Topology Control Algorithms
Davide Bilò
e-mail: [email protected]
What is Topology Control?

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Mechanisms and algorithms to conserve energy in adhoc radio and sensor networks
Primary targets of a topology control algorithm
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abandon long-distance communication links
prevent the network from being partitioned
Secondary targets
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each node has “few” neighbors
routing path does not have to become non-competitively long
topology control algorithms should find a good tradeoff between
connectivity and sparsness
What does a TC Algorihm do?
Let G=(V,E) be an (undirected*) communication graph where

V is the set of devices with |V|=n

E contains edge (u,v) iff u and v can communicate directly

c(u,v) minimum transmission power at which u has to transmit if it wants
to send a msg to v directly (we assume c(u,v)=c(v,u) and c(u,v)pmax)
*
As devices are homogeneous, i.e., they have the same characteristics, we can
assume that if u can communicate with v directly if u transmits at power p(u), then
also v can communicate with u directly if v transmits at power p(v)p(u). This implies
that the devices all have the same maximum transmission power pmax.
Running the TC algorithm A on all the nodes yields a graph
GA=(V,EA) which is a subgraph of G
What is
the Directed Communication Graph G
in the Euclidean Model?
d (u, v)
c(u, v)  
 
if d (u, v)  pmax ;
otherwise.
Observe: for every 1, d(u,v)d(u,v) iff d(u,v)d(u,v ).
For simplicity, we will assume that =1 even though everything we will
see can be generalized to every 1
The (Directed) Communication Graph G
in the Euclidean Model
is a Unit Disk Graph
G has bidirectional symmetric link
The transmission range of any node v is the disk centered at v with radius
W.l.o.g., we assume that 
pmax

1

pmax

Formal Definition of Unit Disk Graph
(UDG)
Given a set V of points in the Euclidean plane,
the Unit Disk Graph induced by V is the (undirected) graph G=(V,E)
where E contains edge (u,v) iff d(u,v)1
What does a TC Algorihm do?
Let G=(V,E) be an (undirected*) communication graph where

V is the set of devices with |V|=n

E contains edge (u,v) iff u and v can communicate directly

c(u,v) minimum transmission power at which u has to transmit if it wants
to send a msg to v directly (we assume c(u,v)=c(v,u) and c(u,v)pmax)
*
As devices are homogeneous, i.e., they have the same characteristics, we can
assume that if u can communicate with v directly if u transmits at power p(u), then
also v can communicate with u directly if v transmits at power p(v)p(u). This implies
that the devices all have the same maximum transmission power pmax.
Running the TC algorithm A on all the nodes yields a graph
GA=(V,EA) which is a subgraph of G
Properties GA should have

Symmetry: GA is symmetric, i.e., u is a neighbor of v in GA
iff v is a neighbor of u in GA
Reason: Asymmetric communications are unpractical
Properties GA should have


Symmetry: GA is symmetric, i.e., u is a neighbor of v in GA
iff v is a neighbor of u in GA
Connectivity: there is a (direct) path from node u to node v
in GA iff there is a (direct) path from u to v in G
Connectivity is not enough
A minimum spanning tree algorithm
yields a connected subgraph GMST
Not a good topology because close-by nodes in G might end too far in GMST
Properties GA should have


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Symmetry: GA is symmetric, i.e., u is a neighbor of v in GA
iff v is a neighbor of u in GA
Connectivity: there is a (direct) path from node u to node v
in GA iff there is a (direct) path from u to v in G
Spanner: if the shortest path from u to v in G w.r.t. some
criteria has cost , then the shortest path from u to v in GA
w.r.t. the same criteria has cost f(). If f() is bounded from
above by a linear function of , then GA is called a spanner
Spanner  Connectivity
Properties GA should have



Symmetry: GA is symmetric, i.e., u is a neighbor of v in GA iff v is a
neighbor of u in GA
Connectivity: there is a (direct) path from node u to node v in GA iff
there is a (direct) path from u to v in G
Spanner: if the shortest path from u to v in G w.r.t. some criteria has
cost , then the shortest path from u to v in GA w.r.t. the same criteria
has cost f(). If f() is bounded from above by a linear function of ,
then GA is called a spanner
Spanner  Connectivity

Sparsness: GA is sparse, i.e., |EA|=O(n)
Reason: Primary target of a topology control algorithm is to abandon
long-distance neighbors
Sparsness is not enough as sparse graphs may have high degree
Properties GA should have



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Symmetry: GA is symmetric, i.e., u is a neighbor of v in GA iff v is a
neighbor of u in GA
Connectivity: there is a (direct) path from node u to node v in GA iff
there is a (direct) path from u to v in G
Spanner: if the shortest path from u to v in G w.r.t. some criteria has
cost , then the shortest path from u to v in GA w.r.t. the same criteria
has cost f(). If f() is bounded from above by a linear function of ,
then GA is called a spanner
Sparsness: GA is sparse, i.e., |EA|=O(n)
Low Degree: Each node in GA has a constant number of neighbors
Spanner  Connectivity
Low Degree  Sparsness
Properties GA should have




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Symmetry: GA is symmetric, i.e., u is a neighbor of v in GA iff v is a
neighbor of u in GA
Connectivity: there is a (direct) path from node u to node v in GA iff
there is a (direct) path from u to v in G
Spanner: if the shortest path from u to v in G w.r.t. some criteria has
cost , then the shortest path from u to v in GA w.r.t. the same criteria
has cost f(). If f() is bounded from above by a linear function of ,
then GA is called a spanner
Sparsness: GA is sparse, i.e., |EA|=O(n)
Low Degree: Each node in GA has a constant number of neighbors
Planarity: GA is planar, i.e., it does not have intersecting edges
Reason: we can use geometric routing algorithms on planar graphs
Spanner  Connectivity
Low Degree  Sparsness
Planar Graphs
A (geometric) graph is planar if it has no intersecting edges
(geometric graphs we consider are graphs whose set of vertices are points on
the Euclidean plane, and edges are straight line segments)
Intersection point
Example of planar graph
Example of non planar graph
red edges intersect
Properties GA should have
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Symmetry: GA is symmetric, i.e., u is a neighbor of v in GA iff v is a
neighbor of u in GA
Connectivity: there is a (direct) path from node u to node v in GA iff
there is a (direct) path from u to v in G
Spanner: if the shortest path from u to v in G w.r.t. some criteria has
cost , then the shortest path from u to v in GA w.r.t. the same criteria
has cost f(). If f() is bounded from above by a linear function of ,
then GA is called a spanner
Sparsness: GA is sparse, i.e., |EA|=O(n)
Low Degree: Each node in GA has a constant number of neighbors
Planarity: GA is planar, i.e., it does not have intersecting edges
Spanner  Connectivity
Low Degree  Sparsness
Which TC Algorihm do we need?

We do not need a global centralized algorithm for sure
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What about a distributed algorithm?
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no central authority in ad-hoc radio and sensor networks
better than the centralized one
not of practical use in case of mobile devices
We do need a local algorithm
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each node is allowed to exchange msg’s with its neighbors a few
times and then must decide which links it wants to keep
Topology Control Algorithms for UDG
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nodes know their coordinates (for instance, nodes use GPS)
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Minimum Spanning Tree
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Delaunay Triangulation
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distributed but not local
symmetry, energy-spanner, low degree, and planarity
Gabriel Graph
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distributed but not local
symmetry, connectivity, low degree, and planarity
local
symmetry, energy-spanner, sparsness, and planarity
nodes can sense signal strength and can perceive from which
direction a signal arrives
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Cone-based
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local
symmetry, energy-spanner, sparsness, and planarity
(an optional distributed (but not local) second phase) satisfies low degree.
Limitations of the Euclidean Model

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signal attenuation is uniform, that is, the Euclidean plane
is flat and free of blocking objects
Radio propagation is as in vacuum
transmission range of v
u
d(v,u)=d(v,u )
u
v
If we add obstacles…
Euclidean Model does not work in realistic environments
transmission range of v
u
d(v,u)=d(v,u )
u
obstacle
v
Algorithm XTC
R. Wattenhofer and A. Zollinger, XTC: A Practical Topology Control
Algorithm for Ad-Hoc Networks, 4th International Workshop on
Algorithms for Wireless, Mobile, Ad Hoc and Sensor Networks, 2004
download link: http://www.dcg.ethz.ch/members/roger.html
Algorithm XTC
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works in every environment (i.e., every undirected graph G)
nodes do not need to know their coordinates
nodes do not need to perceive which direction a signal comes from
it is local
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the system can be
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and fast (every node communicates with its neighborhood twice)
asynchronous
uniform
non-anonymous
satisfies
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symmetry
connectivity
low degree
in UDG’s
planarity
energy-spanner (in random UDG’s)
correctness
efficiency
Algorithm XTC
Three main steps:
1. neighbor ordering
2. neighbor order exchange
3. edge selection (Nu neighborhood of u)
Algorithm XTC

Algorithm XTC (description for node u)
1.
(neighbor ordering)

establish total order <u over u’s neighbors in G
v<uw means that u prefers link (u,v) more than link (u,w),
i.e., link (u,v) is of higher quality than link (u,w)
(for instance, v<uw  c(u,v)c(u,w))
Algorithm XTC

1.
2.
Algorithm XTC (description for node u)
(neighbor ordering)

establish total order <u over u’s neighbors in G
(neighbor order exchange)

3.
broadcast <u to each neighbor in G and
receive orders <v from all neighbors v’s
(edge selection (Nu neighborhood of u))

Nu,Ñu:=

while (<u contains unprocessed neighbors)

v:= least unprocessed neighbor in <u
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if (wNuÑu s.t. w<vu) then

Ñu:=Ñu{v}
i.e., w<uv

else

Nu:=Nu{v}
Graph Yielded After
Execution of Algorithm XTC
Nu is the set of neighbors of u computed by algorithm XTC
GXTC=(V,EXTC)
where
EXTC={(u,v)|u:vNu}
GXTC is Symmetric

Theorem (Symmetry): GXTC is symmetric, i.e., a node u
includes v in Nu iff v includes u in Nv.
Proof: Assume u includes v in Ñu.
We show that v includes u in Ñv.
u includes v in Ñu because wNu Ñu with w <uv and w <vu.
When v processes u, wNv Ñv.
Thus, v includes u in Ñv.
From now on, we will tacitly assume that GXTC is symmetric
Some Assumptions

Weak Assumption (WA): Neighbor orders are based on
function c, i.e.,
u, w<uv  c(u,w)  c(u,v)

Strong Assumption (SA): Every edge (u,v) has a weight
l(u,v)=(c(u,v),min{id(u)*,id(v)},max{id(u),id(v)}).
Neighbor orders are based on the lexicographic order** of
edge weights, i.e.,
u, w<uv  l(u,w) < l(u,v)
*
id(w) is the identifier of node w. Nodes have distinct identifiers.
**(,,)<(,,)  (<) or ((=) and (<)) or ((=) and (=) and (< ))
GXTC Satisfies Connectivity

Theorem (Connectivity): Under SA, two nodes u and v are
connected in GXTC iff they are connected in G.

Corollary: Under SA, GXTC is connected iff G is connected.
GXTC Satisfies Connectivity

Theorem (Connectivity): Under SA, two nodes u and v are connected
in GXTC iff they are connected in G.
Proof:
If u and v are connected in GXTC, then they are connected in G.
(because GXTC is a subgraph of G)
So we have to prove that
Claim: if u and v are connected in G, then they are connected in GXTC.
We prove Claim by contradiction, i.e., we assume that
there exist u and v which are connected in G but not in GXTC.
We use the following scheme:
1.
2.
we choose the “right” u and v
we show that u and v are connected in GXTC
GXTC Satisfies Connectivity
How to choose the “right” u and v
Theorem (Connectivity): Under SA, two nodes u and v are connected
in GXTC iff they are connected in G.
Proof: …
Let Z be the set of all the pair of nodes u and v which are not
connected in GXTC but they are connected in G via a direct edge.

Is Z? YES
w and t are connected in G but not in GXTC
v
Vw: set of nodes connected to w in GXTC
u
Vt: set of nodes connected to t in GXTC
t
w
Vw
G
Vt
VwVt=
GXTC Satisfies Connectivity
How to choose the “right” u and v

Theorem (Connectivity): Under SA, two nodes u and v are connected
in GXTC iff they are connected in G.
Proof: …
Let Z be the set of all the pair of nodes u and v which are not
connected in GXTC but they are connected in G via a direct edge.
Is Z? YES
u and v is the pair of nodes in Z of minimum value l(u,v)
GXTC Satisfies Connectivity
How to prove that u and v are connected in GXTC

Theorem (Connectivity): Under SA, two nodes u and v are connected
in GXTC iff they are connected in G.
Proof: …
What we have shown so far: u and v is the pair of nodes of minimum
value l(u,v) among those pair of nodes which are not connected in GXTC
but connected in G via a direct edge.
u includes v in Ñu because wNu Ñu with w<uv, i.e.,
w<uv
AND
w<vu
l(u,w) < l(u,v)
AND
l(v,w) < l(u,v)
u and w are connected in GXTC
AND
v and w are connected in GXTC
u and v are connected in GXTC
GXTC on UDG’s
(remember that (u,v) is in G iff c(u,v)=d(u,v)1)
GXTC on UDG’s has Low Degree

Theorem (Low Degree): Under WA, if G is a UDG, then
GXTC has degree at most 6.
Proof: Let uV s.t. (u,v),(u,w)EXTC, i.e., d(u,v),d(u,w)1.
We prove that the angle /3 by contradiction.
So, assume for contradiction that </3.
W.l.o.g., assume v<uw, i.e., d(u,v)d(u,w).
Claim: If d(u,v)d(u,w) and </3, then d(v,w)<d(u,w).
 d(v,w)1  (v,w) is in G.
Moreover, v<wu.
Thus, u includes w in Ñu.
By Theorem (Symmetry) (u,w)EXTC.
contradicts (u,w)EXTC
w

u
v
Proof of Claim:
If d(u,v)d(u,w) and </3, then d(v,w)<d(u,w)
(use sin2 +cos2 =1)
d(v,w)2=A2+B2
=d(u,v)2-2d(u,v)d(u,w)cos  +d(u,w)2
(1-2cos )d(u,v)2+d(u,w)2
<d(u,w)2 ([0,/3), cos >0.5)
w
A=|d(u,v)-d(u,w)cos |
B=d(u,w)sin 
B

u
A
v
GXTC on UDG’s is Planar

Theorem (Planarity): Under WA, if G is UDG, then GXTC is planar.
Proof: Let u,v,w,t be any 4-tuple of distinct nodes forming a quadrangle
Q as in figure s.t. d(u,w),d(v,t)1.
The only two intersecting edges of Q may be (u,w) and (v,t).
We prove that (u,w)EXTC or (v,t)EXTC.
(This is almost enough as almost every pair of intersecting edges defines a
quadrangle)
As the sum of the interior 4 angles of Q is 2, one of them is /2.
W.l.o.g., assume /2.
 d(u,v),d(w,v)<d(u,w).
As d(u,w)1, then (u,v),(w,v) are in G.
Moreover, v<uw and v<wu.
When u considers w, vNuÑu.
As v<wu, then u includes w in Ñu.
u
By Theorem (Symmetry) (u,w)EXTC.
t
Q
w

v
GXTC on UDG’s is Planar

Theorem (Planarity): Under WA, if G is UDG, then GXTC is
planar.
Proof: … to complete the proof, we should consider the
case of three aligned points as in figure.
u
v
Exercise: Show that (u,w)EXTC.
w
Experimental Results
Stretch factor of GXTC w.r.t.
energy metric (solid line).
Mean values are plotted in black,
maximum values in gray.
GXTC is an energy-spanner in random UDG’s
Experimental Results
Node degree of GXTC (solid line).
Node degree of G (dotted line).
Mean values are plotted in black,
maximum values in gray.
GXTC has very low degree in random UDG’s
… but we already knew it!!!
A comparison with the Gabriel Graph