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USC Computer Science Technical Reports, no. 652 (1997)
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USC Computer Science Technical Reports, no. 652 (1997)
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Content
Finding Lo wDiameter Lo w EdgeCost Net w orks
Katia Obraczk a
Univ ersit y of Southern California
Information Science Institute
Admiralt yW a y
Marina del Rey CA USA
k atiaisiedu
P eter Danzig
Univ ersit y of Southern California
Computer Science Departmen t
Los Angeles CA danziguscedu
Abstract
This pap er describ es a sim ulated annealing algorithm
to compute k connected graphs that minim i ze a linear
com bination of graph edgecost and diameter Repli
cas of In ternet information services can use graphs with
these prop erties to propagate up dates among themselv es
W e rep ort the algorithms p erformance on and connected no de graphs where edgecost and diame
ter corresp ond to ph ysical distance in a plane F or these
graphs the algorithm nds solutions whose edgecost
and diameter are and lo w er than our graph
construction algorithm that returns feasible but unop
timized solutions When optimizing edgecost only the
resulting graphs sho w and reductions edgecost
and diameter resp ectiv ely In tro duction
This pap er describ es a sim ulated annealing algorithm to
construct lo w diameter lo w edgecost and connected
graphs W e need graphs with these prop erties to prop
agate up dates b et w een replicas of In ternet information
systems F or main taining replicas across the In ter
net w e use connectivit y of or for resiliency lo w
edgecost so the ph ysical net w ork is used ecien tlyand
lo w diameter to limit transien t replica inconsistency Since an y algorithm that minim izes edge cost or di
ameter in or greater connected net w orks is NPcomplete
w e emplo y sim ulated annealing to searc h for ap
pro ximate solutions The algorithm presen ted here si
m ultaneously reduces b oth edgecost and diameter b y
and resp ectiv elyo v er unoptimized feasible so
lutions With resp ect to graphs constructed to minim ize
edgecost alone our algorithm nds solutions that sho w
and reduction in edgecost and diameter resp ec
tiv ely Statemen t of the Problem
Belo w w e formally dene our top ology computation
problem starting with sev eral denitions
k Connected A graph G is said to b e k connected
if no remo v al of an y k v ertices together with
all their inciden t edges disconnects G k Connected Regular Graph A graph G is
said to b e a k connected regular graph if all its
v ertices are exactly k connected
Diameter The diameter of a graph G is dened
as the maxim um shortest path b et w een anyt w o
of Gs v ertices
Degree The degree of a v ertex v in G is the
n um b er of edges of G inciden t with v W e representthe net w ork top ology b y graph G V E
where the set of v ertices V represen ts net w ork no des
and the set of edges E represen t links b et w een no des
In our sim ulations w eev aluate complete graphs where
the edge costs corresp ond to ph ysical distance in the
plane Our In ternet replication system measures and
estimates the endtoend comm unicatio n costs b et w een
no des whic h dep ends on net w ork state ph ysical top ol
ogy and pro cessing load at the no des
The top ology construction problem can b e stated as
follo ws Giv en the graph G V E a cost matrix C E
and an in teger k construct a subgraph G
V E
with
the follo wing prop erties
G
is k connected
G has minim um diameter
G has minim um total edgecost
In the next section w e review related literature and
algorithms for solving similar problems
Related W ork
Plesnik pro v es that an y algorithm that generates a
minim um spanning subgraph of Gsa y G
V E
b y se
lecting E
as subset of E with a giv en budget constrain t
and minim um diameter is NPcomplete
Johnson pro v es that constructing a subgraph whic h
connects all v ertices and minim izes the shortest path
cost b et w een all v ertex pairs sub ject to a budget con
strain t on the sum of its edgecosts is also NPcomplete
Sc h umac her pro vides an algorithm for generat
ing top ologies whichha v e minim um n um b er of edges
are k connected and ha veminim um diameter Ho w
ev er his metho d assumes that all the edges ha v e equal
w eigh ts Wecannot mak e that assumption since our
problem is to build p eertop eer top ologies o v er real net
w orks
Steiglitz et al prop oses a heuristic solution to a
problem similar to ours Their problem consists of nd
ing an undirected graph with the follo wing prop erties
F easibility The redundancy b et w een anyt w o
no des i and j is at least R
ij
Optimalit y No net w ork whic h satises the rst
prop ertyhas lo w er total edgecost
W e cannot completely map our problem on to the
Steiglitz problem While w e can express our connectiv
it y requiremen t in terms of his redundancy matrix R
ij
his optimalit y constrain t cannot express our minim al
diameter requiremen t This pap er extends the Steiglitz
algorithm to searchfor lo w diameter and lo w edgecost
feasible graphs
Steiglitzs algorithm has t w o parts the starting and
the optimizing routines The starting routine generates
a random feasible solution The optimizing routine iter
ativ ely applies heuristics to generate lo w er cost top olo
gies It uses a lo cal transformation called xchange whic h randomly selects four no des connected pairwise
and sw aps the edges connecting them see Figure It
then records the lo w est cost feasible top ology generated
b y these lo cal transformations
The algorithm emplo ys hill clim bing to generate lo
cally optimal solutions from v arious starting congura
tions After determining a set of feasible solutions it
returns the one with lo w est cost
In Rose uses sim ulated annealing to nd net
w ork top ologies with small mean distances b et w een no des
sub ject to a budget constrain t He sho ws that anneal
ing help to equalize the initially unev en distribution of
mean distances His algorithm do es not guaran tee or
connectivit y Roses lo cal transformation algorithm
randomly c ho oses a no de deletes one of its links and
adds a new one to a dieren t no de
Outline
This pap er extends Steiglitzs and Roses algorithms to
searc h for lo w diameter and lo w edgecost top ologies
This pap er is organized as follo ws In Sections and
w e presen t our sim ulated annealing algorithm and
ev aluate its p erformance for v arious graphs and connec
tivities Section presen ts t w o faster but less optimal
heuristics to nd lo w cost feasible solutions
Our T op ology Computation Al
gorithm
Lik e Steiglitzs and Roses our algorithm generates a
feasible starting top ology and then successiv ely applies
one of four transformations to it according to a sim u
lated annealing sc hedule
Generating a Starting T op ology
Figure summarizes ho ww e generate a random feasible
starting top ology The goal is to generate a relativ ely
lo wcost graph that is lik ely to b e feasible The algo
rithms inner lo op creates a graph where all no des ha v e
degree greater or equal to the connectivit y requiremen t
Of course this do esnt guaran tee that the graph is k connected or ev en connected The outer lo op computes
the random graphs connectivit y and accepts the graph
if it is feasible The algorithm randomizes itself b yre lab eling the no des on ev ery execution In practice the
fact that there are N p ossible starting graphs did not
constitute a limitatio n to the algorithm
Steiglitz pro v es that when all no des ha vecon nectivit y requiremen t k the en tire graph is feasible if
an y k no des meet the connectivityrequiremen t There
fore instead of ha ving to c hec k connectivitybet w een all
N
pairs of no des w e only need to c hec k the connec
tivityof kN pairs of no des where k is t ypically or
Toc hec k an individual no des connectivit yweapply
the F ordF ulk erson maxim um o w algorithm whic h
computes the n um b er of disjoin t paths b et w een a source
and a sink in O NE
Figure summarizes ho ww e
p erform top ology feasibilityc hec ks
feasiblestartin t N vector unsatisfiedconne ctivi ty matrix cost
do
k unsatisfiedconnect ivity of node generate Pi a permutation on N
for all nodes iN labeli Pi
do
A lowest labeled node with the highest unsatisfiedconnect ivity unsatisfiedconn ectiv ityA unsatisfiedconnect ivity A B lowest labeled node with the highest remaining
unsatisfiedconne ctivi ty resolving ties in favor of the
lowestlabeled node with the cheapest cost to A
unsatisfiedconn ectiv ityB unsatisfiedconnect ivity B add edge AB to graph
until all nodes have unsatisfiedconnect ivity while not checkfeasibility graph N k
Figure Generating an initial feasible top ology after Steiglitz
Applying Lo cal T ransformations
Next w e searchfor lo w er cost top ologies b y rep etitiv ely
applying four dieren t lo cal transformations to the graph
Steiglitzs used a single transformation dubb ed xchange that op erates on t w o pairs of connected no des and their
resp ectiv e edges Xc hange deletes the pair of edges b e
t w een the connected no des and adds a pair of edges b e
t w een the no des that w ere not formally connected Fig
ure a illustrates Xc hange Since xchange do es not
alter no de degree w e added three more lo cal transfor
mations that w ould allo w the no de degree to increase so
that diameter decreases W e dubb ed these three trans
formations as split deleteand add Split randomly selects a pair of connected no des
breaks the edge connecting them and connects eachof
them to another no de Delete randomly c ho oses a pair of
connected no des with degree greater than the required
connectivit y and deletes the edge connecting them A dd
selects a pair of no des that are not connected and adds
an edge connecting them Figure illustrates all four
transformations
After eac h transformation the resulting top ology is
c hec k ed for feasibilit y If the feasibilit yc hec k fails the
algorithm rejects the resulting top ology and restores the
previous one A t eac h pass through the cost reduction
algorithm w e indep enden tly select a lo cal transforma
tion to apply W e do not necessarily apply the four
transforms with equiprobabilit y In Section w eap ply some transformations more frequen tly than others
dep ending on whether wew an t to reduce edgecost or
diameter
After eac h lo cal transformation w ec hec k feasibilit y In the case of an xc hange transformation this reduces
to c hec king the connectivit y of the four xc hanged
no des F or a split transformation this reduces to
c hec king the feasibilit y of the pair of no des that had
their connecting edge remo v ed No feasibilit yc heckis
needed after add since it do es not remo vean y edges
while a delete transformation requires the full k no de
connectivit y test that is p erformed on starting congu
rations
Annealing
The annealing sc hedule decides whether to accept a fea
sible top ologies generated after eac h transformation Lo w er
cost top ologies are alw a ys accepted while higher cost
ones are accepted according to the Boltzmann prob
abilit y distribution P E exp EkT where
E is the dierence in cost b et w een the new and old
top ologies Figure summarizes our annealing algo
rithm and Section discusses asp ects of tuning the an
nealing sc hedule
boolean checkfeasibilitym atrix topology int N int connectivity
for all nodes iN or noconnected connectivity
for every other node j if disjointpathsst artin gtop ology i j connectivity
then break
else noconnected noconnected return noconnected connectivity
int disjointpathsG s t initialize residual network R to graph G
while shortestpathnew path s t R extract newpath from R
paths paths returnpaths
Figure Chec king feasibilit y Ob jectiveF unctions
Recall that our goal is to generate top ologies o v er whic h
replicas of an In ternet information service can ecien tly
and reliably propagate up dates among themselv es T o
meet this goal b esides b eing or connected the
graphs w e generate ha velo w edgecost and lo w diam
eter This implies that the ob jectiv e functionthatcon trols the annealing sc hedule com bines b oth edgecost
and diameter After a transformation w e compute the
cost dierence E as
E p
edg e cost
new
edg e cost
old
edg e cost
old
p diameter
new
diameter
old
diameter
old
Note that this is a linear com bination of normalized
edge and diameter costs Cho osing p to emphasize edge
costs results in higher diameter graphs c ho osing p to
emphasize diameter results in graphs with more and
p ossibly more exp ensiveedges W e compute diameter
using Dijkstras allpairs shortestpath algorithm at
cost O NE log N
Results
This section summarizes exp erimen ts w e conducted to
ev aluate the p erformance of the algorithm presen ted in
the last section Recall that this algorithm requires a
connectivit y requiremen t k as w ell as an edgecost ma
trix of the top ology Weev aluated our algorithm using random graphs
generated b y NTG a net w ork top ology generator
Links w ere of three dieren t bandwidths no des w ere
placed in an xy plane with higher densit yto w ards the
p eriphery of the plane and all no des had the same de
gree Link cost w as computed as ph ysical distance di
vided b y link bandwidth
Annealing Sc hedule
Initial T emp erature T
F or eac h graph G w e set the
initial T
b y estimating E E p W e then set the
initial temp erature T
so that P E exp nE k T
Note that dieren t ob jectiv e functions result in
dieren tv alues of T
T o study ho w T
aects the annealing pro cess w e
ran the algorithm for a wide range of v alues T
T
X-change
(a) (b)
Delete
(c) (d)
Split Add
Figure P ossible T ransformations
and T
F or a no de graph where the feasible top ol
ogy is connected the graphs in the rst second and
third column of Figure plot total edgecost diameter
in edgecost and diameter in hop coun t resp ectiv ely Eachro w in Figure corresp onds to a dieren tv alue
of T
Note this is not iden tical to generating Hamilto
nian cycles since the top ologies can ha v e higher degree
no des The horizon tal axis refer to iteration through the
annealing pro cess The top ology is accepted when the
annealing pro cess terminates at the far righ thand edge
of eac h graph
In Figure w e observethat as T
increases the algo
rithm allo ws the ob jectiv e function to increase since at
high temp eratures the annealing pro cess accepts p osi
tiv e E more frequen tly Note that since our initial
top ologies are nearly cycles it is hard to eliminate edges
without making the resulting graph infeasible There
fore early in the the annealing pro cess edges are added
reducing diameter but increasing edgecost Later in the
annealing pro cess b oth edgecost and diameter shrink
Note also that at T
increases the diameter and edge
cost uctuate more and that the top ology that mini
mizes diameter in hopcoun t is not usually the top ology
that minim izes diameter in edgecost
Figure rep eats the previous gure but for
connected top ologies Again as T
increases the an
nealing pro cess explores higher edgecost top ologies and
the ob jectiv e function oscillations increase Since connected top ologies ha v e more edges than connected
ones their graphs ha velo w er diameters and their indi
vidual edges are shorter
Figure rep eats the connected no de exp eri
men t with an ob jectiv e function based only on edge
cost Notice ho w the annealing pro cess fails to reduce
the diameter signican tlyF urther note that at lo w T
the algorithm initially fails to nd feasible solutions W e
indicate this b y not sho wing the ob jectiv e function lines
While not sho wn w e rep eated the connected exp eri
men t with an edgecost only ob jectiv e function and sa w
similar results
T emp erature Decrease Rate As the annealing
progresses it needs to slo wly decrease the temp erature
un til it freezes it Ab o v e w e studied the eects of the
initial temp erature T
on the annealing pro cess b e loww e study the m ultipli cativ e temp erature decrease
rate D In the graphs the temp erature hits zero in
m ultiplicativ e steps T
i D T
i
As illustrated
in Figure T decreases after a predened n um ber
of transformations that resulted in a top ology whic h
the annealing pro cess accepted SUCCESS OPS or af
ter a predened n um b er of transformations successful
or not TOT AL OPS Curren tly SUCCESS OPS and
TOT AL OPS are set to and resp ectiv ely Figure rep eats gure Figure except that instead
of studying the eects of dieren t T
it studies ho w
dieren t temp erature decrease rates D aect the an
nealing pro cess Since lo w er decrease rates makethe
temp erature decrease slo w er the annealing pro cess ac
cepts highercost top ologies more frequen tly for a longer
anneal
for i to ITERATIONS oldcost currentcost
select localtransforma tion
newtopology localtransformati oncu rren ttop ology feasible checkfeasibility new topol ogy N connectivity
if feasible continue
currentcost costnewtopology deltacost currentcost oldcost
p expobjfunctiond elta cost temp erat ure
r random
if r p accept newtopology
success success else
undo transformation
operation operation if success SUCCESSOPS or operations TOTALOPS temperature temperature decreaserate success operations Figure The annealing algorithm
p erio d of time during the annealing pro cess W e notice
that on a v erage
lo w er decrease rates result in top olo
gies with lo w er diameter and sligh tly higher edgecost
While not sho wn w e rep eated the decrease rate ex
p erimen t with an edgecost only ob jectiv e function and
observ ed the opp osite eect that is higher decrease
rates pro duce top ologies with lo w er total edgecost and
sligh tly higher diameter
Based on these preliminary results w eset T
and
D for the exp erimen ts describ ed in Section b elo w
W e use T
s lo w est v alue T
since it causes the
sim ulated annealing algorithm to con v erge so oner W e
use b oth D and D
EdgeCost v ersus Diameter
In the next set of sim ulations w e study ho w the prob
abilit y of applying the add delete xchange and split
transformations aects the annealing pro cess Besides
edge cost only p and an equal com bination of edge
cost and diameter in edgecost p w e also use an
W e conducted runs for eac hv alue of D equal linear com bination of edgecost and diameter in
hopcoun t p as ob jectiv e functions F or these ex
p erimen ts w eused P
P
and P
as the dieren t probabilitycom binations of applying the
add delete xchange and split transformations resp ec
tiv elyW e xed the initial no de top ology the anneal
ing pro cess uses so that w e can compare the resulting
top ologies costs
Figure and T able sho w the results of running the
top ology computation algorithm using an equal linear
com bination of total edgecost and diameter in edge
cost in the ob jectiv e function to generate no de connected graphs
F rom T able w e observ e that b ecause the initial
top ology already has lo w total edgecost the annealing
pro cess cannot signican tly impro v e total edgecost but
pro duces top ologies with diameter up to lo w er As
exp ected w e get lo w er total edgecost top ologies with
higher diameter as the probabilityof delete increases
and the probabilityof add decreases Note that using
the delete transformation of the times decreases
the top ologys total edgecost b y and still lo w ers
the diameter byappro ximately W e also observ e
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Connectivity 2 Cost function: SP D=0.01
T0=.08
T0=.8
T0=8
2
2
2
Figure The eects of T
on the annealing pro cess F or T
st ro w T
nd ro w and T
rd
ro w w e plot total edgecost st column diameter in edgecost nd column and diameter in hopcoun t rd
column when generating no de connected graphs using an equal linear com bination of edgecost and diameter
as the ob jectiv e function p The horizon tal axis refer to iteration through the annealing pro cess log scale
that while the results in T able sho wthat high delete
probabilities atten uate the eects of dieren t decrease
rates D the graphs in Figure sho w that D mak es
the annealing algorithm con v erge so oner
W e rep eat the same exp erimen t using only total edge
cost in the ob jectiv e function and rep ort the results in
Figure and T able Since the annealing pro cess only
tries to optimize total edgecost higher delete proba
bilities result in higher total edgecost reductions In
fact according to T able when the delete probabilit y
is the resulting top ologys total edgecost is ap
pro ximately lo w er than the initial top ologys total
edgecost Notice that w e also get a reduction in
diameter The highest diameter reduction is appro xi
mately and as exp ected happ ened when the add
and delete probabilities w ere and resp ectiv ely Ho w ev er w e should p oin t out that except for the cases
where the add probabilit y is the diameter in hop
coun t did not go do wn signican tly This means that our
algorithm generated top ologies with almost the same
n um b er of edges than the initial top ology but the se
lected edges ha velo w er cost
F rom the graphs in Figure w e observethat the
lo w er decrease rate D generates more cost oscil
lation than D Again this is due to the fact that
the lo w er D causes the temp erature to decrease slo w er
and consequen tly allo ws highercost top ologies to b e ac
cepted more frequen tly b y the annealing pro cess
Figure and T able rep orts the results obtained
when running our sim ulated annealing algorithm con
trolled b y an equal linear com bination of total edge
cost and diameter in hopcoun t p F or net w orks
where ph ysical link costs are roughly uniform it ma y
mak e more sense to impro v e diameter in hopcoun tthan
diameter in edgecost
F rom T able w eobserv e that all of the resulting
top ologies ha v e higher total edgecosts but sho wsub stan tial reductions in diameter in hopcoun t Notice
that the resulting top ologies presentlo w er diameter in
hopcoun t than the top ologies obtained when diameter
in edgecost is used in the ob jectiv e function T able F or the same transformation probabilitycom bina
tion the temp erature decrease rate generates lo w er
total edgecost top ologies As the curv es in Figure sho w the higher decrease rate do es not allo w total edge
costs to go v ery high since the probabilit y of accepting
highercost top ologies decreases faster
Summary
Cho osing the ob jectiv e function that con trols the an
nealing pro cess dep ends on the t yp e of optimization
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Connectivity 3 Cost function: SP D=0.01
T0=.08
T0=.8
T0=8
2
Figure Ho w T
aects the annealing pro cess F or T
st ro w T
nd ro w and T
rd
ro w w e plot total edgecost st column diameter in edgecost nd column and diameter in hopcoun t rd
column when generating no de connected graphs using an equal linear com bination of edgecost and diameter
as the ob jectiv e function p The horizon tal axis refer to iteration through the annealing pro cess log scale
problem b eing solv ed Replicas of an In ternet informa
tion service need lo wcost lo wdiameter or connected
graphs for propagating their up dates T o generate suc h
graphs the ob jectiv e function w e use consists of an
equally w eighed linear com bination of total edgecost
and diameter Because the initial top ologies our algo
rithm generates already ha velo w total edgecost the
resulting top ologies sho w total edgecost reductions of
up to Ho w ev er w eac hiev e diameter reductions of
up to Using total edgecost alone in the ob jectiv e function
generates graphs whose total edgecost and diameter
are resp ectiv ely and lo w er than the feasible
but unoptimized starting top ology If wew an t to optimize comm unication cost and prop
agation dela ys in terms of n um b er of net w ork hops
w e could use an ob jectiv e function that com bines total
edgecost and diameter in hopcoun t The corresp ond
ing sim ulation results sho w ed reductions of up to in diameter in hopcoun t at the exp ense of total edge
cost increase of appro ximately This is b ecause re
ductions in diameter in hopcoun t can only b e ac hiev ed
b y adding more edges while sw apping more exp ensiv e
edges with c heap er ones ma y result in reductions in di
ameter in edgecost
With resp ect to running time it tak es appro ximately
min utes for the algorithm to execute iterations
when generating no de connected top ologies on a
Sun SparcStation Practical Issues
The results in Section demonstrate that sim ulated
annealing generates small enough edgecost reductions
that in practice simpler faster less optimal algorithms
can b e used Belo w w e describ e our exp erience with
some of these more practical algorithms
Adding Selected Edges
This algorithm starts b y generating a random feasible
initial top ology using the metho d describ ed in Section
Recall that these initial top ologies tend to ha velo w
total edgecost since they do not include man yextra
edges Therefore in the next step the algorithm adds a
n um b er of extra edges so that it can reduce the graphs
diameter
W e used dieren t edge selection heuristics In the
rst exp erimen t w e add edges connecting no de pairs
whose distance in edgecost is maxim um In other w ords
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Connectivity 2 Cost function: LO D=0.01
T0=200
T0=2000
T0=20000
Figure The eects of T
on the annealing pro cess F or T
st ro w T
nd ro w and T
rd ro w w e plot total edgecost st column diameter in edgecost nd column and diameter in hopcoun t
rd column when generating no de connected graphs using only edgecost as ob jectiv e function p The horizon tal axis refer to iteration through the annealing pro cess log scale
Extra
P
Edge Diameter in Diameter in
Edges Cost EdgeCost Hops
T able Logical top ology costs after adding extra
edges to the initial top ology Links are added according
to diameter in edgecost
w e insp ect the curren t top ologys adjacency matrix and
add an edge connecting the rst pair of no des whose
distance is equal to the diameter in edgecost The
rst en try in T able sho ws the starting solutions costs
The remaining en tries sho w the costs after a new link is
added to the previous top ology As exp ected total edgecost go es up as w e add new
links and diameter go es do wn Notice that after adding
the second link b oth diameter in edgecost and hop
coun t decrease more than with an increase of less
than in total edgecost W e also observe thatasdi ameter in edgecost decreases so do es diameter in hop
coun t Ho w ev er ev ery time a new link is added diameter
in edgecost decreases but diameter in hopcoun tma y
sta y the same
In the next exp erimen t w e based our selection cri
teria in diameter in hopcoun t when adding an edge
w e insp ect the curren t adjacency matrix and add a
link connecting the rst pair of no des whose distance is
equal to diameter in hopcoun t Eac hen try in T able b elo w sho ws total edgecost diameter in edgecost and
diameter in hopcoun tev ery time a new link is added
Again as w e add new links b oth diameter in edge
cost and hopcoun tgo do wn Notice that after adding
the second link diameter in hopcoun t decreases almost
while total edgecost increases less than Finally in the last exp erimen t w e tried to minimi ze
the increase in total edgecost eac h time a new edge is
added So w e use the same edge selection criteria as
b efore except that w ec ho ose the c heap est edge con
necting no des whose distance in hopcoun t is equal to
the diameter in hopcoun t
When w e compare these results with the v alues in
T able w e notice that for the same diameter decrease
10
0
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0
5000
10
0
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5000
Diameter
10
0
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5000
10
0
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20
40
10
0
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40
Diameter (hop count)
10
0
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5
0
5
10
x 10
5
10
0
10
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0
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10
x 10
5
Edge Cost
10
0
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5
0
5
10
x 10
5
10
0
10
5
0
20
40
Connectivity 2 Cost function: SP T0=8
D=.4
D=.1
D=.01
Figure The eects of the temp erature decrease rate D on the annealing pro cess F or D st ro w D nd ro w and D rd ro w w e plot total edgecost st column diameter in edgecost nd column and
diameter in hopcoun t rd column when generating no de connected graphs using an equal linear com bination
of edgecost and diameter as ob jectiv e function p The horizon tal axis refer to iteration through the annealing
pro cess log scale
Extra
P
Edge Diameter in Diameter in
Edges Cost EdgeCost Hops
T able Logical top ology costs after adding extra
edges to the initial top ology Links are added according
to diameter in hopcoun t
Extra
P
Edge Diameter in Diameter in
Edges Cost EdgeCost Hops
T able Logical top ology costs after adding extra
edges to the initial top ology Links are added according
to diameter in hopcoun t The c heap est link is selected
ev ery time
10
0
10
5
10
4
10
6
10
0
10
5
10
4
10
6
Edge Cost
10
0
10
5
0
5000
10000
10
0
10
5
0
5000
10000
Diameter
10
0
10
5
0
5000
10000
10
0
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5
0
50
10
0
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50
Diameter (hop count)
10
0
10
5
10
4
10
6
10
0
10
5
0
50
... D=0.01 ___ D=0.4
Cost function: SP Connectivity 2 T0=0.08
50%,20%,15%,15%
30%,40%,15%,15%
20%,50%,15%,15%
Figure The eects of dieren t transformation probabilities in the annealing pro cess when generating no de
connected graphs W e plot total edgecost st column logscale diameter in edgecost nd column and
diameter in hopcoun t rd column using dieren t transformation probabilities for D and D T
is
set at and the ob jectiv e function is an equal linear com bination of total edgecost and diameter in edgecost
p Ro ws are mark ed with X Y Z W where X Y Z W are the probabilities of applying the add delete xchange and split transformations resp ectiv ely The horizon tal axis refer to iteration through the annealing
pro cess log scale
the total edgecost increase is lo w er F or example after
w e add the second link w e get the same reduction in
diameter in hopcoun t but only a increase in total
edgecost W e also observ e that b ecause w e select the
c heap est link ev ery time the reduction in diameter in
edgecost is higher than in the previous exp erimen t
Instead of using the regular add transformation in
the sim ulated annealing approac h w e can add selected
edges according to one of the criteria presen ted ab o v e
The disadv an tage of using the sele cte d add op eration is
that it is more exp ensiv e than the regular add Deleting Selected Edges
The same w ayasw e use heuristics to c ho ose edges to
add w e can select go o d edges to delete When c ho osing
an edge to delete the goal is to lo w er the top ologys total
edgecost y et k eeping the diameter constan t Recall
that diameter is the maxim um shortest path b et w een
an y pair of no des If w ew anttok eep diameter in edge
cost constan t then w e should nd a directly connected
pair of no des whose connecting edgecost is equal or
higher than the diameter Another p ossibilit yis to nd
all pair of no des that satisfy the ab o v e requiremen ts
and c ho ose the most exp ensiv e link to delete
Similarlyto sele ctedaddw e can use the sele cte d
delete op eration instead of the normal delete op eration
in the sim ulated annealing approac h Lik e sele ctedadd sele cte d delete is also more exp ensiv e than the regular
delete Minim umSpanning T ree with Ad
ditional Edges
Curren tlyour In ternet replication system uses a minim um spanning tree MST algorithm to generate the graphs
o v er whic h replicas propagate their up dates This al
gorithm computes a minim um cost tree connecting all
no des and for eac h no de that do es not ha v e the required
connectivit y degree it adds the c heap est un used edge
Adding the extra edges not only reduces the graphs di
ameter but also impro v es the corresp onding net w orks
resilience to link and site failures
T able sho ws total edgecost diameter in edgecost
and diameter in hopcoun t for the no de graphs this
10
0
10
5
10
4
10
5
Edge Cost
10
0
10
5
0
5000
10000
10
0
10
5
0
5000
10000
Diameter
10
0
10
5
0
5000
10000
10
0
10
5
0
50
10
0
10
5
0
50
Diameter (hop count)
10
0
10
5
10
4
10
5
10
0
10
5
10
4
10
5
10
0
10
5
0
50
... D=0.01 ___ D=0.4
Cost function: LO
Connectivity 2 T0=200
50%,20%,15%,15%
30%,40%,15%,15%
20%,50%,15%,15%
Figure The eects of dieren t transformation probabilities in the annealing pro cess W e plot total edgecost st
column log scale diameter in edgecost nd column and diameter in hopcoun t rd column using dieren t
transformation probabilities for D and D T
is set at and w e use edgecost only as ob jectiv e
function p when generating no de connected graphs Ro ws are mark ed with X Y Z W where
X Y Z W are the probabilities of applying the add delete xchange and split transformations resp ectiv ely The
horizon tal axis refer to iteration through the annealing pro cess log scale
MST algorithm generates W e used as input the same
ph ysical top ology that w as used in the exp erimen ts re
p orted in Section The rst en try in Tablesho ws
the minim um spanning tree costs The remaining en
tries sho w the costs after a new edge is added to the
previous top ology When compared to the the sim ulated annealing al
gorithm the MST algorithm generates top ologies with
reasonable costs and runs times faster W e should
p oin t out that although the top ologies resulting from the
minim um spanni ng tree algorithm ha v e the minim um
no de connectivit y requiremen t they are not necessarily
k connected
Conclusion
This pap er presen ted and ev aluated a sim ulated anneal
ing algorithm to construct lo w edgecost lo w diameter
k connected graphs Graphs with these prop erties are
useful for k eeping consistency among replicas of an In
ternet information service W e use or connectivit y
for resiliencylo w edgecost for ecien t use of the ph ysi
cal net w ork and lo w diameter for limiti ng up date prop
agation dela ys and consequen tly transien t replica in
Extra
P
Edge Diameter in Diameter in
Edges Cost EdgeCost Hops
T able T otal edgecost diameter in edgecost and in
hopcoun t for the graphs resulting from our MST al
gorithm algorithm Eachen try sho ws the costs after
adding a new edge to the previous graph
10
0
10
5
10
4
10
6
10
0
10
5
10
4
10
6
Edge Cost
10
0
10
5
0
5000
10000
10
0
10
5
0
5000
10000
Diameter
10
0
10
5
0
5000
10000
10
0
10
5
0
50
10
0
10
5
0
50
Diameter (hop count)
10
0
10
5
10
4
10
6
10
0
10
5
0
50
... D=0.01 ___ D=0.4
Cost function: HP
Connectivity 2
T=0.08
50%,20%,15%,15%
30%,40%,15%,15%
20%,50%,15%,15%
Figure The eects of dieren t transformation probabilities in the annealing pro cess W e plot total edgecost st
column log scale diameter in edgecost nd column and diameter in hopcoun t rd column using dieren t
transformation probabilities for D and D T
is set at and w e use an equal linear com bination
of total edgecost and diameter in hopcoun t p as ob jectiv e function when generating no de connected
graphs Ro ws are mark ed with X Y Z W where X Y Z W are the probabilities of applying the add delete xchange and split transformations resp ectiv ely The horizon tal axis refer to iteration through the annealing pro cess
log scale
consistencies
Our sim ulated annealing algorithm sim ultaneously
reduces edgecost and diameter b y and resp ec
tiv elyo v er unoptimized feasible solutions When min
imizing edgecost alone our algorithm nds solutions
that sho w edgecost reduction as w ell as di
ameter reduction
Since only mo derate reductions in total edgecost re
sult from suc h sophisticated algorithm in practice sim
pler faster less optimal algorithms can b e used W e
presen ted some of these more practical approac hes in
cluding the minim um spanni ng tree algorithm with ad
ditional edges that our In ternet replication service cur
ren tly uses
References
C Mic Bo wman P eter B Danzig Darren R
Hardy Udi Man b er Mic hael F Sc h w artz and Du
ane PW essels Disco v ery and access system Sub
mittedtoA CMIEEE T r ansactions on Networking Marc h CM Bo wman P B Danzig DR Hardy UMan b er and MF Sc h w artz Harv est A scalable cus
tomizable disco v ery and access system T ec hnical
Rep ort CUCS Departmen t of Computer
Science Univ ersit y of ColoradoBoulder July TH Cormen CE Leiserson and RL Riv est
Intr o duction to A lgorithms The MIT Press and
McGra wHill
Mac hael R Garey and Da vid S Johnson Com
puters and intr actability a guide to the the ory of
NPc ompletenessWH F reeman San F rancisco
S Hotz and R Nagamati Net w ork top ology gener
ator NTG A to ol for generating net w ork top ol
ogy and p olicy for proto col sim ulation purp oses
T ec hnical Rep ort Computer Science Departmen t
Univ ersit y of Southern California Spring DS Johnson The complexit yof net w ork design
problem Networks S Kirkpatric k C D Gelatt Jr and M P V ecc hi
Optimization b y sim ulated annealing SCIENCE Ma y
J Plesnik The complexit y of designing a net w ork
with minim um diameter Networks C Rose Lo w mean in terno dal distance net w ork
top ologies and sim ulated annealing IEEE T r ans
Commun pages Ulric hSc h umac her An algorithm for construc
tion of a kconnected graph with minim um n um
b er of edges and quasiminim al diameter Networks K Steiglitz P W einer and DJ Kleitman The
design of minim um cost serviv able net w orks IEEE
T r ans Cir cuit The ory CT No v em
b er
Probabili ti es Decrease T otal Edge Diameter in Diameter in
Rate D Cost EdgeCost HopCoun t
Add Del Xc h Spt Initial Final Initial Final Initial Final
T able Logical top ology costs b efore and after annealing The initial temp erature is set at and as cost
function w e use total edgecost and diameter in edgecost
Probabili ti es Decrease T otal Edge Diameter in Diameter in
Rate D Cost EdgeCost HopCoun t
Add Del Xc h Spt Initial Final Initial Final Initial Final
T able Logical top ology costs b efore and after annealing The initial temp erature is set at and w e use total
edgecost as ob jectiv e function
Probabili ti es Decrease T otal Edge Diameter in Diameter in
Rate D Cost EdgeCost HopCoun t
Add Del Xc h Spt Initial Final Initial Final Initial Final
T able Logical top ology costs b efore and after annealing The initial temp erature is set at and w e use total
edgecost and diameter in hopcoun tasobjectiv e function when generating no de connected graphs
Abstract (if available)
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Description
Katia Obraczka, Peter Danzig, Kitinon Wangpattanamongkol. "Finding low diameter low edge cost networks." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 652 (1997).
Asset Metadata
Creator
Danzig, Peter
(author),
Obraczka, Katia
(author),
Wangpattanamongkol, Kitinon
(author)
Core Title
USC Computer Science Technical Reports, no. 652 (1997)
Alternative Title
Finding low diameter low edge cost networks (
title
)
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Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA
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USC Viterbi School of Engineering Department of Computer Science
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Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089
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Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA
(publisher)
Copyright
In copyright - Non-commercial use permitted (https://rightsstatements.org/vocab/InC-NC/1.0/