spanning trees in communication networks rui campos inesc porto, faculdade de engenharia,...

Spanning Trees in Communication

Networks

Rui CamposINESC Porto, Faculdade de Engenharia,

Universidade do [email protected]

Outline

1. Graph Theory

2. Spanning Trees

3. Algorithms

4. Spanning Trees Applications

5. Summary

About me• PhD student at University of Porto

PhD Overview

Auto-configuration and Self-management of Personal Area

Networks

considering:

• intra- and extra-PAN connectivity

• heterogeneity in IP networks

• integration of multiple communication technologies

• Researcher at INESC Porto

Participation in FP6 Ambient Networks Project

• 1st phase - protocol design, specification, and simulation

• 2nd phase – automatic and dynamic IP connectivity

establishment between Ambient Networks

1. Graph Theory

Graph formal definition:

A graph G=(V,E,w) is defined as a set of

vertices V connected by a set of edges

E=[V]2 (i.e., the elements of E are 2-

element subsets of V) with weights w that

represent costs assigned to the edges of G.

1. Graph Theory [example graph]

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1. Graph Theory [some definitions]

• Number of vertices in G is n = |V|

• Number of edges in G is m = |E|

• Degree of a vertex v is defined as its number of adjacent edges

(an adjacent edge to v is an edge connecting it to a neighbor vertex)

• Path in G is defined as a sequence of connected adjacent vertices

• Cost of path is equal to sum of weights of edges belonging to path

• Cycle in G is defined as a path where first and last vertex in

sequence are the same

1. Graph Theory [graph types]

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undirected undirected weighted

directed directed weighted

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1. Graph Theory [graph types]

complete graph metric graph

1. Graph Theory [specific types]

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4 5

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4 5

2

2

2

2

2

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4

all vertices have degree n-

1

edge weights satisfy triangle

inequality

spanning tree

1. Graph Theory [specific types]

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4

tree with n-1 edges extracted

from

a graph G with n vertices

treegraph without

cycles

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4 5

2. Spanning Trees [cost functions]

• Path cost

e.g., cT(1,3)=5+2=7

• Total cost

Ctotal(T)=5+2+1=8

• Routing Cost

Cr(T)= cT(1,2)+cT(1,3)+cT(1,4)+cT(2,1)+cT(2,3)+cT(2,4)

+ cT(3,1)+cT(3,2)+cT(3,4)+cT(4,1)+cT(4,2)+cT(4,3)=48

1

1

)()(n

eTtotal ewTC

n

i

n

jTr jijicTC

1 1

),,()(

1

3

2

4

5

12

),( vscT

2. Spanning Trees [types]

Shortest Path Tree (SPT)

Minimum Spanning Tree (MST)

Minimum Routing Cost Tree

(MRCT)

path cost

total cost

routing cost

2. Spanning Trees [SPT]

The Shortest Path Tree (SPT) rooted at a

vertex s defines a tree composed by union

of the shortest paths between s and each of

the other vertices in G such that:

)),(min(),( vscvsc GSPT


Let’s see an example …

SPT root at vertex 1:

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101

2

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14

4

P(1,4) = (1,2,4) ; C(1,4) = 14

P*(1,4) = (1,2,3,4) ; C*(1,4) =

14

P**(1,4) = (1,3,2,4) ; C**(1,2)

= 7

P***(1,4) = (1,3,4) ; C***(1,4)

= 5

P(1,3) = (1,3) ; C(1,3) = 2

P*(1,3) = (1,2,3) ; C*(1,3) =

11

P**(1,3) = (1,2,4,3) ; C**(1,3)

= 17

P(1,2) = (1,2) ; C(1,2) = 10

P*(1,2) = (1,3,2) ; C*(1,2) = 3

P**(1,2) = (1,3,4,2) ; C**(1,2)

= 9


SPT root at vertex 1:

P*(1,2) = (1,3,2)

P(1,3) = (1,3)

P***(1,4) = (1,3,4) 1

2 3

4

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3

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10

101

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14

4

2. Spanning Trees [MST]

The Minimum Spanning Tree (MST)

represents the spanning tree T* such that:

for all spanning trees T that can be

computed from G

1

1

)(min*)(n

eTtotal ewTC

2. Spanning Trees [MST]

Let’s see an example ...

STs that can be computed from graph:

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2 4

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1

2 3

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2

3 5Ctotal = 3+2+4 = 9

Ctotal = 2+4+5 = 11

Ctotal = 3+5+4 = 12

Ctotal = 2+3+5 = 11

2. Spanning Trees [MRCT]

The Minimum Routing Cost Tree (MRCT)

represents the spanning tree T* such that:

for all spanning trees T that can be

computed from G

n

i

n

jTr jijicTC

1 1

),,(min*)(

Note: computation of exact MRCT is a NP-hard problem


Let’s see an example …


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2 4

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2

3 5Cr = 2+4+5+2+6+3

+4+6+9+5+3+9=58

Cr = 2+4+9+2+6+11

+4+6+5+9+11+5=74

Cr = 12+4+9+12+8+3

+4+8+5+9+3+5=82

Cr = 2+10+5+2+8+3

+10+8+5+5+3+5=66


Using other routing cost definition …


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5

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2 3

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3

4

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2 3

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2 4

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2 3

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5

1

2 3

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2

3 5Cr =

2+4+5+6+3+9=29 Cr =

2+4+9+6+11+5=37

Cr = 12+4+9+8+3+5=41

Cr = 2+10+5+8+3+5=33

2. Spanning Trees [other types]

• MRCT considering communication requirement

(MRCT is optimal when communication requirements are equal)

• Steiner Minimal Tree (SMT)

– spans only a given subset of vertices in a graph G

– minimizes same cost functions as MST

• Minimum Diameter Spanning Tree (MDST)

– diameter = cost of longest path between any two vertices in a tree

– defines spanning tree of minimum diameter among all possible

spanning trees

Note: other types and further details can be found in [1]

2. Spanning Trees [2 interesting cases]

• When edge weights of G are highly heterogeneous

MST = MRCT = SPT (*)

• When there is a single source s

MRCT = SPTs

(*) P. Mieghem, S. Langen, Influence of the link weight structure on the shortest path, Physical Review E 71 (2005) 056113-1–056113-13.

P. Mieghem, S. M. Magdalena, Phase transition in the link weight structure of networks, Physical Review E 72 (2005) 056138-2–056138-7.

2. Spanning Trees [MST=MRCT=SPT]



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410

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1

Cr=64; Ctotal=21

Cr=46; Ctotal=21

Cr=46; Ctotal=21

Cr=64; Ctotal =21

Cr=36; Ctotal =12

Cr=36; Ctotal =12

Cr=10; Ctotal =3 Cr=64; Ctotal =21SPT

3. Algorithms [Prim’s algorithm]

• Algorithm to compute MST for a graph G

• Proposed by Robert Prim in 1957

• Time complexity

O(n.m)

– in its more naive implementation

O(m.log(n))

– using binary heaps in conjunction with adjacency lists

O(m+n.log(n))

– using Fibonacci heaps in conjunction with adjacency

lists

Algorithm: PRIM

Input: A weighted, undirected graph G=(V,E,w)

Output: A minimum spanning tree T

T ← 0

Let uV be an arbitrary vertex

U ← {u}

while |U| < n do

find uU and v(V - U) such that edge (u,v) is the smallest edge between U and V –

U and does not form a cycle

T ← T {(u,v)}

U ← U {v}


Steps in a nutshell:(T ← 0)

1. Pick up arbitrary vertex v

2. Select edge (i,j) adjacent to v with lowest weight and add it

to T

3. Select adjacent edge (i,j) to T with lowest weight that does

not form a cycle and add it to T

4. Repeat step 3 until all vertices have been spanned

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input graph MST


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input graph MST

Other possible result ...

2 2

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3. Algorithms [Kruskal’s algorithm]

• Algorithm to compute MST for a graph G

• Proposed by Joseph Kruskal in 1956

• Time complexity

• O(m.log(m))

– using priority queues

Algorithm: KRUSKAL

Input: A weighted, undirected graph G=(V,E,w)

Output: A minimum spanning tree T

Sort edges in E in nondecreasing order by weight

T ← 0

Create one set for each vertex

for each edge (u,v) in sorted order do

if FIND_SET(u) FIND_SET(v) then T ← T {(u,v)} UNION(u,v)

MAKE_SET(v) create a new set whose only member is v

FIND_SET(v) returns a pointer to set containing v

UNION(u,v) unites dynamic sets that contain u and

v into new set that is union of these

two sets

Legend


Steps in a nutshell:(T ← 0)

1. Pick up edge (i,j) with lowest weight that does not belong

to T and does not form a cycle

2. Add edge (i,j) to T

3. Repeat step 1 and 2 until all vertices have been spanned

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input graph MST


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3. Algorithms [Dijkstra’s algorithm]

• Algorithm to compute SPT for a graph G

• Proposed by Edsger Dijkstra in 1959

• Time complexity

• O(n2)

– in its more naive implementation

• O(m.log(n))

– using binary heaps

• O(m+nlog(n))

– using Fibonacci heaps

Algorithm: DIJKSTRA

Input: A weighted, directed graph G=(V,E,w);

a source vertex s.

Output: A shortest path tree T rooted at s

for each vV do

δv ← ; pv ← NIL

δs ← 0; T ← 0; S ← V

while S 0 do

choose uS with min(δu)

S ← S – {u}

if u s then T ← T {(pu,u)}

for each vertex v adjacent to u do

if (δv δu + w(u,v)) then

δv ← δu + w(u,v)

pv ← u


Steps in a nutshell:

(T ← 0; δv ← ; δs ← 0)

1. Select vertex v with min shortest path estimate and add edge

(pv,v) to T

2. Re-evaluate shortest path estimates of vertices adjacent to v and

update them and parent vertex in T if lower estimates are

obtained

3. Repeat steps 1 and 2 until all vertices have been spanned

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input graph


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SPT rooted@vertex 1

v δ

v

2 1

3 1

0

v δ

v

3 1

0

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v

3 1

0

4 6

v δ

v

3 9

6 7

v δ

v

3 9

3. Algorithms [Bellman-Ford algorithm]

• Algorithm to compute SPT for a graph G

• Proposed by Richard Bellman in 1958

• Works in more general cases• finds SPT even when graph has negative weights

• Time complexity• O(n.m)

Algorithm: BELLMAN-FORD

Input: A weighted, directed graph G=(V,E,w);

a source vertex s.

Output: A shortest path tree T rooted at s

for each vV do

δv ← ; pv ← NIL

δs ← 0

for i ← 1 to n-1 do

for each (u,v)E do


δv ← δu + w(u,v); pv ← u

for each (u,v)E do


Output “A negative cycle exists”; Exit

T ← 0

for v(V-s) do

T ← T {(pu,u)}


Steps in a nutshell:(T ← 0; δv ← ; δs ← 0)

1. For each edge (u,v) re-evaluate shortest path

estimate δv and update it and pv if (δu + w(u,v) δv)

2. Repeat step 1 (n-1) times

3. The union of edges (pv,v) gives shortest path tree T

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input graph

Let’s see an example ... SPT rooted@vertex 2

v 1 2 3 4

δv 0

pv NI

L

NI

L

NI

L

NI

L

v 1 2 3 4

δv 2 0 3

pv 2 NI

L

NI

L

2

E={(1,2),(2,1),(1,3),(3,1),(2,4),(4,3)}

v 1 2 3 4

δv 2 0 12 3

pv 2 NI

L

1 2Re-evaluation:for each (u,v)E do if (δu + w(u,v) δv)

then δv ← δu + w(u,v) SPT_rooted@2

v 1 2 3 4

δv 2 0 6 3

pv 2 NI

L

4 2

3. Algorithms [Add algorithm]

• Algorithm for finding approximate MRCT for a

graph G

• Proposed by Vic Grout in 2005

• Vertex oriented version of Prim’s algorithm• minimizes number of relay nodes in final spanning tree

• Time complexity• O(n.log(n)) time in the worst case

– usually, faster due to tendency for many nodes to be added to

T at each step

Algorithm: ADD

Input: undirected graph G=(V,E)

Output: approximate minimum routing cost tree T

T ← 0

for all v V

sv ← 0

for all u,v V

wuv ← 0

find u such that

du ← max(dv), v V

su ← 1

while there exists j such that sj = 0 do

for each v V adjacent to u do

wuv ← 1; sv ← 1; T ← T {(pu,u)}

find u such that (du − δu)← max(dj − δj) where sj = 1

Legend:sv=1 vTwuv – weight of edge (u,v)du – degree of vertex uδu – degree of vertex u in T


Steps in a nutshell:1. Pick up vertex v adjacent to maximum number of

unspanned vertices

2. Add to T all edges adjacent to v that do not form a cycle

3. Repeat steps 1 and 2 until all vertices are added to T

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input graph approximate MRCT


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3

3. Algorithms [Wong’s algorithm]

• Proposed by Richard Wong in 1980

• 2-approximate MRCT algorithm

• Based on Dijkstra’s algorithm

• Time complexity• O(n2.log(n)+m.n)


Steps:

1. Compute n SPTs using Dijkstra’s

algorithm

2. Compute routing cost for each SPT

3. Select SPT with lowest routing cost

2-approximate MRCT

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SPT_rooted@1

SPT_rooted@2

SPT_rooted@3

SPT_rooted@4

SPT_rooted@5

SPT_rooted@6

SPT_rooted@7

SPT_rooted@8


1

23

5

4

67

8Let’s see an example ...

SPT root Routing

Cost

1 124

2 136

3 118

4 122

5 144

6 126

7 134

8 142

1

23

5

4

67

8

approximate MRCT


Minimum Spanning Tree (MST)

cable TV network

• cost-efficient connection of several end clients to

infrastructure

islands connection

• cost-efficient connection of islands using bridges

travelling salesman problem

• NP-hard problem

• MST can be used to find 2-approximate solution

4. Spanning Trees ApplicationsShortest Path Tree (SPT)

routing protocols

• Routing Information Protocol (RIP)

• Open Shortest Path First (OSPF)

• Ad-hoc On-demand Distance Vector (AODV)

• Optimized Link State Routing (OLSR)

multicast routing

• minimization of path cost between source and destinations

road network

• find shortest path between a city and other cities in a

country


Minimum Routing Cost Tree (MRCT)

Ethernet networks

• minimization of routing cost between

bridges/switches

road network

• connection of cities in a country with minimum

distance between cities

5. Summary

• Different types of graphs defined

• undirected, undirected weight, directed, directed weighted

• Specific graph types

• complete graph, metric graph, tree, spanning tree

• Different types of spanning trees

• Shortest Path Tree (SPT)

• Minimum Spanning Tree (MST)

• Minimum Routing Cost Tree (MRCT)

↓

their computation by exhaustive search is impratical

5. Summary

• Spanning tree algorithms

• Enable efficient computation of spanning trees

• Spanning trees have several applications

Spanning

Tree

Algorithm

MST • Prim

• Kruskal

SPT • Dijkstra

• Bellman-Ford

MRCT • Add

• Wong

References

[1] B. Wu and K. Chao, Spanning Trees and

Optimization Problems, Chapman & Hall, 2004.

[2] R. Diestel, Graph Theory, Springer-Verlag

Heidelberg, New York, Electronic Edition 2005.

[3] V. Grout, Principles of Cost Minimization in

Wireless Networks, J. of Heuristics 11 (2005)

115-133.

Contact

Rui Campos

(+351) 222 094 268

[email protected]

http://telecom.inescporto.pt/~rcampos

mailto:[email protected]

http://telecom.inescporto.pt/~rcampos

spanning trees in communication networks rui campos inesc porto, faculdade de engenharia,...

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