!"
#$
%$ # &'(&) *''*
Energy, Congestion and Dilation in Radio Networks
[Extended Abstract]
Friedhelm Meyer auf der Heide
Christian Schindelhauer
Matthias Grünewald
Klaus Volbert
Heinz Nixdorf Institute, Paderborn University
ABSTRACT
% + $, $ (
" + (
, $ , + $(($ ,, ,
- , ,$ (
" ,, "
% $, $ (
" ,$
$, , . (
+ / 0 1 $$
, (
$$ 2/ 0 # , $$ " ($ ,(
, " $$3, (
$, ,, " / / 0 0
, +3 $, $, $,4
, % (5
+ 1 $ -
$$3, ,,4 $,(
$, $,
"
Categories and Subject Descriptors
*& 6
78 !"
# 9 *) 6 $,
, ,$ $$
(
);< , . $,, = , 1.(&>>>(&?&@< /#AB(.0
$, 1(
, . , C .
: $$
(
);<
:
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, to
republish, to post on servers or to redistribute to lists, requires prior specific
permission and/or a fee.
SPAA’02, August 10-13, 2002, Winnipeg, Manitoba, Canada.
Copyright 2002 ACM 1-58113-529-7/02/0008 ...D5.00.
78 .
5 ,$3(
9 ** 6
78 $ .(
General Terms
#, .
Keywords
" ,,
1. INTRODUCTION
1 $$ ,
"
,
,$ " $ ,,4 , B ,E (5 +
8 ,,4 , $ % " ,(
, + ( $" 3$ F
$ ,
" . $$ " $ , +, , G " (, $(
+
1 ($$
+,
1 " (3$+ (
+
+ (
,, /$+ $ 0
,,4
+ . + " $ Æ + +
Æ + . F, 3$
,(
F8 $ .
(
$ #B 6&>7
$ $ .
3 $( $ 5
, / ,
$ $" + $ +0 3
, +
" 6** *) )7
. , $ + , F"(
$ / # 6H70 + "
$ " 3(
$
F
1 " ,, $ " B , $$ , $, " + + / , +3 + +
6&
&'70 , , ,$(
(" , $, " % $ ,
+ $(
, " F / $"
" ,
, " ,
(
6& *? &' * &? &&70 G
, 1 * $
$" + $, "
. $ $ , + (
- ,, " . $
,, " , + + ,
,, " B - + ,(
6&7 5 $$
A" 6&7 " $(
, $, . - , ,$(
,$ , + I I
. , $ "8 + F -
$ J (
,$ ,,4
% +
+3 +
1 + ,
$ $ 5 $,
,, "
"$ , . " +
$$
1 ) $ $ $+ $,4 ,$ + / / 0 0(
$$3, 1 ? ,
,
,,4
, .
(5 , +
H "
2.
MODELING RADIO NETWORKS
% (
, + F + *(
, $ A
K ,3 , , # ,
6&@7 E , , ' $" . ,, (
" K / 0 K . "
$" + E , , . ,(
, $ +
(
"
$" K / 0 /0 K /0 / 0 /0 " . / 0
/$ &0 E , $ $" ,
,,
radio station
edge
interference
!" 1 6>7 3
- ,
,, . ,,(
,
1 $" ,
"
,
+ ,$ % , + 3
+
, . " $ (5 ,$
$$ ,
(
% - 1/0 8K / 0 ! $"
, 1/0 (
. " / 0 1 +3 / 0 /ÁÒØ/ 00 5 1/0 . (
$ ,, " . ,3,, ,(
,3,, , + . , " ,3,, , ! $, 8 / 0 $" + , % +
$8
8 , $ ,
$" $ . $ , +
" ,, " K / 0
8 # + $"
1/0 , ,
$
$ , A L/0 & , $ L/0 E
/0 , $" $
ÁÒØ / 0 % , F /0M ÁÒØ / 0 / F -
$
, 0 /0 . -
/ 0 8K ,3 /0 ¼
¼
È
% / 0 $ " $(
G -(
$, , - / 0
+ , 8
& !
" # " !
$ % " $ &
$
$ ,3&* K N/ M 0
$ ' # ( / M 0" $
)$ &
/ M 0" $
$$
& A K / 0 ,3,, (
% $ $ < 3 ( ( ) , $ $ < (
-
8K / 0 1/0 ! (
, , 1/ 0 1/ 0
. , + + F A 8K /0 M / 0 .
¼
,3
&*& &*
. $$ * ) $+
, , ., *&* .,
*&) 6&7 ! 6&7 5 ,
$$
. + , $ (
,$ + $ (
. + + /
0 1 $
,
/
0 + /
0 .(
$$ ,
/
0 + , ) $$ (
3$
% ,
1 - ,
, , ,, " $$ / 0 . *!+ " + =( / 0 8K
3. MINIMIZING ENERGY AND CONGESTION
3.1 Energy
. $ , " -
,$ + $" ,, " 1 ,, $ $,4 ! 6&? ?7 $$ ,
, E
$"
* &
!
$
$
-
. ,,, (
" :,P "P ,
,,, $ ! , ,$ ,
. ,, "
,,, $ O ,
,, " (
+ ,$ ,, O
" $, , 1 O $ 6<7 / 0 $ " ,(
, . 8
) , # !
- -
. $
È 1 -
,(
/ 0 3 +3 / 0 / 0
. ., . . ,, " 5 * . , -
$ + ($ $+
(
, , + $ $ -
. O $,
" , (
($ $ = ($(($
, + $, "
! (
$ $
(, $ +
" . $ $ $,
. # O (
,
, $" $ (
,$ ,, " /0 8
( / 0 8K
/0 È 1 $$ F8 + , $ $ , (
,$ , $, " $(
,$ $
+ 3.2 Congestion
3.2.1 Diversity of a Vertex Set
,, (
(
+3 K , 9 $
4 , b
a
3.2.2 Approximating Congestion
c
$ % & M '
K * . / 0 / 0 ) . , " & $$ ,, $ 4 / 0 B $ ,,
( ) % % " 1 , $, 4(
$, ,, " "
, ,
+
5 ,
5 , *
& &
/ 0 + / / 0 8K
! 8 K ! ! + + , ,3(
,, + /
,0 $ , + , , 1 , +3 , / / 00
& &
N/ 0 /0$ , !
0 / 0 1$$" & /# " '2$ ,!
"
/ 0
& M ,,3
. F , - 1
,$ , + , $ ½
$ +3 ¾ (
+
$ / 0 / 0 , . $ $$ 6*'7
. , +
,
/ 0 . $$3, ($, ,, (
" " 3
%
-
6>7 #
$
, 6; @7 4 $$ 6&7 $ $ # # # . $ + A
$ . # + .
+3 +3 +3
K /# 0 /# 0 /# 0 K . $$ #
+ ,,, 8 /# 0 8 , 3( , +
8 /# 0 /# 0 8
B $, $ % " &
, , 8K % - # /# 0 /# 0 8K
/ 0 /# 0 $ +3 + / 0 +
,3,, , K // 00 % ! $$ ,$ ,(
,4 ! - ($ K / 0 " ($
* K / 0 !" # 12 $ !" # $
?
3
% -
456 ' $ " * ! 1 !2 K
,3 % $
* , # / 0 !
3
% -
/ / 00$
. , A
$ , G -
+ , , K
/ / 00
# $ ,, (
, , " ,
% ,4 + - $ , $$ $
6&'7 A
&/0 , ) &
'/0 / 7 ' '/0 8K /0 &/0 &/0 $ $ 8
/ '/0 8K '/ 0 . - ,$ $ $ p
A,, ? + $ /0* G A,, ) +(
$ , '/0 K * / M&0 .
+ , /0 ¼ ¼ /0 ?/) M &0 G - , / 0 (,$ ¼ . ,$ ,
/0 ¼ / 08
u
e
v
/0 / 0?/) M &0 ¼
K / 0
8K ?/) M &0 * ) & ' + , " / 0
) % $ &
" '/0 &/0 $
+ /0 ,$ $ +
("
? /0 /0 " '$
. $ , - H
% !
# $ &
" ! /0 /0 / 0 $
+ $ $ , $
+ K / 0 3 " (
$ $ , ,$ " /0 / 0 / M 0
$ , , + ,$ (
, ,,
" /0 / M 0 / ?0
% ,$ / 0 $- / 0 ($ 1 K / 0 / 0 , Æ K / 0 / 0 $ ( 8K / M 0 ( 8K
/ M 0 ( ( / 0
! ¼ / 0 6* * 7 Æ . $ " / 0* * ( . + /0 1 $ * ,$ " * / M &0 ( . * / M &0 H % !
# $ &
3
% -
/ / 0 £ / 00$
, ., ? (
A $ " ($ . A,, H 3 /0 / 0 / 0
A,, * , (
" / / 00 ,$ / 0 K
/ / 0 / 00
£
£
$ + ,
, A $ $+
// 0 0(
$$3, 4. TRADE-OFFS
% + Æ $ $(
,4 $$3, B , (
, ,$ $ , " $,4 , , 1 4.1 Congestion versus Dilation
+3 $
(
,,4 $ + ,
/ 0 K
* + / H0 . /* 0 + / 0 1 6&'7 $ " # 4 + $(
/ 0 /* 0 / ( (
$, $
6*&70 1 + / 0 / 0 N/* 0
. $ 8
< - # Æ *
!9 / 0 / 0 #7
/ 0 / 0
N/* 0 K > $ )
$
+3 + + + ,
% ,
+ A ) " , +3 +3
=( 1$ . (
2 #7
/# 0 *!+ /# 0
/# 0 ,!+ /# 0
A / 0 K / 0 , O $ + . $
, / 0 . $ '/ 0
$ '/ 0 / 0 . , ,,4
$ ,3,, $(
, / 0 F ½ ¾ . + '/ 0 ¾ . , $ + $ 3
$ /)0 /)0(F $
. +8
È
'/ 0 >
½ ¿
, F , / 0 / 0
A ! , $
" / , (4 , O /0 " '0 ,
, $
, / , (
0 % +
,,4 =( /0 8K / 0 -
K , ,, K K K K +
=( /0 /0 . O - ) È
N/ 0 N/ * 0 4.3 The Incompatibility of Congestion and Energy
% , +3 ,$ . (5( $ ,$,
. +3 + $ 6' 7 (
4 $ $ # !
/
$$0
+ + . ,
*
+ $ $$ + $ . + F
$
+ $ (
, + $ + / ;0
* @&
4.2 Dilation versus Energy
. ,$ +3 #
+
6&?7 < + # K
$
F K B , , 3(
, / 0 K * / 0 K ' $ / 0 . ($, " O $ / 0 + ¾
=( /# 0 K O ( /# 0 K
¾ . ) #
. " 4 / 0 $(
& /
O * 0
. $ , + ,$, 3, + $
8
; - # # !9 / 0 +
. ,,, $ + + 2/ 0 . =(ÅËÌ /+ 0 K / 0
ÅËÌ /+ 0 K /* 0 . O /,0 ,,, " + (ÅËÌ /+ 0 K /* 0 . $, $ , +(
(4 ,
1 /+ 0 K /* 0
=( /+ 0 K / 0 . O + ( /+ 0 K /* 0 ¼
¼
¼
< , $ 6' 0 # + , ' - 6' * 7 . $ &
" " . 7
*!+ /+ 0
/+ 0
,!+ /+ 0
/+ 0
M , *
,M&
- M /* - 0
/&0
/*0
- M /* - 0 /)0
/?0
. ,,, " + . =($
$ ! 4 $ . , /,0 4
+ /
$ +
0 #
+ ,$ ¾
,$ ¾ , M & 4 /
0 Æ
O , @ &
# # !
: " : *!+ . !
,!+ # *!+ / 0 / 0 ,!+ / 0 / 0
N/ 0 N/ *!+ 0 N/ 0 N/ ,!+ 0 . , A,, < $ K + $ "
$,4 , $, , . (
" + , (
,$
5.
CONCLUSIONS AND FURTHER WORK
. , 5 " " ,, " (
" O F % ,
$ O $ +
+ $ " 1 + " $ , $$ " ,(
$
+ , 5 (
,, " , , , , , $ " $(($ ,,
/ $, "0 " +
/ 6&70 + ,(
$-
,
8 . $(($ ,, -3
Q ,
$ , , ,
/ E 6*70
% + F $, ,, " + $ "
% , F $(
+ $ , $" $ - " , O + " $
"
# $ + $
" $, " 6&7 % , $$
, (
,3,, , % $+ /
$
$" O0 $, $ , ,$
$, , , $+ " (
$ ,, " $ , , $$3, / / 0 0 ,,
/ 0 + +3
% , 4 ,,
+3 $ $$ + / 0 K
2/ 0 K +3 , ,3,, ,,, , $, /. ,,4
. &0
+ $ $(
,4 , , , . *
% $+ (5 + + + $ , $$3, , ./ 0 $, (
5 ,$ N/ 0 + + ,,4 " ,$
"
# $ ,(
$ F / G 6&<7 1 @'*&&
6&*70 # , - F , /
$ +,P F $0 , ,$+ - + F $$ ,$ + (5 ,$
$, G
,
,(
,, + ,
+ / 0
,, $ (
$ ,$3 F " $(
/ 6&H70 . F
$$ ,
6>7
# ,, " ,$,
$$ ,, , (
,, . $$ , -3
, 5 E , $ $
1 3 ,
, ,
$ /6&; &)70 . " $
$, 3$,
#$$3(
= A $ ,$ !"
.
($
/ 0
$,
$,
$,
) %
= N/* 0
/ 0 = / 0 N/ 0
/ 0
/ 0 / 0
/ 0 N/ * 0
/ 0
= / 0
/ 0
/ 0
R
N/ / 00 N/ = / 00
N/ / 00 N/ / 00
) $ )12 ,
, G
,$ , 6*H7
+
,, (
$ " ,
,$
$ $ , $ $ ,, "
! $$ , $ (
$+
,,4 "
6. REFERENCES
6&7 #
Æ ,,
#
( % !" /3
#0 1 ;5< $ *H>S*<@ &>>@
6*7 G # : G" # G", ,$ #
+ , 1 , 1,8;2 $ &H@S&<; *''&
6)7 T A . +
% #
!" 1 '+++
'=,88 $ **S)& *'''
6?7 # , : : + B $ , $, "
+ 0 #
1+;2 /'H?0 *'''
6H7 (Q % .$
(G
#
$+ /#0 #
!" '+++ $ )<S?H &>>;
6<7 " # $$ $ + 1 >
1<2 $ *H>S*;@ &><>
6;7 T A T T A U # U , 1 - $
&@@S&>< *''&
6@7 T A T T A U # U , $ ,
" 1 3
= $ ?HSHH *''&
6>7 . A"+4" Æ % !" .$ /3 #0
$$ $ $
,$ /= B(:# P'*0 *''*
6&'7 : $ : , . $ %
!" 1 '+++ & '
&
" ? @12 $ )@@S?'? *'''
6&&7 14" !" :$" #
Æ ,, #
( !" 1 $ A
B 1B';2 $ *@HS*>>
*''&
6&*7 1 1 @'*&& ( % A#! ,
# /#0 : A /:Q0
$4-" 1 '+++ &>>;
6&)7 (., # C
*'''
$8JJ"(,,JJ"$
6&?7 A " 4 # :
: ,$ $" "
&
1)7!2 $
*@>S)'H *'''
6&H7 Q(G "", ! , # : = # #
!" 1 '+++ '=,88
$ &)S*& *'''
6&<7 G G"
1
. 1 '+'+ &$
," $ +<)!" $ *'''
6&;7 4 # .
+$, "$ 1 ' C
D $ ;S&)
:
, , &'(&& &>>>
6&@7 T " G+ %( # :
$ # : %
:" !" 1 '+++ ' " "
$ &S&& *''&
6&>7 :" 3 = #
(% *''&
6*'7 ,, !" :,
,
=+ :
*''*
6*&7 G % .(#$$3,
%
( :, /3
#0 1 '$ 1';2 *''&
6**7 +
:## ( $ ,( $ " E
1A2 $ HS*< &>>@
6*)7 % :(# #
!" 1 $ F'+++
= $ &@&S&>' &>>@
6*?7 =" Q $ , 1 '+++ &$ $"
? @1@2 $ ;@?S;>@ &>>@
6*H7 # , +, #
!" = + 1 +
D " B = !
1B;2 $ ?&>S?*< *''*
© Copyright 2026 Paperzz