost vertex) and at most |E| decreaseKey (for updates) are performed each taking O(log|V|) time) using binary heaps,which is good for sparse graphs Izmir University of Economics 22

Izmir University of Economics Kruskal’s Algorithm A second greedy strategy is continually to select the edges in order of smallest weight and accept an edge if it does not cause a cycle.Formally, Kruskal's algorithm maintains a forest - a collection of trees. Initially, there are |V| single-node trees. Adding an edge merges two trees into one. When the algorithm terminates, there is only one tree, and this is the minimum spanning tree. Izmir University of Economics 23

Kruskal’s Example By Joe James Minimum Spanning Tree Kruskal’s Example By Joe James

Kruskal’s Algorithm for MST

Time Complexity of Kruskal’s Algoritim

First: Sort all edge costs

2nd: Chose the Edge of Lowest Cost

Chose Other Edges of Lowest Costs and join them

Chose Edges of Lowest Costs and join them

Now Finalize the MST

An example run of Kruskal’s Algorithm Izmir University of Economics 32

Kruskal’s Algorithm: Pseudo Code I // Create the list of all edges E solution = { } // will include the list of MST edges while ( more edges in E) do // Selection select minimum weight(cost) edge remove edge from E // Feasibility if (edge closes a cycle with solution so far) then reject edge else add edge to solution // Solution check if |solution| = |V | - 1 return solution Izmir University of Economics 33

Kruskal’s Algorithm: Pseudo Code II The worst-case running time of this algorithm is O(|E|log|E|), which is dominated by the heap operations. Notice that since |E| = O(|V|2), this running time is actually O(|E| log |V|). In practice, the algorithm is much faster. Izmir University of Economics 34

Izmir University of Economics Homework Assignments 9.15, 9.16 You are requested to study and solve the exercises. Note that these are for you to practice only. You are not to deliver the results to me. Izmir University of Economics 35