1. TCP Traffic and Congestion Control in ATM Networks
Chapter 14
Overview of Graph Theory and Least-Cost Paths
Chapter 14 Overview of Graph Theory and Least-Cost Paths
Chapter 13

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,V
V1,V2,V3,V5,V
V1,V2,V3,V
V1,V2,V4,V3,V5,V
V1,V2,V4,V5,V
V1,V3,V2,V4,V5,V
V1,V3,V
V1,V4,V5,V
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 V1 with 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