Spanning Trees in Communication Networks

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Spanning Trees in Communication Networks

• Rui Campos
• INESC Porto, Faculdade de Engenharia,
• Universidade do Porto
• rcampos_at_inescporto.pt

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Outline

1. Graph Theory
2. Spanning Trees
3. Algorithms
4. Spanning Trees Applications
5. Summary

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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

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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 EV2
  (i.e., the elements of E are 2-element subsets of
  V) with weights w that represent costs assigned
  to the edges of G.

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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

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1. Graph Theory graph types
undirected
undirected weighted
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1. Graph Theory graph types
directed
directed weighted
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1. Graph Theory specific types
complete graph
metric graph
all vertices have degree n-1
edge weights satisfy triangle inequality
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1. Graph Theory specific types
spanning tree
tree
tree with n-1 edges extracted from a graph G
with n vertices
graph without cycles
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2. Spanning Trees cost functions

• Path cost
• e.g., cT(1,3)527
• Total cost
• Ctotal(T)5218
• 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

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2. Spanning Trees types

• Shortest Path Tree (SPT)
• Minimum Spanning Tree (MST)
• Minimum Routing Cost Tree (MRCT)

path cost
total cost
routing cost
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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

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2. Spanning Trees SPT

• Lets see an example
• SPT root at vertex 1

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
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
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2. Spanning Trees SPT

• SPT root at vertex 1
• P(1,2) (1,3,2)
• P(1,3) (1,3)
• P(1,4) (1,3,4)

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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

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2. Spanning Trees MST

• Lets see an example ...
• STs that can be computed from graph

Ctotal 324 9
Ctotal 354 12
Ctotal 235 11
Ctotal 245 11
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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

Note computation of exact MRCT is a NP-hard
problem
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2. Spanning Trees MRCT

• Lets see an example
• STs that can be computed from graph

Cr 12491283 48593582
Cr 245263 46953958
Cr 2492611 465911574
Cr 2105283 108553566
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2. Spanning Trees MRCT

• Using other routing cost definition
• STs that can be computed from graph

Cr 124983541
Cr 24563929
Cr 249611537
Cr 210583533
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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
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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) 056113-1056113-13. P.
Mieghem, S. M. Magdalena, Phase transition in the
link weight structure of networks, Physical
Review E 72 (2005) 056138-2056138-7.
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2. Spanning Trees MSTMRCTSPT

• Lets see an example ...
• STs that can be computed from graph

Cr64 Ctotal21
Cr46 Ctotal21
Cr46 Ctotal21
Cr64 Ctotal 21
Cr36 Ctotal 12
Cr36 Ctotal 12
Cr10 Ctotal 3
Cr64 Ctotal 21
SPT
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3. Algorithms Prims 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(mn.log(n))
• using Fibonacci heaps in conjunction with
  adjacency lists

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• Algorithm PRIM
• Input A weighted, undirected graph G(V,E,w)
• Output A minimum spanning tree T
• T ? 0
• Let u?V be an arbitrary vertex
• U ? u
• while U lt n do
• find u?U 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

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3. Algorithms Prims algorithm

• Steps in a nutshell
• (T ? 0)
• Pick up arbitrary vertex v
• Select edge (i,j) adjacent to v with lowest
  weight and add it to T
• Select adjacent edge (i,j) to T with lowest
  weight that does not form a cycle and add it to T
• Repeat step 3 until all vertices have been spanned

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3. Algorithms Prims algorithm
Lets see an example ...
input graph
MST
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3. Algorithms Prims algorithm
Other possible result ...
input graph
MST
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3. Algorithms Kruskals algorithm

• Algorithm to compute MST for a graph G
• Proposed by Joseph Kruskal in 1956
• Time complexity
• O(m.log(m))
• using priority queues

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• 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)

Legend
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
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3. Algorithms Kruskals algorithm

• Steps in a nutshell
• (T ? 0)
• Pick up edge (i,j) with lowest weight that does
  not belong to T and does not form a cycle
• Add edge (i,j) to T
• Repeat step 1 and 2 until all vertices have been
  spanned

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3. Algorithms Kruskals algorithm
Lets see an example ...
input graph
MST
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3. Algorithms Dijkstras 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(mnlog(n))
• using Fibonacci heaps

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• 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 v?V do
• dv ? ? pv ? NIL
• ds ? 0 T ? 0 S ? V
• while S ? 0 do
• choose u?S with min(du)
• S ? S u
• if u ? s then T ? T ? (pu,u)
• for each vertex v adjacent to u do
• if (dv ? du w(u,v)) then
• dv ? du w(u,v)
• pv ? u

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3. Algorithms Dijkstras algorithm

• Steps in a nutshell
• (T ? 0 dv ? ? ds ? 0)
• Select vertex v with min shortest path estimate
  and add edge (pv,v) to T
• Re-evaluate shortest path estimates of vertices
  adjacent to v and update them and parent vertex
  in T if lower estimates are obtained
• Repeat steps 1 and 2 until all vertices have been
  spanned

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3. Algorithms Dijkstras algorithm
Lets see an example ...
SPT rooted_at_vertex 1
v dv
2 1
3 10
v dv
3 10
5 4
v dv
3 10
4 6
v dv
3 9
6 7
input graph
v dv
3 9
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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)

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• 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 v?V do
• dv ? ? pv ? NIL
• ds ? 0
• for i ? 1 to n-1 do
• for each (u,v)?E do
• if (dv ? du w(u,v)) then
• dv ? du w(u,v) pv ? u
• for each (u,v)?E do
• if (dv ? du w(u,v)) then
• Output A negative cycle exists Exit
• T ? 0
• for v?(V-s) do
• T ? T ? (pu,u)

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3. Algorithms Bellman-Ford algorithm

• Steps in a nutshell
• (T ? 0 dv ? ? ds ? 0)
• For each edge (u,v) re-evaluate shortest path
  estimate dv and update it and pv if (du w(u,v)
  ? dv)
• Repeat step 1 (n-1) times
• The union of edges (pv,v) gives shortest path
  tree T

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3. Algorithms Bellman-Ford algorithm
Lets see an example ...
SPT rooted_at_vertex 2
E(1,2),(2,1),(1,3),(3,1),(2,4),(4,3)
v 1 2 3 4
dv ? 0 ? ?
pv NIL NIL NIL NIL
v 1 2 3 4
dv 2 0 ? 3
pv 2 NIL NIL 2
v 1 2 3 4
dv 2 0 12 3
pv 2 NIL 1 2
v 1 2 3 4
dv 2 0 6 3
pv 2 NIL 4 2
input graph
Re-evaluation for each (u,v)?E do if (du
w(u,v) ? dv) then dv ? du w(u,v)
SPT_rooted_at_2
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3. Algorithms Add algorithm

• Algorithm for finding approximate MRCT for a
  graph G
• Proposed by Vic Grout in 2005
• Vertex oriented version of Prims 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

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• 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 - du)? max(dj - dj)
  where sj 1

Legend sv1 ? v?T wuv weight of edge (u,v) du
degree of vertex u du degree of vertex u in T

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3. Algorithms Add algorithm

• Steps in a nutshell
• Pick up vertex v adjacent to maximum number of
  unspanned vertices
• Add to T all edges adjacent to v that do not form
  a cycle
• Repeat steps 1 and 2 until all vertices are added
  to T

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3. Algorithms Add algorithm
Lets see an example ...
approximate MRCT
input graph
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3. Algorithms Wongs algorithm

• Proposed by Richard Wong in 1980
• 2-approximate MRCT algorithm
• Based on Dijkstras algorithm
• Time complexity
• O(n2.log(n)m.n)

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3. Algorithms Wongs algorithm

• Steps
• Compute n SPTs using Dijkstras algorithm
• Compute routing cost for each SPT
• Select SPT with lowest routing cost

2-approximate MRCT
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3. Algorithms Wongs algorithm
Lets see an example ...
SPT_rooted_at_3
SPT_rooted_at_1
SPT_rooted_at_2
SPT_rooted_at_8
SPT_rooted_at_6
SPT_rooted_at_4
SPT_rooted_at_5
SPT_rooted_at_7
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3. Algorithms Wongs algorithm
Lets see an example ...
SPT root Routing Cost
1 124
2 136
3 118
4 122
5 144
6 126
7 134
8 142
approximate MRCT
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4. Spanning Trees Applications

• 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

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4. Spanning Trees Applications

• Shortest 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

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4. Spanning Trees Applications

• 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

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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
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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
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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.

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Contact

• Rui Campos
• (351) 222 094 268
• rcampos_at_inescporto.pt
• http//telecom.inescporto.pt/rcampos