O(n logn) implementation of an approximation
for the Prize-Colleting Steiner Tree Problem
PauloFeolo
CristinaG.Fernandes y
CarlosE. Ferreira z
JoséCoelhodePina
February 2002
Abstrat
We give alow-leveldesriptionof an O(n 2
logn)implementation of Johnson,Minko and
Phillips'approximationalgorithmforthePrize-ColletingSteinerTreeProblem.
1 Introdution
ThePrize-Colleting SteinerTree Problemis anextension of theSteinerTree Problemwhere eah
vertex left out of the tree pays a penalty. The goal is to nd a tree whih minimizes the sum
of its edge osts and the penalties for the verties left out of the tree. Johnson, Minko and
Phillips [2 ℄ presenteda 2-approximation for the this problembasedon the primal-dualsheme. In
this manusript, we desribe indetailsan O(n 2
logn)implementation of this algorithm.
Weadoptthenotationused[1 ℄,whihissummarizedbelow. Westartwithaformaldenitionofthe
problem. Consider a graphG=(V;E), afuntion from E into Q
(non-negative rationals) and
afuntion fromV into Q
. For anysubset F ofE andanysubset W of V,let(F) :=
P
e2F
e
and (W) :=
P
w2W
w
. The Prize-Colleting Steiner Tree Problem(pst) onsists of the
following: given G,,and ,nda treeT inGsuh that
(E
T
)+(V nV
T
) isminimum:
(V
H
and E
H
denotethe vertexand edgesets ofa graph H.)
An edge is internal to a partition P of V if both of its ends are in the same element of P. All
other edges areexternal to P. For anyexternal edge, there aretwo elements of P ontaining its
ends. Weall these two elementstheextremes of theedgeinP.
AolletionLofsubsetsofV islaminarif,foranytwoelementsL
1 andL
2
ofL,eitherL
1
\L
2
=;
orL
1 L
2 orL
1 L
2
. TheolletionofmaximalelementsofalaminarolletionLwillbedenoted
by L
. So,L
isa olletionofdisjointsubsets ofV. Let S
Ldenote theunionof allsets inL.
DepartamentodeCiêniadaComputação,InstitutodeMatemátiaeEstatístia,UniversidadedeSãoPaulo,Rua
doMatão1010,05508-090 SãoPaulo/SP,Brazil. E-mail: {pf,ris,ef,oelho}ime.usp. br. Researhsupported
inpartbyPRONEX/CNPq664107/1997-4(Brazil).
y
ResearhsupportedinpartbyCNPqPro.301174/97-0(Brazil).
z
ResearhsupportedinpartbyCNPqPro.300752/94-6(Brazil).
ForanyolletionLofsubsetsofV andanysubsetX ofV,letX:=VnX,L :=fL2L:LXg
and L
X
:=fL 2L:LXg. When X =fvg, we write L v
and L
v
instead, andwhen X =V
T or
X=V
T
,wewriteT orT instead. ForanyeinE,letL(e):=fL2L:e2Æ
G
Lg;where Æ
G
Lstands
for thesetofedges ofG withone end inLand the otherinL. For anyfuntion y fromL into Q
and anysubolletion MofL,let y(M):=
P
L2M y(L).
We saythaty respets afuntion dened onE (relative to L)if
y(L(e))
e
for eaheinE. (1)
An edgeeistight for y ifequalityholds in(1).
We sayy respets afuntion dened on V (relative to L)if
y(L L
) (L) for eah L inL. (2)
(3)
2 Johnson, Minko and Phillips' algorithm
In itshigh-level desriptionbelow, we refer to analgorithm Pruning whose high-leveldesription
we omit. It orresponds to the seond phase of the primal-dual sheme, where edges are deleted
fromthetree produed inthe rst phase.
Johnson, Minko and Phillips' algorithm reeives G, , and returns a tree T in G suh that
(E
T
) +(V
T
) 2opt (pst (G;;)): Eah iteration starts with a spanning forest F in G, a
laminar olletion L of subsets ofV with S
L =V,a subolletion S of L, and afuntion y from
L into Q
. The rstiteration startswith F =(V;;), L=ffvg:v2Vg, S =;, and y =0. Eah
iteration onsistsof thefollowing:
Case 1: jL
nSj>1.
Let"bethe largest numberinQ
suhthatthe funtion y 0
dened by
y 0
L
=
y
L
+"; ifL2L
nS
y
L
; otherwise
respets and.
Subase 1A:some edgeeexternal to L
istight for y 0
.
Let L
1 and L
2
be the extremes of e in L
. Set y 0
L
1 [L
2
:= 0 and start a new
iterationwithF+e,L[fL
1 [L
2 g,S,y
0
intherolesofF,L,S,yrespetively.
Subase 1B:some element L ofL
nS istight fory 0
.
Start a new iteration with F, L, S [fLg, y 0
in the roles of F, L, S, y
respetively.
Case 2: jL
nSj=1.
Let M be the only element of L
nS. Call subalgorithm Pruning with arguments
F \M, L M
, and S M
. The subalgorithm returns a subolletion Z of S M
. Return
T :=(F \M) S
Z and stop.
Hereis thelistof variables and funtionsusedbythe algorithm:
1. L
1
;:::;L
N
arenonemptysubsetsofV
G
suhthatL
1
[[L
N
=V
G
and,foreahpairi<j,
eitherL
i L
j or L
i
\L
j
=;,whene N <2n,where n:=jV
G
j. Eah L
i
is representedbya
bit vetor aswell asbya linked list. (In thehigh-level version of thealgorithm given in[1 ℄,
fL
1
;:::;L
N
g isdenotedbyL.)
2. Asubset F of E
G
,represented asa doubly-linked list(abit vetorwould be toolong). Sine
(V
G
;F) is aforest, jFj<n.
3. A bitvetor [1::N℄ suhthat [i℄=1 i L
i
is amaximal element offL
1
;:::;L
N
g. (Inthe
high-levelversionof thealgorithm, thisset ofmaximal elementsis denotedby L
.)
4. An array d indexed by V
G
with values in Q
. (In terms of the high-level notation, d[v℄ :=
L
v
P
L2L:v2L y
L
for eah vertexv.)
5. AfuntionResidualCostthattakesedgesintoQ
: uponreeivinganedgeuv,thefuntion
returns the number
uv
d[u℄ d[v℄. Of ourse this an be implemented to run in O(1)
time. (We do not treat ResidualCost as an array beause we annot aord to update
ResidualCost every timed hanges.)
6. An array [1::N℄ with values in Q
.
1
(In terms of the high-level notation, [i℄ =
P
v2L
i [v℄
P
SL
i y
S .)
7. Abitvetor [1::N℄suh thatif[i℄=0 then[i℄=0. WesaythatL
i
isative i[i℄=1.
(Interms ofthe high-level notation,[i℄=0 iL
i 2S.)
8. Avariable mxAtive reordstheardinalityof thesetfi:1iN;[i℄=1;[i℄=1g.
9. AnarrayA[1::N;1::N℄whoseelementsaresetsof atmostone edgeeah. Morespeially,
for i6=j suhthat(i)=(j)=1,
ifÆ(L
i
)\Æ(L
j
)=; thenA[i;j℄=A[j;i℄=;;
otherwise, A[i;j℄ = A[j;i℄ = fuvg where uv is an element of Æ(L
i
)\ Æ(L
j ) that
minimizes ResidualCost (uv).
10. AfuntionKeydened onf1;:::;Ngf1;:::;Ngasfollows: ifA[i;j℄=;thenKey(i;j)=
1; else Key(i;j) =ResidualCost(uv), where uv is the unique edge in A[i;j℄. Of ourse
this funtionan beimplemented to runinO(1) time.
11. For eahisuhthat[i℄=1,therearetwosubsetsof f1;:::;Ng denotedbyH
0
[i℄andH
1 [i℄.
For eah h,the setH
h
[i℄onsistsof allj 6=isuh that
[j℄=1; [j℄=h; A[i;j℄6=;:
EahsetH
h
[i℄isorganizedasamin-heap, thekeyofeahelement jbeingKey (i;j).
2
Hene,
therstelement ofH
h
[i℄minimizes Key(i;).
1
Johnson,MinkoandPhillipssaythisisthesurplusofLi.
2
Johnson,MinkoandPhillipssaythatthekeyofjisthedeit oftheonlyedgeinA[i;j℄.
p2H
h
[i℄ istrueor false. Moreover, ifthestatement istrue,we assumethatthedeletionof
pfromH
h
[i℄ande arriedoutinO(logn) time. (Thisiseasy toimplement: foreahi, eah
h,and eah pinf1;:::;Ng,maintaintheloation of pinH
h [i℄.)
4 Main funtions
Theore ofthe algorithm is givenbythenext funtions.
PCST-low-level (G;;)
1 Iniialization()
2 N mxAtive n
3 while mxAtive >1 do at most 2niterations
4 OneIteration()
5 (X;F) Pruning ()
6 return X and F
The number of iterations is 2n beause the sum 2mxAtive +mxInative,where mxInative
istheardinality of fi:1iN;[i℄ =1;[i℄=0g,starts at 2nand stritly dereases witheah
iteration.
Iniialization()
01 n jV
G j
02 i 0
03 for eahv inV
G do
04 d[v℄ 0
05 i i+1
06 L
i
fvg
07 o[v℄ i
08 [i℄ [i℄ 1
09 [i℄
v
10 for eahiinf2;:::;ngdo
11 for eahj inf1;:::;i 1g do
12 A[i;j℄ ;
13 Key(i;j)=1
14 for eahiinf2;:::;ngdo
15 for eahuv inÆ(L
i ) do
16 ifo[u℄=i
17 thenj o[v℄
18 else j o[u℄
19 ifKey(i;j)>ResidualCost(uv)
20 thenA[i;j℄ A[j;i℄ fuvg
22 H
0 [i℄ ;
23 for eahiinf1;:::;ngdo
24 H
1 [i℄ ;
25 for eahj inf1;:::;ng figdo
26 ifA[i;j℄6=;thenH
1
[i℄ H
1
[i℄[fjg
The total time spent exeuting lines 1420 is O(m) = O(n 2
). The total time spent building the
heap H
1
[i℄inlines2021isO(n). Thetotal spentbyIniialization is O(n 2
).
OneIteration() eah all takes O(nlogn) time
01 "
0
"
00
1
02 for eahp inf1;:::;Ng suh that[p℄=[p℄=1do
03 if"
0
>[p℄
04 then"
0
[p℄
05 p
0
p
06 ifH
0 [p℄6=;
07 then let q be the rst element ofH
0 [p℄
08 if"
00
>Key (p;q)
09 then"
00
Key (p;q)
10 p
00
p
11 q
00
q
12 ifH
1 [p℄6=;
13 thenlet q be therst element ofH
1 [p℄
14 if"
00
>
1
2
Key(p;q)
15 then"
00 1
2
Key (p;q)
16 p
00
p
17 q
00
q
18 " min("
0
;"
00
)
19 for eahp inf1;:::;Ng suh that[p℄=[p℄=1do
20 [p℄ [p℄ "
21 for eahv inL
p do
22 d[v℄ d[v℄+"
23 no need to rebuildheapsH
0
and H
1
24 if"="
0
25 thenSubCase1B(p 0
) takestimeO(nlogn)
26 else SubCase1A (p 00
;q 00
) takestime O(nlogn)
Taken together,allexeutionsofline 22onsume O(n)time. Thetotal spentbyOneIteration is
O(nlogn).
p q
01 letuv be the unique element ofA[p;q℄
02 F F [fuvg
03 L
N+1 L
p [L
q
04 [p℄ [q℄ 0 L
p and L
q
are nolonger maximal
05 [N+1℄ 1 nowL
N+1
ismaximal
06 if[q℄=1 thenmxAtive mxAtive 1
07 [N+1℄ [p℄+[q℄
08 [N+1℄ 1 now L
N+1
isative
09 for eahiinf1;:::;Ng suhthat[i℄ =1do
10 ifKey(p;i)Key(q;i)
11 thenA[N+1;i℄ A[i;N +1℄ A[p;i℄
12 else A[N+1;i℄ A[i;N +1℄ A[q;i℄
13 for eahh inf0;1g do
14 H
h
[N +1℄ H
h
[p℄ time O(1)
15 H
h
[N +1℄ H
h
[N+1℄ fqg time O(logn)
16 for eahiinH
h [q℄do
17 ifi2= H
h
[N+1℄
18 thenH
h
[N+1℄ H
h
[N+1℄[fig time O(logn)
19 ifi2H
h
[N+1℄andKey (N+1;i)>Key(q;i)
20 thenDerease-Key(H
h
[N+1℄;i;Key(q;i))
21 for eahiinf1;:::;Ng suhthat[i℄ =1do
22 H
0
[i℄ H
0
[i℄ fqg time O(logn)
23 H
1
[i℄ H
1
[i℄ fp;qg timeO(logn)
24 ifKey(i;N +1)<1
25 thenH
1
[i℄ H
1
[i℄[fN+1g timeO(logn)
26 N N +1
SubCase1B(p) deativateL
p
01 [p℄ 0
02 mxAtive mxAtive 1
03 for eahiinf1;:::;Ng fpg do
04 ifp2H
1 [i℄
05 thenH
1
[i℄ H
1
[i℄ fpg timeO(logn)
06 H
0
[i℄ H
0
[i℄[fpg time O(logn)
Pruning() O(n ) time
01 for i N downto 1do reverse delete
02 if[i℄=0 L
i
is inative
03 thendegree 0
04 for eah uv inF do O(n)time
05 ifjfu;vg\L
i
j=1 O(1)time
06 then degree degree+1
07 degreee is theardinalityof F \Æ(L
i )
08 ifdegree 1
09 thenfor eah uv inF do
10 ifjfu;vg\L
i j1
11 thenF F fuvg
12 ifF 6=;
13 thenletX be the setof verties ofG[F℄
14 else letx be a vertexthatmaximizes
x
15 X fxg
16 return(X;F)
Referenes
[1℄ P. Feolo, C.G. Fernandes, C.E. Ferreira, and J. de Pina. Approximation algorithms for
the prize-olleting Steiner tree problem. Available at http://www.ime.usp.br/ris/publ/.
Submitted, 2004.
[2℄ D.S. Johnson, M. Minko, and S. Phillips. The prize olleting Steiner tree problem: theory
and pratie. InSymposium on Disrete Algorithms,pages 760769, 2000.
