Understanding Graph Theory Basics and Paths

1. Chapter 14 Overview of Graph Theory and Least-Cost Paths Chapter 14 Overview of Graph Theory and Least-Cost Paths

2. Introduction • Comms networks can be represented by graphs • Switches & routers are vertices • Comms lines are edges • Routing protocols use shortest path algorithms • This chapter is background to chapters on routing Chapter 14 Overview of Graph Theory and Least-Cost Paths

3. Elementary Concepts • Graph G(V,E) is two sets of objects • Vertices (or nodes) , set V • Edges, set E • Defined as an unordered pair of vertices • Shown as dots or circles (vertices) joined by lines (edges) • Vertex i is adjacent to vertex j if (i,j)  E • Magnitude of graph G characterised by number of vertices |V| (called the order of G) and number of edges |E|, size of G • Running time of algorithm measured in terms of order and size Chapter 14 Overview of Graph Theory and Least-Cost Paths

4. Example Graph Chapter 14 Overview of Graph Theory and Least-Cost Paths

5. Adjacent Matrix • Used to represent graph • Number vertices • Arbitrary • 1,2,3,…,|V| • The |V| x |V| adjacent matrix A=(ai,j) defined by: • ai,j = 1 if (i,j)  E 0 otherwise • Matrix symmetrical about upper left to lower right diagonal • Because edge defined as unordered pair Chapter 14 Overview of Graph Theory and Least-Cost Paths

6. Adjacent Matrix Example Chapter 14 Overview of Graph Theory and Least-Cost Paths

7. Terminology • Two edges incident on same pair of vertices are parallel • Edge incident on single vertex is a loop • Graph with neither parallel edges nor loos is simple • Path from vertex i to vertex j is: • Alternating sequence of vertices and edge • Starting at i and ending at j • Each edge joins vertices immediately before and after it • Simple path – no vertex nor edge appears more than once • In simple graph, simple path may be defined by sequence of vertices • Each vertex adjacent to preceding and following vertices • No vertex repeated Chapter 14 Overview of Graph Theory and Least-Cost Paths

8. Simple Paths (1) • From V1 to V6 (incomplete list) • V1,V2,V3,V4,V5,V6 • V1,V2,V3,V5,V6 • V1,V2,V3,V6 • V1,V2,V4,V3,V5,V6 • V1,V2,V4,V5,V6 • V1,V3,V2,V4,V5,V6 • V1,V3,V6 • V1,V4,V3,V6 • Total of 14 paths (Work out the rest yourself) Chapter 14 Overview of Graph Theory and Least-Cost Paths

9. Simple Paths (2) • V1,V3,V6 is shortest • Distance between vertices is minimum number of edges on all paths • Cycle is path staring and ending on same vertex • E.g. V1,V3,V4,V1 Chapter 14 Overview of Graph Theory and Least-Cost Paths

10. Digraphs • Directed graph • G(V,E) with each edge defined by ordered pair of vertices • Lines, representing edges, have arrow head to indicate direction • Parallel edges allowed if in opposite directions • Good for representing comms networks • Each directed edge represents data flow in one direction • Still use adjacent matrix • Not symmetrical unless each pair of adjacent vertices connected by parallel edges Chapter 14 Overview of Graph Theory and Least-Cost Paths

11. Weighted Graph • Or weighted digraph • Number associated with each edge • Used to illustrate routing algorithms • Adjacent matrix defined as • ai,j =wi,j if (i,j)  E 0 otherwise Where wi,j is weight associated with edge (i,j) • Length of path is sum of weights • Shortest-distance path not necessarily shortest-length (see next two slides) Chapter 14 Overview of Graph Theory and Least-Cost Paths

12. Weighted Graph and Adjacent Matrix Chapter 14 Overview of Graph Theory and Least-Cost Paths

13. Path Distances and Lengths V1 to V6 Path Distance Length V1,V2,V3,V4,V5,V6 5 11 V1,V2,V3,V5,V6 4 8 V1,V2,V3,V6 3 10 V1,V2,V4,V3,V5,V6 5 10 V1,V2,V4,V5,V6 4 7 V1,V3,V2,V4,V5,V6 5 16 V1,V3,V6 2 10 V1,V4,V5,V6 3 4 Chapter 14 Overview of Graph Theory and Least-Cost Paths

14. Trees • Subset of graphs • Equivalent definitions: • Simple graph such that if i and j vertices in T, there is a unique simple path from i to j • Simple graph of N vertices is tree if it has N-1 edges and no cycles • Simple graph of N vertices is tree if it has N-1 edges and is connected • One vertex may be designated root • Root drawn at top • Vertices adjacent to root drawn at next level • Can reach root on path distance 1 Chapter 14 Overview of Graph Theory and Least-Cost Paths

15. Family Tree • Each vertex (except root) has one parent vertex • Adjacent vertex closer to root • Each vertex has zero or more child vertices • Adjacent vertices further from root • Vertex without children is called a leaf • Root assigned level 1 • Vertices immediately under root level 1 • Children of vertices on level 1 are on level 2 Chapter 14 Overview of Graph Theory and Least-Cost Paths

16. E.g. Tree Chapter 14 Overview of Graph Theory and Least-Cost Paths

17. Subgraph • Subgraph of graph G obtained by selecting number of edges and vertices from G • For each edge, the two vertices incident on that edge must be selected • Give graph G(E,V), graph G’(E’,V’) is a subgraph of G iff • V’  V and E’  E and •  e’  E’, if e’ incident on v’ and w’ then v’, w’  V’ Chapter 14 Overview of Graph Theory and Least-Cost Paths

18. Spanning Tree • Subgraph T of graph G is a spanning tree if • T is a tree • T includes all vertices of G • In other words remove edges from G such that: • Remove all cycles • Maintain connectivity • Not usually unique Chapter 14 Overview of Graph Theory and Least-Cost Paths

19. E.g. Spanning Trees For Previous Graph • Also previous tree example (slide 16) Chapter 14 Overview of Graph Theory and Least-Cost Paths

20. Breadth First Search (BFS) for Spanning Tree • Partition vertices of graph into sets at various levels • Process all vertices on given level before proceeding to next level • Start at any vertex, x • Assign it level 0 • All adjacent vertices are at level 1 • Let Vi1, Vi2,Vi3,…Vij,be vertices at level i • Consider all vertices adjacent Vi1 not at level 1,2,…,i • Assign these level (i+1) • Consider all vertices adjacent Vi2 not at level 1,2,3,…,i, (i+1) • Assign these also level (i+1) • Until all vertices processed Chapter 14 Overview of Graph Theory and Least-Cost Paths

21. E.g. Using Previous Graph • Choose order • Obvious one is V1,V2,V3,V4,V5,V6 • Select root • Again, obvious one is V1 • Let tree T consist of single vertex V1with no edges • Add to T each edge (V1,x) and vertex x • Such that no cycle is produced • Gives edges (V1,V2), (V1,V3), (V1,V4) and vertices V1,V2, V3 • This is first level • Repeat for all level 1 vertices to give level 2 • All vertices now added • If not repeat for level 2 to give level 3 … Chapter 14 Overview of Graph Theory and Least-Cost Paths

22. BFS of Previous Graph Chapter 14 Overview of Graph Theory and Least-Cost Paths

23. Shortest Path Distance • BFS finds shortest path distance from given source vertex to all other vertices • Minimum number of edges in any path from s to v, δ(s,v) Chapter 14 Overview of Graph Theory and Least-Cost Paths

24. Estimated Running Time • After initialization each vertex is used exactly once as a starting point for adding the next layer • Time take is order of |V| • Each edge already in tree is rejected if examined again • Each edge not in tree is checked to see if it produces a cycle • If not it is included • Bulk of edge processing is once per edge • Time take is order of |E| • Total time taken is linear with |V| and |E| Chapter 14 Overview of Graph Theory and Least-Cost Paths

25. Shorted Path Length Determination • Packet switching, frame relay or ATM network can be viewed as digraph • Each node is a vertex • Each link is a pair of parallel edges • For an internet (Internet or intranet) • Each router is vertex • If routers directly connected (e.g. LAN or WAN) two way connection corresponds to pair of parallel edges • If more than two routers, network represented by multiple pairs of parallel edges • One pair connecting each pair of routers • In both cases, routing decision needed to pass packet from source to destination • Equivalent to finding path through a graph Chapter 14 Overview of Graph Theory and Least-Cost Paths

26. Routing Decisions • Based on least cost • Minimum number of hops • Each edge (hop) has weight 1 • Corresponds to minimum path distance • Or, cost associated with each hop (next slide) • Cost of path is sum of costs of links in path • Want least cost path • Corresponds to minimum path length in weighted digraph Chapter 14 Overview of Graph Theory and Least-Cost Paths

27. Cost of a Hop • Inversely proportional to path capacity • Proportional to current load • Monetary cost of link etc. • Combination • May be different in different directions Chapter 14 Overview of Graph Theory and Least-Cost Paths

28. Dijkstra’s Algorithm (1) –Definitions • N = set of vertices in network • s = source vertex (starting point) • T = set of vertices so far incorporated • Tree = spanning tree for vertices in T including edges on least-cost path from s to each vertex in T • w(i,j) = link cost from vertex i to vertex j • w(i,i) = 0 • w(i,j) =  if i, j not directly connected by a single edge • w(i,j)  0 of i,j directly connected by single edge • L(n) = cost of least cost path from s to n currently known • At termination, this is least cost path from s to n Chapter 14 Overview of Graph Theory and Least-Cost Paths

29. Dijkstra’s Algorithm (2) –Steps • Initialization • T = Tree = {s} - only source is so far incorporated • L(n) = w(s,n) for n  s - initial path cost to neighbors are link costs • Get next vertex • Find x  T such that L(x) = min L(j), j  T • Add x to T and Tree • Add edge to T incident on x and has least cost • Last hop in path • Update least cost paths • L(n) = min[L(n), L(x) + w(x,n)]  n  T • If latter term is minimum, path from s to n is now path from s to x concatenated with edge from x to n Chapter 14 Overview of Graph Theory and Least-Cost Paths

30. Dijkstra’s Algorithm (3) –Notes • Terminate when all vertices added to T • Requires |V| iterations • At termination • L(x) associated with each vertex is cost of least cost path from s to x • Tree is a spanning tree • Defines least cost path from s to each other vertex • One step adds one vertex to T and defines least cost path from s to that vertex • Running time order of |V|2 Chapter 14 Overview of Graph Theory and Least-Cost Paths

31. Dijkstra’s Algorithm on Example Graph Chapter 14 Overview of Graph Theory and Least-Cost Paths

32. Bellman-Ford Algorithm (1) –Definitions • s = source vertex (starting point) • w(i,j) = link cost from vertex i to vertex j • w(i,i) = 0 • w(i,j) =  if i, j not directly connected by a single edge • w(i,j)  0 of i,j directly connected by single edge • h = max number of links in path at current stage • Lh(n) = cost of least cost path from s to n such that no more than h links Chapter 14 Overview of Graph Theory and Least-Cost Paths

33. Bellman-Ford Algorithm (2) –Steps • Initialization • L0(n) =   n  s • Lh(s) = 0  h • Update • For each successive h  0 • For each n  s, compute: Lh+1+(n) = min[Lh(j)+ w(j,n)],  j • Connect n with predecessor vertex j that achieves minimum • Eliminate any connection of n with different predecessor vertex from previous iteration • Path from s to n terminates with link from j to n Chapter 14 Overview of Graph Theory and Least-Cost Paths

34. Bellman-Ford Algorithm (3) –Notes • Results agree with Dijkstra • Running time order of |V| x |E| Chapter 14 Overview of Graph Theory and Least-Cost Paths

35. Bellman-Ford Algorithm on Example Graph Chapter 14 Overview of Graph Theory and Least-Cost Paths

36. Results of Dijkstra and Bellman-Ford Chapter 14 Overview of Graph Theory and Least-Cost Paths

37. Comparison of Information Needed – Bellman-Ford • Calculation for vertex n involves knowledge of link cost to all neighbors of n plus total path cost to each from source • Each vertex can keep set of costs and paths for every other vertex in network • Exchange information with direct neighbors • Each vertex can use use Bellman-Ford step 2 based on information from neighbors and knowledge of link costs to update its costs and paths Chapter 14 Overview of Graph Theory and Least-Cost Paths

38. Comparison of Information Needed – Dijkstra • Step 3 requires each vertex must have complete topology • Must know link costs of all links in network • Information must be exchanged between all other vertices • Evaluation must also consider calculation time Chapter 14 Overview of Graph Theory and Least-Cost Paths

39. Other Notes • Both Algorithms converge under static conditions of topology and link cost • Give to same solution • If link costs change, algorithms will attempt to catch up • If link costs depend on traffic, which depends on routes chosen: • Feedback condition exists • Instability may result Chapter 14 Overview of Graph Theory and Least-Cost Paths