GRAPHS AND GRAPH TRAVERSALS 9/26/2000

1. COMP 6/4030 ALGORITHMS GRAPHS AND GRAPH TRAVERSALS9/26/2000

2. Graphs - Definitions • Graph is a set of vertices (nodes) and edges where edges connect certain vertices • Undirected graph: no direction of edges • Directed graph (di-graph): edges are directed from one vertex to another • NOTE: a graph is defined by an binary relation upon the set of vertices V: • If node v and w are connected, then they are related as vRw. For di-graphs the binary relationship is ordered.

3. SxS aRb R Graphs - as Relationships • RELATION: • Two members a,b of a set S either satisfy a certain relationship R or not. If satisfy, we write aRb. • NOTE: aRb defines a subset of all pairs (a,b). • Example: • ‘larger’: 4<5 True: 4R5 • 3<2 False

4. Relationships • Example: Equivalence relation • A relationship with the properties below (=) • Reflective: sRs is True • Symmetric: If sRt true then tRs also True • Transitive: If sRt and tRu then sRt. • Note: The equivalence relationship R partitions S in to disjunct subsets Si

5. Introduction to Graphs • Approaches to graphs: • Discrete math<------> Continuous math • Combinatorics Calculus from 17th century, • Graph theory Newton, Leibnitz,… • Digital computers Limitations of equations • Dilemmas: • Are natural processes discrete or continuous? • Is continuity is a mathematical abstraction?

6. Graphs - Introduction • Graphs as Networks <---> Neural Networks • Low-dimensional High dimensional • ‘Few’ nodes (100, 1000) Billions of nodes • Firm math foundations Empirical: mostly • How to link mathematical rigor and high dimensionality?? • ===> Random Graphs are mathematically solid and can handle high-dimensional issues

7. Graphs -Examples • Traffic routes: airlines • Flowcharts • Proper factor relation • Electrical networks • Computer networks • ----------------------------------- • ====> What are the nodes and what are the edges? • ====> Which are undirected and directed graphs?

8. Graph Definitions cont’d • Subgraph G’=(V’,E’) of Graph G=(V,E) • V’ are from V and E’ from E, and E’ from V’xV’ • Symmetric di-graph: if (v,w) is an edge then (w,v) is also an edge • Complete graph: all possible edges are contained, each pair of vertices are connected • Path: from v node to w node is a sequence of edges that starts at v and ends in w. • Path: {v1, v2, v3,…, vn}, and v1=v, vn=w

9. Graph - Definitions cont’d • Connected undirected graph: Iff there is a path for each pair of vertices v and w. • Strongly connected directed graph : A digraph iff there is a path from each v to each w (directed!) • Connected component: of a undirected G graph, is a maximal connected subgraph of G. (there could be several connected components).

10. Graph - Definitions cont’d • Cycle: in directed graph:a path (>0) with the first and last vertices are the same; in undirected graph: as above + all the edges have a unique orientation in the path! • Forest (undirected): an undirected acyclic graph • Tree: a connected undirected acyclic graph

11. How to define a graph? • Adjacency matrix (A) • In a graph with n nodes, then A is nxn • A has 1 in the A[i][j] element if the ith and jth node are adjacent (linked by an edge) and all the other elements are 0 • Weighted graph: • edges have numeric values assigned (cost or capacity). • Those values are shown in the adjacency matrix. • The adjacency matrix components w/o links are infinity (if cost) or zero (is capacity).

12. How to define a graph? /2 • Give nodes and list edges • Give edges in an adjacency list structure • List nodes • List all edges linked to the node • When to use adjacency matrix A and when list? • ==> A is efficient for connected/strongly connected graphs. • ==> For sparse graphs adjacency list is better