chapter 14 overview of graph theory and least-cost paths 1 chapter 14 overview of graph theory and...

Chapter 14 Overview of Graph Theory and Least-Cost Paths1

Chapter 1Chapter 144Overview of Graph Theory and Least-Cost Paths


IntroductionIntroduction

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


Elementary ConceptsElementary 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


Example GraphExample Graph


Adjacent MatrixAdjacent 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


Adjacent Matrix ExampleAdjacent Matrix Example


TerminologyTerminology

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


Simple Paths (1)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)


Simple Paths (2)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


DigraphsDigraphs

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


Weighted GraphWeighted 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)


Weighted Graph and Adjacent Weighted Graph and Adjacent MatrixMatrix


Path Distances and Lengths Path Distances and Lengths VV11

to to VV66Path Distance LengthV1,V2,V3,V4,V5,V6 5 11V1,V2,V3,V5,V6 4 8V1,V2,V3,V6 3 10V1,V2,V4,V3,V5,V6 5 10V1,V2,V4,V5,V6 4 7V1,V3,V2,V4,V5,V6 5 16V1,V3,V6 2 10V1,V4,V5,V6 3 4


TreesTrees

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


Family TreeFamily 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


E.g. TreeE.g. Tree


SubgraphSubgraph

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’


Spanning TreeSpanning 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


E.g. Spanning Trees For E.g. Spanning Trees For Previous GraphPrevious Graph

Also previous tree example (slide 16)


Breadth First Search (BFS) for Breadth First Search (BFS) for Spanning TreeSpanning 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


E.g. Using Previous GraphE.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 …


BFS of Previous GraphBFS of Previous Graph


Shortest Path DistanceShortest 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)


Estimated Running TimeEstimated 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|


Shorted Path Length Shorted Path Length DeterminationDetermination 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


Routing DecisionsRouting 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


Cost of a HopCost of a Hop

Inversely proportional to path capacity Proportional to current load Monetary cost of link etc. Combination May be different in different directions


Dijkstra’s Algorithm (1) –Dijkstra’s Algorithm (1) –DefinitionsDefinitions 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


Dijkstra’s Algorithm (2) –Dijkstra’s Algorithm (2) –StepsSteps1. Initialization

a. T = Tree = {s} - only source is so far incorporatedb. L(n) = w(s,n) for n s - initial path cost to neighbors are link

costs

2. Get next vertexa. Find x T such that L(x) = min L(j), j Tb. Add x to T and Treec. Add edge to T incident on x and has least cost

Last hop in path

3. Update least cost pathsa. 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


Dijkstra’s Algorithm (3) –Dijkstra’s Algorithm (3) –NotesNotes 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


Dijkstra’s Algorithm on Dijkstra’s Algorithm on Example GraphExample Graph


Bellman-Ford Algorithm (1) –Bellman-Ford Algorithm (1) –DefinitionsDefinitions 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


Bellman-Ford Algorithm (2) –Bellman-Ford Algorithm (2) –StepsSteps1. Initialization

a. L0(n) = n s

b. Lh(s) = 0 h

2. Updatea. For each successive h 0

i. For each n s, compute:

Lh+1+(n) = min[Lh(j)+ w(j,n)], jii. Connect n with predecessor vertex j that achieves minimumiii. Eliminate any connection of n with different predecessor

vertex from previous iterationiv. Path from s to n terminates with link from j to n


Bellman-Ford Algorithm (3) –Bellman-Ford Algorithm (3) –NotesNotesResults agree with DijkstraRunning time order of |V| x |E|


Bellman-Ford Algorithm on Bellman-Ford Algorithm on Example GraphExample Graph


Results of Dijkstra and Results of Dijkstra and Bellman-FordBellman-Ford


ComparisonComparison of Information of Information Needed – Bellman-FordNeeded – 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


ComparisonComparison of Information of Information Needed – DijkstraNeeded – DijkstraStep 3 requires each vertex must have

complete topology– Must know link costs of all links in network– Information must be exchanged between all

other verticesEvaluation must also consider calculation

time


Other NotesOther Notes

Both Algorithms converge under static conditions of topology and link cost

Give to same solutionIf link costs change, algorithms will

attempt to catch upIf 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 1 chapter 14 overview of graph theory and...

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