Minimum Average Delay of Routing Trees

0.56 MB·

by  Saber Mirzaei

Average Delay of Routing Trees
SaberMirzaei
BostonUniversity
January13,2016
6 Abstract
1
The general communication tree embedding problem is the problem of mapping a set of communicating
0
2 terminals,representedbyagraphG,intothesetofverticesofsomephysicalnetworkrepresentedbyatreeT.
InthecasewheretheverticesofGaremappedintotheleavesofthehosttreeT theunderlyingtreeiscalled
n
a routing tree and if the internal vertices of T are forced to have degree 3, the host tree is known as layout
a
J tree. Differentoptimizationproblemshavebeenstudiedintheclassofcommunicationtreeproblemssuchas

2 well-knownminimumedgedilationandminimumedgecongestionproblems. Inthisreportwestudytheless
1 investigatemeasurei.e. treelength,whichisarepresentativeforaverageedgedilation(communicationdelay)
measureandalsoforaverageedgecongestionmeasure. WeshowthatfindingaroutingtreeT foranarbitrary

]
graphGwithminimumtreelengthisanNP-Hardproblem.
C
C
. 1 Definitions and Introductory Points
s
c
[ ConsideragroupofterminalscommunicatingviaafinitenetworkG=(V,E),wherethesetofvertices(finite

1 setV)andedges(finitesetE),respectivelyrepresentthecollectionofterminalsandtheirdirectcommunication
v paths. Weshow∣V∣bynand∣E∣asm.
7
9 The general communication tree embedding problem is the problem of mapping the set of terminals into the
6 set of vertices of some physical network represented by a tree T. Accordingly, the two vertices v,u ∈ V(G)
2
that are directly connected via e ∈ E(G), are connected indirectly via some path P (v,u) in T. In the case
0 T
. where the vertices of G are mapped to the leaves of the host tree, the underlying tree is called a routing tree.
1
0 Inthisreportwemostlyfocusonthecasewheretheinternalverticesofthehosttreehavedegree3(knownas
6 treelayout problem). WedenotethesetsofleafnodesandinternalnodesoftreeT respectivelybyV (T)and
L
1 V (T). 1
: I
v For a graph G and a communication tree T for G, there are different measures defined in literature. In
i
X followingwedefinethetwomeasuresthatweareintrustedinthisreport. Foracomprehensivelistofmeasures,
r aninterestedreadercanreferto[3].
a
Definition1.1(EdgeDilation). ConsideragraphGandacommunicationtreeT andabijectionϕfromvertices
of G to leaf nodes of T. The dilation λ(uv,T,ϕ,G) of an edge {u,v} ∈ E(G) is the distance between ϕ(u)
andϕ(v)inT. ◻
Werepresentthedistanceoftwovertices{u,v}inagraphGwithd (u,v)
G
Definition 1.2 (Edge Congestion). Give a graph G and a communication tree T and and a bijective mapping
ϕ∶V(G)→V (T). Thecongestionδ(xv,T,ϕ,G)andofanedge{x,y}∈E(T)isthethenumberofedges
L
in{u,v}∈E(G)thatinT,thepathP (ϕ(v),ϕ(u))traversetrough{x,y}. ◻
T
1Wetrytousethetermnodeincaseoftreesasopposedtothetermvertex,whichweuseforgeneralgraphs.
1
Based on the definition of the communication tree for a graph G, removal of every edge {x,y} ∈ E(T)
partitions the set of vertices of G into two component. Hence every edge of tree corresponds to a cut in G.
Thereforethecongestionof{x,y}∈E(T)isthesizeofthecutitcorrespondsto.
Several optimization problems can be defined based on these two measures. Minimum tree layout dilation is
the problem of finding a tree layout for a given graph G such that the maximum edge dilation is minimized,
wherethemaximumistakenoveralledgeofG. In[7]itisshownthattheproblemoffindingatreelayoutwith
minimumdilationisNP-hard,whenthelayouttreeisrooted.
Similarly, given a graph G, in minimum tree layout congestion problem the goal is to find a tree layout T,
such that the maximum edge congestion is minimized. In [8] Seymour and Thomas show that the minimum
tree layout congestion problem is polynomially solvable for the case of planer graphs, and is NP-hard when
consideringgeneralgraphs. Inthisreportwestudytheminimumtreelayoutlengthproblem(shortlycalledMin
TreeLength),formallydefinedasitfollows.
Definition1.3(Minimumtreelayoutlength). ConsiderthefiniteundirectedgraphG = (V,E). Theminimum
treelayoutlengthproblemistheproblemoffindinglayouttreeT andabijectivemappingϕ∶V(G)→V (T)
L
suchthatL(T,ϕ,G)=∑{u,v}∈E(G)λ(uv,T,ϕ,G)isminimized. ◻
It is not hard to see that ∑{u,v}∈E(G)λ(uv,T,ϕ,G) = ∑{x,y}∈E(T)δ(xv,T,ϕ,G). Hence, in the rest of this
reportwemayusetheminterchangeably.
Accordingly, in the communication graph embedding problems, the dilation of an edge {u,v} ∈ E(G)
abstractly represent the communication delay between vertices u and v. Similarly the congestion of an edge
e∈E(T)isarepresentativeforthetrafficonthephysicallinke. Hencetreelengthmeasurecorrespondstothe
averagedelaybetweentheverticesofGandalsototheaverageedgecongestionofthehosttree.
2 Minimum Length of Tree Layout
Inthespecialcaseoftreelayoutproblem,theunderlyinghostgraphisatreeT wherethedegreeofeverynode
is either 1 or 3 and the vertices of G are being mapped to leaves of T. In this section we study minimum tree
layoutlength. WeshowthatMinTreeLengthproblemisNP-hardformulti-graphs2,andlateronweshowthe
problemstaysNP-hardwhenrestrictedtotheclassofsimplegraphs.
2.1 MinTreeLengthofCompleteGraphs
ConsiderthecompletegraphG=(V,E)where∀u,v ∈V(G),{u,v}∈E(G). Itisnothardtoseethatalayout
treeT isasolutionfortheMinTreeLayoutproblemforG, iff∣V (T)∣ = n(andhence∣V(T)∣ = 2n−1), and
L
thesummationofdistanceofleafnodesofT isminimized. Wedenotethesummationofdistancesofleavesof
atreeT by:
1
σ (T)= ∑ d (x,y)
LL T
2
x,y∈VL(T)
Leaf to leaf distance summation measure σ is very similar to the definition of Wiener index (proposed by
LL
chemistWiener[10]),whichisthesummationofdistancesofallverticesofagivengraphGasrepresentedin
followingequation.
1
σ(G)= ∑ d (u,v)
G
2
u,v∈V(G)
2Bymulti-graphwerefertofinitegraphswithpossibilityofparalleledgesandnoloop.
2
Wiener index is widely studied both in mathematical and chemical literature. In [2] Fischermann et al. study
Wienerindexoftrees. Inthisworksauthorsrepresentthestructureofthefamilyofthetreesthathaveminimum
(ormaximum)Wienerindexamongallthetreesofthesameorderwithmaximumnodedegree∆ ≥ 3. Dueto
similarityofσ measureandWienerindex,intherestofthissubsectionweborrowsomeofthenotationsand
LL
definitionsfrom[2]inordertostudyσ fortreeswithmaximumdegree∆.
LL
Definition2.1(T(R,∆)treefamily). Considerintegers∆andR∈{∆,∆−1}. Foragivenn∈N,thefamily
T(R,∆)oftreeswithnnodeshasauniquememberTuptoisomorphism, definedusingaplanarembedding
asitfollows.
LetMk(R,∆)≤n<Mk+1(R,∆)where:
⎧
⎪⎪1 k =0
M (R,∆)=⎨
k ⎪⎪1+R+R×(∆−1)+...+R×(∆−1)k−1 k ≥1
⎩
Figure1depictstheembeddingoftreeTwiththefollowingproperties:
1. allnodesofTlieonsomelineL for0≤i≤k+1
i
2. exactlyonenodev liesonthelineL whichhasmin{n−1,R}childrenonlineL
0 1
3. fori=1tok−1everynodeonlineLi isconnectto∆−1nodesonlineLi+1 andonenodeonlineLi−1
4. theonlylinethatmaybeincomplete,islineLk+1. Letn−Mk(R,∆)=m×(∆−1)+rfor0≤r<∆−1,
wheren−Mk(R,∆)isthenumberofremainingnodesonlineLk+1. Alsolet{v1,..vm+1}bethesetofm
leftmostnodesonlineLk wherevm+1 istherightmostoneintheset. Eachofv1,..vm nodesisconnected
to∆−1nodesonlineLk+1,whilevm+1 isconnectedtor nodesfromlineLk+1 (seefigure1).
◻
… … Lk+1
Lk
V1 V2 Vm Vm+1
… … …
L2
L1
L0
Figure1: PlanarembeddingofTrepresentedastheembeddingofnodesonk+2linesL for0≤i≤k+1.
i
We defined the family T(R,∆) for the general case of trees with maximum degree ∆, while we focus
on the case where the degree of every internal node is ∆ = 3, but all the results presented in the rest of this
subsectionextendtoalltreeswitharbitrarymaxdegree∆≥3.
3
Lemma 2.2. Consider tree T ∈ T(R,∆) of order n, and assume Mk(R,∆) < n < Mk+1(R,∆) for k ≥ 1.
LetT beanarbitrarytreeofordernconstructedfromtreeT ∈T(R,∆),withM (R,∆)nodes,byattaching
0 k
n−M (R,∆)nodestotheleafnodesofT (whichlieonthelineL ). Thenitisthecasethateitherσ (T)>
k 0 k LL
σ (T)orT isisomorphicwithT.
LL
Proof. WeproofthislemmausinginductionontheheightoftreeT,whenembeddedontheplaneasexplained
in definition 2.1. It is easy to check the correctness of theorem for trees of height 1 and 2. Assume tree T of
heightk andtreeT ∈T(R,∆)arenotisomorphic. Letv ∈V(T)bethenodeonlineL . Nodev isconnected
0
toRsubtrees{T1,...,TR}. BasedontheassumptionofinductioneverysubtreesTiisamemberofthefamily
T(∆−1,∆) of height k or k−1. Since T and T are not isomorph, there are at least two subtrees T and T
i j
for i ≠ j where are incomplete on line L . Formally speaking ∃1 ≤ i ≠ j ≤ R,T ,T ∈ T(∆−1,∆) and
k i j
∣V(Ti)∣−Mk−1(∆−1,∆)=ri >0∧∣V(Tj)∣−Mk−1(∆−1,∆)=rj >0. Withoutlossofgeneralityweassume
i=1andj =1andalso∣V(T )∣≤∣V(T )∣. Figure2aabstractlyrepresentstreeT.
1 2
L L
k k
L L
T T k-1 T T k-1
1 2 1 2
L
L 1
1
L
L 0
v 0 v
′
(a)InitialtreeT. (b)TreeT constructedfromT.
Figure2: Figure2arepresentstreeT ∉T(R,∆)ofheightk+1,constructedfromtreeT ∈T(R,∆)ofheight
0
k. Nodev isconnectedtoatleasttwosubtreesT ,T ∈T(R,∆)ofheightk whichareincompleteonlineL .
i j k
′
AlternativetreeT depictedinfigure2b,isconstructedbyrelocatingsomeleafnodesofsubtreeT thatareon
1
lineL . Asyouseeinthisconstructionr leftmostleavesofT arerelocatedtocompletesubtreeT . Inthis
k 2 1 2
examplethenumberofleafnodesofT onL ,islargeenoughtocompletethesubtreeT .
1 k 2
′
Let L (T) denote the set of nodes of tree T on line l. We present an alternative tree T , by relocating some
l
leaf nodes of L (T ) (in order from left to right) to complete the line L of T (in order from right to left).
k 1 k 2
′
Let L ⊆ L (T ) be the set of leaf nodes of T on line L , candidate for relocation. Figure 2b depicts tree T
k 1 1 k
constructedfromT.
′ ′
InthealternativetreeT ,considerabijectivemappingϕ fromthenodesofT tonodesofsubtreeT ,where
0 1 2
′
T is the modified version of T in T. More specifically, mapping ϕ is a reflection form T to T , which
2 2 0 1 2
′
reflectsnodesofLl(T1)tothenodesofLl−1(T2)for2≤l≤k,suchthattheleftmostnodeonlineLl ofT1 is
′
mappedtotherightmostnodeofT onlineL . Accordinglyϕ mapseveryleafnodeofL=L (T )−L(the
2 l 0 k 1
set of remaining leaf nodes of T on line L ) to one leaf node of T on line L . On the other hand every leaf
1 k 2 k
nodeofLk−1(T1)ismappedtoonenode(leaforinternal)ofT2 onlineLk−1.
′
From ϕ we construct a bijective mapping ϕ from leaf nodes of T to sets of leaf nodes of T . For every
0 1 2
′
w ∈V (T),ϕ(w)={ϕ (w)}ifϕ (w)∈V (T )isaleafnodenode,otherwise(ϕ (w)isaninternalnodeof
L 0 0 L 2 0
′
T2 onlineLk−1),ϕmapsw tothesetofdirectchildrenofϕ0(w)onlineLk 3.
′
Using the bijective mapping ϕ, we analyze the change in value of σ in the process of constructing T from
LL
T asitfollows.
1. ClearlytheinternalsummationofdistancesofnodesinLstaysunchanged.
3Inthiscase∣ϕ(w)∣≤∆−1.
4
2. Foreveryleafnodew1 ∈L,w2 ∈VL(T)−VL(T1)⊎VL(T2),itisthecasethatdT(w1,w2)=dT′. Hence,
thesummationofdistancesamongnodesinLandleafnodesofV (T)−V (T )⊎V (T )alsodoesnot
L L 1 L 2
change.
3. Weshowthatforeveryw ∈Lthesummationofdistancesofw fromleafnodesofV (T )⊎V (T )−
1 1 L 2 L 1
′ ′
{w }isgreaterthanthesummationofdistancesofϕ(w )fromleafnodesofV (T )⊎V (T )−{ϕw }.
1 1 L 1 L 2 1
Notation 2.3. Let L ,L ⊂ V(T), for an arbitrary tree T. By σ (L ,L ) we denote the summation of
1 2 T 1 2
distancesofnodesofL andL . Formallyspeaking:
1 1
1
σ (L ,L )= ∑ ∑ d (v,u)
T 1 2 T
2
v∈L1u∈L2
Foreveryw ∈V (T )−L:
2 L 1
• Ifw2 isonlineLk,itisthecasethatdT(w1,w2)=dT′(ϕ(w1),ϕ(w2)). Therefore:
(1) σ ({w },{w }⊎ϕ(w ))=σ ({ϕ(w )},{w }⊎ϕ(w ))
T 1 2 2 T 1 2 2
′
• Ifw2 isonlineLk−1 andϕ(w2)mapsw2 toaleafnodeofT2 onlineLk−1,similartopreviouscase
wehave:
(2) σ ({w },{w }⊎ϕ(w ))=σ ({ϕ(w )},{w }⊎ϕ(w ))
T 1 2 2 T 1 2 2
′
• Otherwisew2 isonlineLk−1 andϕ(w2)mapsw2 toanon-emptysetofleafnodeofT2 onlineLk.
Assumethesizeofthissetis1 ≤ ∇ ≤ ∆. Sinceforsomenodesw where∣ϕ(w )∣ = ∇ = ∆−1 > 1,
2 2
thenforsuchw ,thefollowingequallyholds:
2
(3) σT({w1},{w2}⊎ϕ(w2))−σT′({ϕ(w1)},{w2}⊎ϕ(w2))=
(d (w ,w )+2k×∇)−(2k+(d (ϕ(w ),ϕ (w ))+1)×∇)=
T 1 2 T′ 1 0 2
(d (w ,w )+2k×∇)−(2k+(d (w ,w )+1)×∇)=
T 1 2 T 1 2
(2k−d (w ,w ))×(∇−1)−d (w ,w ))≥
T 1 2 T 1 2
2k−2d (w ,w ))>0
T 1 2
′
Putting the results of previous cases and equations 1, 2 and 3, we conclude that σ (T) > σ (T ). If
LL LL
′ ′ ′′
T ∉T(R,∆),thenbasedontheassumptionofinduction,replacingthesubtreeT withsubtreeT ∈T(R,∆)
1 1 1
′′ ′′ ′
ofthesameorder,resultsintreeT ,whereσ (T )<σ (T ).
LL LL
′′
UsingthesameapproachandcontinuingwithT ,inasequenceofleafnoderelocations,wecanconstructthe
finaltreeT̃, suchthatinT̃, nodev onlineL hasexactlyoneincompletesubtreeT̃ whereT̃ ∈ T(∆−1,∆).
0 i i
Morespecifically, ∀1 ≤ l < i, allleafnodesofT̃ leionlineL and∀i < l ≤ ∆, allleafnodesofT̃ leionline
l k l
Lk−1,i.e. T̃∈T(R,∆).
Notation2.4. GivenanarbitrarytreeT,wedefinetheplanarlineembeddingofT similartothetheapproach
inthedefinition2.1. Startingfromadesignatedv ∈ V(T),weembedT onlinesofplane,wherev liesonline
L anddirectneighboursofv areplacedonlineL . Similarlyallthenodesindistancedfromv areplacedon
0 1
line L . Also for u ∈ V(T) on line L , the subtree rooted at u, where all its nodes are on lines L for l′ ≥ l is
d l l′
denotedbyT . Formallyspeakingw∈V(T )iffuisontheshortestpathfromw tov onlineL .
u u 0
5
Notation2.5. ConsideratreeT ofordernandanodev ∈V(T)ofdegree∇. RemovingvformT partitionsT
n
intoasetofsubtrees{T1,...,T∇}. Wecallnodev centralif∀1≤i≤∇,∣V(Ti)∣≤ . Thesetofcentralnodes
2
oftreeT isrepresentedbyC .
T
Theorem2.6. EveryarbitrarytreeT hasatleastoneandatmosttwocentralnodes. Inotherword1≤∣C ∣≤2.
T
For proof see [9]. Using this theorem, we proof the main result of this subsection as we present in the theo-
rem2.7.
Theorem 2.7. Consider tree T with maximum node degree ∆. where σ (T) ≤ σ (T′) for every tree T′
LL LL
(that∣V (T′)∣=∣V (T)∣). ThenintheplanarlineembeddingofT withcentralnodev ∈C onfixedlineL ,it
L L T 0
isthecasethat:
• T ∈T(∆,∆)
• T ∈T(∆−1,∆)foru≠v
u
Proof. The proof of this theorem is carried out using induction on the height of planar line embedding of tree
T. Itisnothardtocheckthecorrectnessoftheoremfortreesofheight1and2. LetT beagraphofheighth≥3,
andletu ,...,u bethedirectneighboursofv onlineL andT ,...,T respectivelybetheircorresponding
1 ∆ 0 1 ∆
subtrees.
Case 1: ∃w ,w ∈ V (T),d (w ,v) ≥ d (w ,v)+2. Based on the assumption of induction w and w
1 2 L T 1 T 2 1 2
can not be on the same subtree. Without loss of the generality assume w and w are the two leaf nodes with
1 2
maximumdistanceandw1 ∈T1 andw2 ∈T2. LetT11,T12,...,T1,∆−1 ⊂T1 besubtreesrespectivelywithroots
u11,...u1,∆−1 connectedtou1 (nodesu11,...u1,∆−1 lieonlineL2). Alsoassumew1 ∈VL(T11).
Case1.1: ∣V (T )∣>∣V (T )∣. WeconstructanalternativetreeT̃byremovingedges{u ,u }and{v,u }
L 11 L 2 1 11 2
and introducing two new edges {v,u } and {u ,u }. The structures of initial tree T and the alternative tree
11 1 2
̃
T are represented in figure 3. One can verify that the following equation 4 correctly represents the relation
T T
11 12
T T
L
u u 2 12
11 12 T T 2
2 3 L
T T 2
u1 u2 u3 LL1 u11 u2 u1 u12 u3 L1
v 0 11 3
L
0
v
(a)InitialtreeT whered (w ,v)≥
T 1
d (w ,v)+2forw ∈V (T )and (b) Alternative tree T̃ constructed
T 2 1 L 11
w ∈V (T ). fromT.
2 L 2
̃
Figure3: RepresentationofinitialtreeT anditsmodifiedversionT.
6
betweenσ (T)andσ (T̃).
LL LL
(4) σ (T)−σ (T̃)=
LL LL
∣V (T )∣×( ∑ ∣V (T )∣− ∑ ∣V (T )∣)
L 11 L 1i L i
2≤i≤∆−1 3≤i≤∆
+∣V (T )∣×( ∑ ∣V (T )∣− ∑ ∣V (T )∣)
L 2 L i L 1i
3≤i≤∆ 2≤i≤∆−1
=(∣V (T )∣−∣V (T )∣)×( ∑ ∣V (T )∣− ∑ ∣V (T )∣)
L 11 L 2 L 1i L i
2≤i≤∆−1 3≤i≤∆
Itisthecasethat∑2≤i≤∆−1∣VL(T1i)∣<∑3≤i≤∆∣VL(Ti)∣otherwiseitmustbethecasethat∣V(T1)∣>∑2≤i≤∆∣V(Ti)∣>
∣V(T)∣,whichisincontradictionwiththecentralityofnodev. Thereforeσ (T)−σ (T̃)>0,whichcon-
LL LL
2
tradictstheoptimalityofT.
Case 1.2: ∣V (T )∣ ≤ ∣V (T )∣. Hence, d (w ,v) = d (w ,v)+2 and w ∈ V (T ) is located on line L
L 11 L 2 T 1 T 2 1 L 11 k
andw2 ∈ VL(T2)liesonlineLk−2. AlsononofsubtreesT11 andT2 canbecompleterespectivelyonlinesLk
andLk−1.
LetL (T)denotethesetofnodesoftreeT onlinel. Similartotheproofoflemma2.2,wepresentanalternative
l
treeT̃,byrelocatingsomeleafnodesofLk(T11)(inorderfromlefttoright)tocompletethelineLk−1ofT2(in
order from right to left). Let L ⊆ L (T ) be the set of leaf nodes of T on line L , candidate for relocation.
k 11 11 k
Basedonanexactreasoningasinlemma2.2(case3),whichweomit,itcanbeinferredthatthesummationof
distanceofleafnodesinLfromleafnodesofT ⊎T reducesgoingfromT toT̃. Formallyitcanbededuced
11 2
that:
(5) σT(L,VL(T11))+σT(L,VL(T2))>σT̃(L̃,VL(T̃11))+σT̃(L̃,VL(T̃2))
WhereT̃ andT̃ respectivelycorrespondtoT andT afterrelocatingleaf nodesofL(representedbyL̃in
11 2 11 2
̃
T).
Ontheotherhand,relocatingL,increasesthedistanceofeveryleafnodeinLfromleafnodeofT12,...,T1,∆−1
by 1 unit, while it decreases the distance of every node of L from every leaf node of T ,...,T . Since we
3 ∆
assumedthatw ∈V (T )andw ∈V (T )havethemaximumdistanceamongallleafnodes,then:
1 L 11 2 L 2
(6) ∣VL(T12)∣+...+∣VL(T1,∆−1)∣<∣VL(T3)∣+...+∣VL(T∆)∣
Fromequations5and6weconcludethefollowingcontradictoryresult:
(7) σ (T)−σ (T̃)=
LL LL
σT(L,VL(T11))−σT̃(L̃,VL(T̃2))+
σT(L,VL(T2))−σT̃(L̃,VL(T̃11))+
σT(L,VL(T)−VL(T11)⊎VL(T2))−σT̃(L̃,VL(T̃)−VL(T̃11)⊎VL(T̃2))
>0
Case2: ∀w ,w ∈V (T),∣d (w ,v)−d (w ,v)∣<2. SincebasedontheassumptionofinductionT ,...,T ∈
1 2 L T 1 T 2 1 ∆
T(∆−1,∆), then T can be constructed from some tree T ∈ T(R,∆) of order ∣V(T )∣ = M (R,∆), by
0 0 k
attaching n − M (R,∆) nodes to the leaf nodes of T . Therefore, based on lemma 2.2, T is optimal iff
k 0
T ∈T(∆,∆).
Corollary2.8. ConsiderthecompletegraphGofordern=∣V(G)∣. TreeT isanoptimaltreelayoutforGiff
∣V (T)∣=nandT ∈T(3,3).
L
7
Example 2.9. Let G be a complete graph of order n = 2l for some l > 1. Based on the result of theorem 2.7,
alayouttreeT forG(oforder2n−1)hasminimumvalueσ (andaccordinglyisasolutionforMinLayout
LL
Length)iffitisisomorphictosometreewithastructuresimilartothetreeinfigure4.
Figure 4: Two planar embeddings of an optimal layout tree T of size 2n−1, corresponding to the complete
graphGwithn=2l vertices.
2.2 MinTreeLengthofMulti-Graphs
Graph G is a multi-graph if either it is a simple undirected graph, or it can be constructed from a simple
undirectedgraphbyaddingparalleledges. InourmainresultofthisreportweshowthattheMinTreeLength
problem is NP-hard for class of multi-graphs. Finally we show that this result can be extended to the class of
simplegraphs.
Definition2.10(EqualSize4-CliqueCover). GivengraphG,EqualSize4-CliqueCoverproblemistheprob-
n
lem of partitioning V(G) into four disjoint subsets V ,...,V s.t. G(V ) is a clique of size , for 1 ≤ i ≤ 4 .
1 4 i
4
◻
Lemma 2.11. Equal Size 4-Clique Cover problem is NP-complete, even for the class of graphs of order 2l
verticesforsomel∈N.
Forproofyoucanreferto[].
Theorem2.12. MinTreeLengthproblemisNP-hardfortheclassofmulti-graphs.
Proof. The correctness of the theorem can be represented using a polynomial reduction form Equal Size 4-
CliqueCover.
ConsideranarbitrarygraphG,asaninstanceinputofEqualSize4-CliqueCoverproblem,where∣V(G)∣=2l
for some l ∈ N. Let G′ be the multi-graph obtained from G by introducing M = m×(2n−2) parallel edges
between every two vertices u,v ∈ V(G). Notice that every tree layout T for graph G has 2n−2 edges where
the congestion of each edge is less than m = ∣E(G)∣. Considering graph G as an vertex induced subgraph of
G′,thenG′ =G⊎G̃,whereG̃isanothervertexinducedsubgraphofG′. SubgraphG̃iscompletemulti-graph.
HenceforeverytreelayoutT andϕ∶V(G)→V (T)forG′ wehave:
L
(8) L(T,ϕ,G′)=L(T,G)+L(T,ϕ,G̃)
From corollary 2.8 we know that a layout tree T̃ for G̃ is optimal iff T̃ ∈ T(3,3). Also for every layout tree
T ∉T(3,3),itisthecasethatL(T, G̃)≥L(T̃, G̃)+M. Ontheotherhandforn>2andeverylayouttreeT
, ,
where∣V (T)∣=n,itisalwaysthecasethatL(T,G)<M.
L
8
′ ̃ ′
Therefore, a layout tree T for G is optimal iff T and T are isomorphic. In other words T for G is optimal
iff T ∈ T(3,3). Hence the Min Tree Length problem for G′ reduces to the problem of finding an optimal
bijection ϕ form vertices of G to leaf nodes of T̃ ∈ T(3,3), such that the summation of edge dilations for all
{u,v}∈E(G)isminimized. Formallyspeaking:
(9) argmin L(T,ϕ,G′)=(T̃,argminL(T̃,ϕ,G))
(T,ϕ) ϕ
LetGdenotethecomplementofgraphG. Onecaneasilycheckthat:
(10) L(T̃,ϕ,G)=L(T̃,ϕ,K )−L(T̃,ϕ,G)
n
WhereK isacompletegraphofsizen.Therefore:
n
(11) argminL(T̃,ϕ,G)=argmaxL(T̃,ϕ,G)=argmax ∑ λ(uv,T̃,ϕ,G)
ϕ ϕ ϕ
{u,v}∈E(G)
AlsobasedonthestructureofT̃,∃c1,c2 ∈CT̃(infigure4shownbydoublelinedcircles). Sincen=2l forl∈N,
̃
removingcentralnodesc andc partitionleafnodesofT into4equallysizedpartitionsL ,...,L .
1 2 1 4
n
Finally, we can deduce that the original graph G can be partitioned into 4 complete sub-graphs of size , iff
4
thereexistbijectionϕsuchthat∀{u,v}∈E(G),ϕ(u)∈L ∧ϕ(v)∈L for1≤i≠j ≤4. Inotherwords,graph
i j
G can be partitioned into 4 complete sub-graphs G ,...,G of size n, iff (T̃,ϕ) is a solution for Min Tree
1 4
4
LengthofG′,whereϕmapsverticesofG toleafnodesofL for1≤i≠j ≤4. WhichinferstheNP-hardness
i i
ofMinTreeLengthproblemforthemulti-graphs.
2.3 MinTreeLengthofSimpleGraphs
Finite graph G′ is simple, if for u ≠ v ∈ V(G′) there is at most one edge {u,v} ∈ E(G′) and ∀u ∈
V(G),{u,u}∉E(G). Considercompletemulti-graphG,whereeverytwoverticesuandvareconnectedvial
paralleledges. Havingmulti-graphGonecanobtainasimplegraphG′bysubdividingveryedge{u,v}∈E(G)
andintroducinganewvertexxofdegree2.
ConsideratreelayoutT′ andbijectivemappingϕ′ ∶ V(G′) → V (T′)forthesimplegraphG′. Foreveryx ∈
L
V(G′),ϕ′(x)∈V (T′)isdirectlyconnectedtosomeinternalnodew∈V (T′). Removingwresultsintheset
L I
ofthreesubtree{T′ ,ϕ′(u),T′ }. Foreveryx∈V(G′),byT′(x)andT′(x)wereferrespectivelytosubtree
w,1 w,2 1 2
T′ andT′ , wherew isthedirectneighbourofϕ′(x)inT′. Itiseasytoseethatσ({ϕ′(x),w},T′,ϕ′,G′)
w,1 w,2
isequaltothedegreeofxinG,andalsoifxisofdegree2thenthecongestionofthetwoedgesconnectingw
toT′ andT′ isequaliffϕ′(u)∈V(T′ )andϕ′(v)∈T′ ,whereuandv arethedirectneighborsofxin
w,1 w,2 w,1 w,2
G.
Consider tree layout T and bijective mapping ϕ ∶ V(G) → V (T) for the multi-graph G. Starting from tree
L
′ ′
T, we can constructed a new tree T for the simple graph G by subdividing some edge of T and introducing
′
sub-treelayoutscontainingonlyverticesofdegree2. Inotherwords,T isconstructedfromT byintroducing
m = ∣E(G)∣internalnodeandmleafnodes(eachcorrespondingtoonevertexofG′ ofdegree2). Figures 5a
and 6a respectively depict the multi-graph G and its corresponding tree layout T, while in figures 5b and 6b
′ ′ ′
youcanseesimplegraphG obtainedfromGandonepossibletreelayoutT forsimplegraphG constructed
fromT.
Lemma2.13. LetGbeacompletemulti-graphwhere∀u,v ∈V(G)thereexistlparalleledge{u,v}∈E(G).
′
LetT andϕbearbitrarytreelayoutandthecorrespondingmappingforG. AlsoassumeG isthesimplegraph
′
obtainedbysubdividingeveryedgeofG. WeshowtheclassofallpossiblelayouttreesforG ,constructedfrom
T byF(T). ConsidertreelayoutT′ ∈F(T),where∀T′′ ∈F(T),LA(T′,G)≤LA(T′′,G). Thenitisthecase
′
that T is constructed by subdividing only edges of T that each one is adjacent to a leaf node (which we call
0
externaledges).
9
1 2
12
1 2
21
23 14
13 31 24 42
41 32
3 4
34
3 4
43
(a) Complete multi-graph G where
everytwoverticesareconnectedvia (b) Simple graph G′ obtained from
l=2paralleledges. G.
′
Figure5: SimplegraphG obtainedfromGbysubdividingeveryedgeofGandintroducingavertexofdegree
2.
Proof. We know that for every edge e ∈ E(T),σ(e,T,G) ≥ l×(n−1) where n = V(G) and equality holds
onlyifeisadjacenttoaleafnodew ∈V (T). NowconsideranarbitraryT′′ ∈F(T),obtainedbysubdividing
L
atleastoneinternaledgee∈E(T)4. Itiseasytocheckthat∀e∈E (T),σ(e,T,G)≥l×2×(n−2). Asappose
I
′′ ′ ′
to tree layout T for G , we suggest tree layout T constructed by relocating the subtree (possibly more than
one subtree) that subdivides e (and hence removing all the subdivisions of e) and creating a new subdivision
(possiblymorethanonesubdivision)inaanexternaledge{w,w′}(assumew∈V (T)). Wechooseanexternal
L
edge{w,w′}suchthatfortheeveryleafnodexoftherelocatedsubtree, {ϕ−1(x),ϕ−1(w)} ∈ E(G′). Hence
′ ′′
basedontheconstructionofT formT ,thefollowingequationholds,whichinturnevidencesthecorrectness
ofthelemma.
LA(T′′,G′)−LA(T′,G′)>(l×2×(n−2))−(l×(n−1))>0
From the result of lemma 2.13 one can infer the fact that given layout tree T and mapping ϕ for complete
′
multi-graph G, an optimal tree layout (based on T) for simple graph G can be constructed by subdividing
onlyexternaledgesofT andintroducingsub-treelayoutscorrespondingtothenewverticesofdegree2. Butit
doesnotprovideanyinformationregardingtheexactstructureoftheoptimaltree. Inwhatfollowsandwithout
′
providingallthedetailsoftheproof,wepresentthestructureoftheoptimallayouttreeforG ,constructedfrom
T. Notethatforthesakeofthemaintheoreminthissubsection,wedonotneedtoknowtheexactstructureof
thethreelayoutwithminimumtreelength.
n(n−1)
ThesimplegraphG′containsexactlyl× verticesofdegree2. Everyvertexv ∈G′ofdegreel×(n−1)is
2
directlyconnectedtol×(n−1)verticesofdegree2. TheoptimaltreelayoutT′obtainsfromT,bysubdividing
every external edge {w,w′} ∈ E (T) (where w ∈ V (T)) exactly once with a sub-tree layout containing
E L
l
×(n−1) leaf nodes5. Every leaf node of this subtree correspond to one vertex x ∈ V(G′) with degree 2
2
wherexisdirectlyconnectedtow.
4 Edge e ∈ E(T) is external if it is adjacent to a leaf node of T, and internal otherwise. Let E (T) and E (T) respectively
I E
representthesetofinternalandexternaledgesofT
5Forthesakeofthemaintheoreminthissection,weassumelisaneveninteger.
10