Graphs and MSTs

1. Graphs and MSTs • Sections 1.4 and 9.1

2. Partial-Order Relations • Everybody is not related to everybody. Examples? • Direct road connections between locations • How about using a tree to represent the connections?

3. Edges (arcs) Vertices (nodes) {(v,v’) : v and v’ are in V} Graphs: Definition G = (V, E) Edges are a subset of V  V We also write v v’ instead of (v.v’)

4. A directed graph or digraph is one were the order matters v v’ Directed Graphs Our PA map is undirected because if (v,v’)  E then it is assumed that (v’,v)  E (but not included in E) Can you think of a situation were the graph is directed? That is, we may have (v,v’)  E but not (v’,v)  E City streets. Some are one way

5. Complete Graphs A complete graph is one where there is an edge between every two nodes If the graph has N nodes and the graph is complete and undirected, how many edges are there? (N-1)N/2 If the digraph has N nodes and the digraph is complete, how many edges are there? (N-1)N

6. Every element in the left set is an element in the right set Subgraphs • Given a graph G = (V, E) and a graph G’ = (V’, E’), G is a subgraph of G’ if: • V  V’ • E  E’

7. Quiz: Obtain all the paths in the following digraph: 2 3 7 Paths Given a graph G = (V, E), a path between two elements v, v’ in V is a sequence of elements: v v1 v2 vN v’ … such that v v2,…, v1, v1 vN v’ are all edges in E The length of the path is the number of edges on it

8. Weighted Graphs A weighted graph G = (V, E), is a graph such that each edge v v’ has an associated number (called the weight) Given a weighted graph G = (V, E), a path between two elements v, v’ in V : v  v1  …  v’, the weight or cost of the path is the summation of the weight of the edges in the path

9. Connected? 2 3 7 5 Connected Graphs A graph G = (V, E) is connected if there is at least one path between any two elements Yes

10. Acyclic? 2 3 7 5 Acyclic Graphs Given a graph G = (V, E), a cycle is a path of at least length 1 starting with a node and ending in the same node A graph G = (V, E) is acyclic if it has no cycles No

11. Trees A tree is a connected acyclic graph G = (V, E) Number of arcs in a tree with |V| nodes: |E|-1

12. Techniques for Design of Algorithms • Brute Force: follows directly from the definition • Exhaustive search: particular form of Brute Force • Greedy Algorithm: Obtain solution by expanding partial solution constructed so far. On each step a decision is made that is: • Feasible • Locally optimal • Irrevocable

13. Example: Greedy algorithm for TSP Consider the following undirected graph: 2 D F 2 2 4 2 A 3 7 E 7 C 6 2 2 B 8 G 2 2 H Use a Greedy technique to obtain a solution of the TSP Complexity (n = # arcs)? O(n) Does using a Greedy technique always solves TSPs? No

14. The Minimum Spanning Tree • Give a graph G = (V, E), the minimum spanning tree (MST) is a weighted graph G’ = (V, E’) such that: • E’  E • G’ is connected • G’ has the minimum cost

15. V V’ Why does it works? Property: Suppose that we divide the nodes of a graph G = (V, E) in two groups V, V’: 12 22 7 20 20 5 6 66 10 13 If there is an edge e between the 2 groups with minimum cost, then there is an MST containing e

16. Visited Non visited Example of a “Greedy” Algorithm Dijkstra-Primis an example of a greedy algorithm since it looks at a subset of the larger problem 12 22 7 20 20 5 6 66 10 13 Unfortunately, not all problems can be solved with greedy algorithms Example of a non greedy algorithm? Our naïve algorithm for MST

17. … … e2 e1 Complexity • Actual complexity is O((|E|+|V|)log2 |E|) • It is easy to see that it is at most |E||V| e1 + (e1 + e2) +…+ (e1 + e2 + … +e2)  |E||V|