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A Boolean Algebraic Analysis of Fire Protection
Annals of Discrete Mathematics 19 (1984) 215-228 0 Elsevier Science Publishers B.V. (North-Holland)
A BOOLEAN ALGEBRAIC ANALYSIS
B.L. Hulme and A.W.
Shiver
Sandia N a t i o n a l L a b o r a t o r i e s A1 buquerque, NM 87185 U.S.A.
OF FIRE PROTECTION P.J. S l a t e r U n i v e r s i t y o f Alabama H u n t s v i l l e , AL 35899 U.S.A.
I n a comolex f a c i l i t y , such as a n u c l e a r power p l a n t , t h e d e s t r u c t i o n by f i r e o f c e r t a i n c r i t i c a l combinations o f equipment can have p o t e n t i a l l y s e r i o u s consequences. T h i s paper d e s c r i b e s a computational procedure which can be used t o f i n d minimum c o s t ways t o p r o t e c t t h e c r i t i c a l combinat i o n s o f equipment f r o m a s i n g l e - s o u r c e f i r e by p r o t e c t i n g c e r t a i n areas and s t r e n g t h e n i n g c e r t a i n b a r r i e r s a g a i n s t f i r e . The procedure y i e l d s a complete s e t o f optimum s o l u t i o n s by i t e r a t i v e l y computing upper and l o w e r bounds on t h e minimum c o s t . The f i r e p r o t e c t i o n s e t s e v o l v e f r o m Boolean a l g e b r a i c computations which o b t a i n minimum c o s t b l o c k i n g s e t s a s s o c i a t e d w i t h t h e l o w e r bounds w h i l e t h e upper bounds a r e produced by maxflow-mincut c a l c u l a t i o n s i n a network. The problem can be viewed as one o f m i n i m i z a t i o n o v e r t h e cobases o f an independence system.
1.
THE FIRE PROTECTION PROBLEM
The f i r e spread p o s s i b i l i t i e s i n a f a c i l i t y a r e modeled by a f i r e spread network n (V,A),
a weighted d i r e c t e d graph c o n s i s t i n g o f a s e t V o f v e r t i c e s , a s e t A o f
d i r e c t e d a r c s , d i s j o i n t f r o m V b u t j o i n i n g one v e r t e x o f V t o another, and nonn e g a t i v e w e i g h t s a s s o c i a t e d w i t h t h e a r c s and v e r t i c e s .
The v e r t e x s e t V i s
determined by p a r t i t i o n i n g t h e f a c i l i t y l a y o u t i n t o s u i t a b l e areas and a s s i g n i n g one v e r t e x t o each area.
The a r c s e t A i s determined by f i r s t i d e n t i f y i n g which
f i r e areas a r e p h y s i c a l l y a d j a c e n t and t h e n d o i n g combustion c a l c u l a t i o n s [2]
for
each f i r e area t o see whether t h e f u e l load, v e n t i l a t i o n r a t e , room c o n f i g u r a t i o n , e t c . , can s u p p o r t a f i r e s u f f i c i e n t t o p e n e t r a t e t h e s u r r o u n d i n g w a l l s , c e i l i n g o r f l o o r i n t o an a d j a c e n t area.
The a r c s a r e d e f i n e d i n p a i r s .
I f f i r e can
spread i n one o r b o t h d i r e c t i o n s between t h e v e r t i c e s i and j , t h e n b o t h t h e arcs ( i , j )
and ( j , i )
E
A a l t h o u g h t h e y may have d i f f e r e n t weights.
V e r t i c e s where v i t a l components o f s a f e t y systems a r e l o c a t e d a r e c a l l e d t a r g e t vertices.
A minimal s e t o f t a r g e t v e r t i c e s whose d e s t r u c t i o n c o u l d have unaccep-
t a b l e consequences i s c a l l e d a t a r g e t s e t .
The c o l l e c t i o n T = IT1,T 2,...,Tk3
of
t a r g e t s e t s t o be p r o t e c t e d i s t y p i c a l l y determined by f a u l t t r e e a n a l y s i s [1,9]. Being minimal, no t a r g e t s e t Ti c o m p l e t e l y c o n t a i n s a n o t h e r .
Since a l l v e r t i c e s
6.L. Hulme et al.
216
i n a t a r g e t s e t must be d e s t r o y e d i n o r d e r t o i n i t i a t e an u n a c c e p t a b l e e v e n t , p r o t e c t i n g one t a r g e t v e r t e x i s s u f f i c i e n t t o p r o t e c t any t a r g e t s e t t o which i t belongs. The a r c s and v e r t i c e s o f
11
a r e weighted w i t h f i r e p r o t e c t i o n costs.
The v e r t e x
weight, c ( i ) , i s t h e c o s t o f p r e v e n t i n g f i r e f r o m p r o p a g a t i n g t h r o u g h area i as
For example,
w e l l as p r e v e n t i n g f i r e damage t o any v i t a l equipment l o c a t e d t h e r e . t h i s m i g h t be t h e c o s t o f a n adequate s p r i n k l e r system. i n d i v i d u a l arc weight, c ( i , j ) , through the b a r r i e r ( i , j ) ,
Each a r c has an
w h i c h i s t h e c o s t o f p r e v e n t i n g t h e spread o f f i r e
f o r example by i m p r o v i n g t h e f i r e r a t i n g o f t h e
b a r r i e r o r by r e d u c i n g t h e f u e l l o a d i n a r e a i. The a r c s come i n p a i r s , as mentioned above, even though one a r c m i g h t have a z e r o w e i g h t i n d i c a t i n g t h a t i t i s already protected against f i r e . j o i n t arc weight, b ( i , j ) ,
Each p a i r of a r c s , ( i , j ) and ( j , i )
t h e c o s t o f p r o t e c t i n g b o t h ( i , j ) and ( j , i ) .
has a Since
t h e r e may be economies p o s s i b l e i n p r o t e c t i n g both, we have max { c ( i , j ) ,
A network n ' ( V ' , A ' ) in
17'
c(j,i)l 4 b(i,j) 6 c(i,j)
i s a subnetwork o f n ( V , A )
a r e simply r e s t r i c t i o n s from
sequence vOalvla
11 t
o n'.
+
.
c(j,i)
i f V ' s V, A '
C_
(11
A, and t h e w e i g h t s
A path i n n i s an a l t e r n a t i n g
?... akvk o f d i s t i n c t v e r t i c e s vi and d i s t i n c t a r c s ai such t h a t
t o vi, 1 6 i 4 k. D e l e t i o n o f a v e r t e x v f r o m M means t h e removal ai j o i n s v . 1-1 n o t o n l y o f v b u t a l s o o f a l l a r c s i n c i d e n t t o o r f r o m v, w h i l e d e l e t i o n o f a n a r c means removal o f o n l y t h e a r c . The most l i k e l y a c c i d e n t a l f i r e s a r e s i n g l e - s o u r c e f i r e s .
They may s t a r t a t any
v e r t e x and f l o w a l o n g p a t h s o f p o s i t i v e l y w e i g h t e d a r c s and v e r t i c e s .
I f every
v e r t e x i n some t a r g e t s e t has a common predecessor v e r t e x a l o n g such paths, t h e n t h e f a c i l i t y i s vulnerable t o a f i r e propagating from t h a t vertex.
This motivates
the following definitions.
An s - t n e t f o r (Iz,T)
i s a minimal, p o s i t i v e l y weighted, subnetwork a' i n M con-
s i s t i n g o f a source v e r t e x s, a t a r g e t s e t Ti II
such t h a t f o r e v e r y t a Ti
set f o r
(II,T)
E
T, and o t h e r v e r t i c e s and a r c s o f
t h e r e i s a p a t h i n M' f r o m s t o t .
A f i r e protection
i s a s u b s e t R o f a r c s and v e r t i c e s whose d e l e t i o n ( p r o t e c t i o n )
y i e l d s a subnetwork
11-R
n o t c o n t a i n i n g any s - t n e t .
The c o s t o f a f i r e p r o t e c t -
e c t i o n s e t R i s t h e sum o f t h e w e i g h t s o f i t s elements, u s i n g t h e j o i n t w e i g h t b(i,j)
instead o f c ( i , j )
+
c ( j , i ) when b o t h ( i , j )
and ( j , i ) E R.
The minimum c o s t f i r e p r o t e c t i o n problem i s t o f i n d a l l o f t h e minimum c o s t f i r e protection sets f o r
(M,r).
211
A boolean algebraic analysis of fire protection
2.
A BLOCKING SYSTEM
F o l l o w i n g t h e t e r m i n o l o g y o f Edmonds and F u l k e r s o n [4],
we d e f i n e a c l u t t e r s on a
f i n i t e s e t E t o be a c o l l e c t i o n o f subsets o f E, no one o f which c o n t a i n s a n o t h e r . The c l u t t e r R on E c o n s i s t i n g o f t h e minimal subsets o f E having nonempty i n t e r s e c t i o n w i t h e v e r y member o f S i s c a l l e d t h e b l o c k i n g c l u t t e r , o r t h e b l o c k e r , of S.
We s h a l l c a l l t h e members o f R b l o c k i n g s e t s a n d t h e p a i r (S,R) a b l o c k i n g
sys tern
.
F o r p r e s e n t purposes we l e t E = V
t h e s e t o f a r c s and v e r t i c e s i n n.
An s - t
The c o l l e c t i o n S o f a l l t h e s - t n e t s f o r ( n , T ) i s a c l u t t e r
n e t i s a subset o f E.
on E .
u A,
The b l o c k i n g c l u t t e r R o f S i s a c o l l e c t i o n o f minimal subsets o f a r c s and
v e r t i c e s such t h a t each member o f R c o n t a i n s a t l e a s t one element o f e v e r y s - t net. O b v i o u s l y t h e b l o c k i n g s e t s i n R a r e minimal f i r e p r o t e c t i o n s e t s .
Our t a s k i s t o
f i n d t h e minimum c o s t b l o c k i n g s e t s i n R , and we propose t o do t h i s w i t h o u t constructing either
s
o r R s i n c e t h e s e c o l l e c t i o n s can be q u i t e l a r g e .
We a r e i n d e b t e d t o t h e r e f e r e e who p o i n t e d o u t t h a t t h i s problem can a l s o be p l a c e d i n t h e framework o f o p t i m i z a t i o n o v e r independence systems
P O , p.191.
An
independence system S = ( E , I ) i s a f i n i t e s e t E t o g e t h e r w i t h a c o l l e c t i o n I o f if I
subsets o f E c l o s e d under i n c l u s i o n , i . e . , elements o f I a r e c a l l e d independent s e t s .
E
I and J
G
I , then J
E
I.
A maximal independent s e t i s a
The
&.
A subset o f E n o t i n 1 i s c a l l e d a dependent s e t , and a minimal dependent s e t i s
a circuit. E\Bi
The independence system S* = (E,I*)
whose bases a r e t h e complements
o f t h e bases Bi o f S i s c a l l e d t h e dual independence system o f S.
S* i s a cobase o f S, and a c i r c u i t o f S* i s a c o c i r c u i t o f S. system S i s c a l l e d a m a t r o i d i f I and J
E
A base o f
An independence
I w i t h IJI=II 1+1 i m p l i e s t h a t t h e r e
e x i s t s an element j e J \ I such t h a t I IJ { j }e I .
Among t h e many p r o p e r t i e s o f
m a t r o i d s a r e t h e f a c t s t h a t a l l bases have t h e same c a r d i n a l i t y and t h a t t h e greedy algorithm
[lo,
p.3061 w i l l always f i n d a maximum w e i g h t independent s e t
g i v e n any nonnegative w e i g h t f u n c t i o n on E. I t i s known f o r m a t r o i d s [7,
pp.36-371 and more g e n e r a l l y f o r independence
systems t h a t t h e c i r c u i t s and t h e cobases f o r m a b l o c k i n g system. problem, where E =
VU
independence system.
I n the f i r e
A, l e t S, t h e c o l l e c t i o n o f s - t nets, be t h e c i r c u i t s o f an The cobases, which a r e b l o c k i n g s e t s o f S, a r e minimal f i r e
p r o t e c t i o n s e t s , whose complements, t h e maximal p r o t e c t e d subnstworks, a r e bases. A subset (subnetwork)
X
C
E i s independent ( p r o t e c t e d ) i f X does n o t c o n t a i n any X i s dependent ( u n p r o t e c t e d ) . T h i s independence
c i r c u i t ( s - t n e t ) , otherwise
system i s n o t a m a t r o i d because t h e bases can have d i f f e r e n t c a r d i n a l i t i e s . Therefore, t h e greedy a l g o r i t h m w i l l n o t always s o l v e t h e f i r e problem.
Our
i t e r a t i v e a l g o r i t h m , which uses Boolean a l g e b r a t o produce t h e b l o c k i n g s e t s , i s
B. L. Hulme et al
218
d e s c r i b e d i n t h e language o f t h e f i r e problem t o m i n i m i z e o v e r t h e m i n i m a l f i r e p r o t e c t i o n s e t s o b t a i n e d as b l o c k i n g s e t s o f t h e s - t n e t s .
It i s e a s i l y t r a n s -
l a t e d t o an a l g o r i t h m f o r m i n i m i z i n g o v e r t h e cobases o f any independence system, where t h e cobases a r e o b t a i n e d a l g e b r a i c a l l y as b l o c k i n g s e t s o f t h e c i r c u i t s .
3.
3.1.
A SOLUTION ALGORITHM
The B a s i c Prdcedure
The b a s i s f o r o u r a l g o r i t h m can be e x p l a i n e d by a b r i e f d e s c r i p t i o n o f a t h e o r e t i c a l a l g o r i t h m w h i c h o m i t s a l l t h e r e f i n e m e n t s needed f o r a p r a c t i c a l computation. The t h e o r e t i c a l a l g o r i t h m c o n s i s t s o f g e n e r a t i n g a n e s t e d sequence o f subc l u t t e r s Si
in
s,
sl= s* '=... c s i ==... c-s,
(2)
Every b l o c k i n g s e t R e Ri i n t e r s e c t s e v e r y
and t h e i r c o r r e s p o n d i n g b l o c k e r s Ri.
From ( 2 ) i t f o l l o w s t h a t R a l s o i n t e r s e c t s e v e r y member o f
member o f Si.
T h i s means t h a t e v e r y b l o c k i n g s e t R E Ri i s a s u p e r s e t o f Si-,,Si-2,...,S,. some b l o c k i n g s e t f o r e v e r y p r e v i o u s Sj, 1 Q j Q i - 1 , so t h a t min c ( R ) 6 m i n c ( R ) 4 RER R6R
... .c min
R6Ri
c ( R ) .c m i n c ( R )
R e
I n each b l o c k e r Ri t h e l e a s t c o s t b l o c k i n g s e t s Ri,j c ( R ~ , ~=) m i n c ( R ) z Li R=Ri
.
(3)
and t h e i r c o s t ,
,
(4)
Thus, ( 3 ) shows t h a t Li i s a l o w e r bound on t h e minimum c o s t o f
are i d e n t i f i e d . f i r e protection.
Every l e a s t c o s t b l o c k i n g s e t Ri,j
E
Ri
i s t e s t e d t o see i f i t b l o c k s n o t o n l y
e v e r y s - t n e t i n Si b u t a l s o e v e r y s - t n e t i n S.
T h i s means a s k i n g i f M-R
i,j c o n t a i n s any s - t n e t s , a q u e s t i o n answerable by s t a n d a r d p a t h f i n d i n g t e c h n i q u e s . If
11-R.
1 ,j
c o n t a i n s s - t n e t s , some o f them s h o u l d be saved t o u n i t e w i t h Si a t t h e
n e x t stage. Moreover, Ri,j
If
11-R. . c o n t a i n s no s - t n e t s , t h e n Ri,j i s a b l o c k i n g s e t i n R. 1 ,J i s a minimum c o s t member o f R because o f ( 4 ) and ( 3 ) , and hence
R . . i s one o f t h e minimum c o s t f i r e p r o t e c t i o n s e t s which we seek. 1 ,J I f t h e r e a r e any o t h e r minimum c o s t members o f R , t h e y w i l l a l s o b e l o n g t o R i and be i d e n t i f i e d a l o n g w i t h Ri,j.
T h i s can be seen as f o l l o w s .
I f R i s another
member o f R h a v i n g t h e same minimum c o s t , c ( R ) = c ( R ~ , ~ ) ,t h e n R i n t e r s e c t s e v e r y
R i s minimal w i t h respect t o t h e p r o p e r t y o f i n t e r s e c t i n g e v e r y member o f Si. I f t h i s were n o t so, t h e n R would c o n t a i n an element i n a d d i t i o n t o a b l o c k i n g s e t o f Si. From t h e p o s i t i v i t y o f
member o f S and hence e v e r y member o f Si.
A boolean algebraic analysis of fire protection
219
t h e w e i g h t s o f t h e elements i n s - t n e t s , R would have a c o s t c(R) > c ( R ~ , ~ ) a, Being a minimal subset o f E which i n t e r s e c t s every member o f S .
contradiction.
R
E
Ri.
1’
T h e r e f o r e , whenever one optimum s o l u t i o n i n R i s found among t h e l e a s t a l l t h e o t h e r s w i l l be found t h e r e a l s o .
c o s t members o f Ri,
The t h e o r e t i c a l a l g o r i t h m must converge t o a complete s e t o f optimum s o l u t i o n s because S i s f i n i t e and hence t h e number o f i t e r a t i o n s i s f i n i t e . l a r g e t h i s process can r e q u i r e enormous amounts o f t i m e .
When S i s
B e f o r e d i s c u s s i n g ways
t o make t h i s a l g o r i t h m more p r a c t i c a l , we summarize i t s s t e p s as f o l l o w s . Theoretical Algorithm
8, Z
8.
1.
Set i = 0, So =
2.
I f n has s - t n e t s , then save some i n AS
=
a1 ready p r o t e c t e d .
e l s e stop, h a v i n g shown t h a t n i s
3.
Set i
4.
C o n s t r u c t t h e b l o c k e r Ri o f Si.
5.
O b t a i n a l l o f t h e l e a s t c o s t b l o c k i n g s e t s R i,j i n R 1. .
6.
F o r e v e r y Ri,j
=
i+l, Si = S i - l u
0
i f M-R.
ASi.
7.
3.2
If Z =
0,
1 ,j
ASi-,.
o b t a i n e d i n 5, do has no s - t nets, t h e n save R .
1
,j
i n Z e l s e save some s - t n e t s i n
t h e n go t o 3 e l s e stop, having a l l t h e optimum s o l u t i o n s i n Z .
Aspects o f a P r a c t i c a l A l g o r i t h m
The f i r s t p r a c t i c a l m a t t e r t o be c o n s i d e r e d i s t h e presence o f s i n g l e t o n s i n T. Any t a r g e t v e r t e x t which forms a s i n g l e t o n Ti = I t } i n T must be p a r t o f any f i r e p r o t e c t i o n set.
I n p a r t i c u l a r , a l l s i n g l e t o n t a r g e t v e r t i c e s must belong t o
a l l minimum c o s t f i r e p r o t e c t i o n s e t s .
I n o r d e r t o s i m p l i f y t h e computations we
assume t h a t a l l s i n g l e t o n s have been d e l e t e d from T , t h a t t h e c o r r e s p o n d i n g t a r g e t v e r t i c e s have been e i t h e r d e l e t e d f r o m n o r g i v e n z e r o weights, and t h a t these v e r t i c e s and t h e i r c o s t s w i l l be combined w i t h t h e a l g o r i t h m i c r e s u l t s t o f o r m complete s o l u t i o n s t o t h e f i r e p r o t e c t i o n problem. The procedure s t a t e d above i s r e a l l y n o t an a l g o r i t h m u n t i l we s t a t e s p e c i f i c a l l y (a)
how t o s e l e c t t h e new s - t n e t s &‘i-l
(b)
how t o c o n s t r u c t t h e b l o c k e r Ri o f Si.
i n f o r m i n g Si and
These i s s u e s a r e r e l a t e d by t h e f a c t t h a t t h e more s - t n e t s t h a t a r e produced a t On t h e
each stage, t h e l a r g e r i s I S i I and t h e more d i f f i c u l t i t i s t o o b t a i n R i .
B. L. Hulme et al.
220 o t h e r hand, i f ISi/
grows t o o s l o w l y t h e method w i l l be slow t o converge.
Our
approach t o i s s u e ( b ) d e t e r m i n e s o u r approach t o ( a ) , so we s h a l l d e a l f i r s t w i t h (b). T h e o r e t i c a l l y t h e b l o c k e r Ri o f a c l u t t e r Si can be c o n s t r u c t e d a l g e b r a i c a l l y by f o r m i n g t h e Boolean p r o d u c t o v e r t h e members o f Si o f t h e Boolean sum of t h e elements i n t h e members, expanding t h e p r o d u c t by t h e d i s t r i b u t i v e l a w ( a ( b = ab
v ac),
v
c)
and s i m p l i f y i n g t h e sum-of-products by idempotence (aa = a, a V a = a )
and a b s o r p t i o n ( a
v
ab = a ) .
The terms i n t h e r e s u l t i n g d i s j u n c t i v e normal f o r m
correspond t o t h e b l o c k i n g s e t s i n Ri. Both t h e t i m e and space r e q u i r e m e n t s o f t h i s a l g e b r a i c t e c h n i q u e can grow exponent i a l l y w i t h ISi o f Ri,
1.
We have chosen t o combat t h i s g r o w t h by n o t c o n s t r u c t i n g a l l
b u t i n s t e a d o n l y a s u b s e t R i G Ri of b l o c k i n g s e t s h a v i n g a c o s t no g r e a t -
e r t h a n a c u r r e n t upper bound, Ui,
a s s o c i a t e d w i t h some f e a s i b l e s o l u t i o n .
The Boolean a l g e b r a i c m a n i p u l a t i o n language SETS [8] has t h e c a p a b i l i t y n o t o n l y o f expanding and s i m p l i f y i n g Boolean f o r m u l a s b u t a l s o o f t r u n c a t i n g sums-ofp r o d u c t s on any g i v e n n u m e r i c a l t h r e s h o l d f o r t e r m v a l u e s , where t e r m v a l u e s can be computed as sums o f v a r i a b l e v a l u e s .
We a s s o c i a t e f i r e p r o t e c t i o n c o s t s w i t h
t h e Boolean v a r i a b l e s r e p r e s e n t i n g t h e v e r t i c e s and a r c s o f t h e s - t n e t s .
I n any
Boolean sum c o n t a i n i n g t h e v a r i a b l e I - J f o r a r c ( i , j ) we i n c l u d e 1 - 4 r e p r e s e n t i n g t h e j o i n t a r c s ( i , j ) and ( j , i ) , ensures t h a t , i f ( i , j ) and ( j , i )
provided t h a t b ( i , j )
c c(i,j)
c(j3i).
t
This
a r e a t t r a c t i v e choices f o r a b l o c k i n g set, then
t h e c o r r e s p o n d i n g Boolean t e r m w i l l c o n t a i n t h e v a r i a b l e 1 - 4 h a v i n g c o s t b ( i , j ) rather than the product ( I - J ) ( J - I )
having c o s t c ( i , j )
t
c(j,i).
Also, i t i s
e f f i c i e n t t o o m i t f r o m a Boolean sum t h e v a r i a b l e I - J f o r an a r c ( i , j ) , l e a v i n g 1 - 4 f o r t h e a r c p a i r , whenever c ( i , j )
= b(i,j).
a r c s can be p r o t e c t e d f o r t h e same c o s t as t h e s i n g l e a r c . c(j,i) c(j,i),
while
T h i s i s because b o t h
O f course, i f
= 0 so t h a t I - - J has been o m i t t e d f r o m t h e sum because b ( i , j )
= c(i,j)
t
t h e n I - J i s l e f t i n t h e sum.
Thus, we need an upper bound Ui on which t o t r u n c a t e terms so t h a t t h e r e s u l t i n g Boolean f o r m u l a F i , whose terms a r e t h e b l o c k i n g s e t s i n R t , m i g h t be more manageable i n s i z e .
N o t i c e t h a t t h e s e t o f s - t n e t s Si does n o t need t o be saved
f o r t h e n e x t stage.
I t i s enough t o save t h e t r u n c a t e d f o r m u l a Fi.
A t the
i + l - s t stage a Boolean p r o d u c t o v e r t h e new s - t n e t s ASi can be m u l t i p l i e d by Fi, expanded, t r u n c a t e d on Uitl
Q
Ui,
and s i m p l i f i e d t o f o r m Ryt1.
In t h i s way o n l y
b l o c k i n g s e t s w i t h c o s t s l o w enough t o p o s s i b l y produce minimum c o s t s o l u t i o n s a r e c o n s t r u c t e d and saved a t each stage. I t i s i m p o r t a n t i n p r a c t i c e t o n o t i c e t h a t t h e c o m p u t a t i o n o f RY+,
i s much f a s t e r
A boolean algebraic analysis of fire protection
i f , b e f o r e t h e m u l t i p l i c a t i o n by Fi, expanded, t r u n c a t e d on Uitl,
221
t h e Boolean product-of-sums c v e r aSi i s
and s i m p l i f i e d w h i l e repeated use i s made o f t h e
d i s t r i b u t i v e law ( a v b ) ( a V c ) = a V bc. T h i s r e p e a t e d l y saves p r o d u c i n g t h e c r o s s p r o d u c t s ac by e i t h e r t r u n c a t i o n o r a b s o r p t i o n .
v ab and t h e n
d e l e t i n g them
Our need d u r i n g t h e Boolean computation f o r an upper bound c o s t Ui a s s o c i a t e d w i t h a f e a s i b l e s o l u t i o n determines how we deal w i t h i t e m ( a ) .
Constructing s - t
n e t s s i m p l y by p a t h f i n d i n g i n M-R. . does n o t produce a f e a s i b l e s o l u t i o n . We use 1 ,J a s u b r o u t i n e FIRE [5] which computes b o t h f e a s i b l e s o l u t i o n s and s - t n e t s . By u s i n g D i n i c ' s a l g o r i t h m [3] t o f i n d a minimum c u t Q . . s e p a r a t i n g two b l o c k i n g 133
s e t s o f T i n a network r e l a t e d t o n-Ri,j
and weighted w i t h v e r t e x and j o i n t a r c
(M,J)
Qi,j
i s a f i r e protection set f o r
Qi,j
i s m i n i m a l , i t may n o t be a minimum c o s t supplement t o RiYj.
c(Ri , j )
+
u
f o r (M-R. . , T ) . Hence, R . . 1 ,J 133 h a v i n g c o s t c(R. .) + c ( Q ~ , ~ ) A . lthough
weights, FIRE produces a f i r e p r o t e c t i o n s e t Qi,j
1 ,J
However,
c(Qi , j ) i s an upper bound which may be used t o o b t a i n Ui+l.
See [6]
f o r t h e d e t a i l s o f t h e minimum c u t c a l c u l a t i o n s . FIRE produces t h e supplementary s - t n e t s
. 1 ,J
Q.
f o r minimality.
as an immediate byproduct o f t e s t i n g
I n t h e network G = M - R ~ , -~ Qi,j
-
{ z e r o weighted a r c s and
v e r t i c e s } , which c o n t a i n s no s - t nets, each element e e Q . . i s p u t back i n , one 1 ,J a t a time, and a p a t h f i n d e r t e s t s G + e f o r t h e e x i s t e n c e o f an s - t n e t . I f G + e has no s - t n e t , e i s n o t e s s e n t i a l t o Qi,j,
so e i s d e l e t e d f r o m QiYj.
Otherwise,
e i s essential t o Q
because t h e s - t n e t j u s t f o u n d c o n t a i n s e and w i l l be uni, j p r o t e c t e d i f e i s n o t p r o t e c t e d . Thus, FIRE produces o n l y one s - t n e t f o r each
a r c and v e r t e x i n Q. ., h e l p i n g t o keep I A S ~ fI r o m b e i n g t o o l a r g e , and each s - t 1 ,J n e t c o n t a i n s i t s corresponding a r c o r v e r t e x . Consequently, p r o t e c t i n g Q . . p r o t e c t s a l l t h e new s - t n e t s , ASi,
and p r o t e c t i n g Ri,j
i n Si,
so t h a t Ri,jU Qi,j
cost.
The Boolean computation o f Ri+l
c o s t p r o t e c t i o n s e t s Ritl
p r o t e c t s Si+l
,j
= Si
u ASi,
1,J
protects a l l o f the s - t nets b u t perhaps n o t w i t h minimum
a t t h e n e x t stage w i l l produce minimum
f o r Sitl.
The f i n a l p r a c t i c a l m a t t e r concerns Step 6 o f t h e t h e o r e t i c a l a l g o r i t h m . sometimes i n e f f i c i e n t t o t e s t
fl o f
3.3
., p a r t i c u 7 1J I t i s always s u f f i c i e n t t o t e s t o n l y
t h e l e a s t c o s t blocking sets R.
l a r l y when t h e r e i s a l a r g e number o f them. t h e f i r s t one, Ri,,,
It i s
u n t i l t h e f i n a l s t a g e m, and o n l y t h e n t o t e s t a l l Rm,j.
THE ALGORITHM
With t h e above d e t a i l s i n mind we can now s t a t e a much more p r a c t i c a l a l g o r i t h m .
22 2
1. 2.
B. L. Hulme et al,
Set i = 0 , S 0 = 0, Z = 0, L0 = 0 , Fo = 1 (Boolean). Compute Qo = a minimal f i r e protection s e t f o r ( n , T ) , U1 = c ( Q 0 ) = c o s t of Qo, aS0 = an associated c o l l e c t i o n of s - t nets i n by
3.
If U, = 0 , then go t o 16.
4.
Set i = i + l .
5.
Form the Boolean products =
11
t h a t would be protected
Q,, one s-t net f o r each a r c a n d vertex in Qo.
A Ve SenSi-, eeS
,
Fi-1 A Pi-1
Fi
si
( i n e f f e c t constructing
=
si-l U AS^-^).
6.
Expand F i , truncating terms on t h e value U i , and simplify t o form R Y , the family of blocking s e t s R e R i having c(R) 6 U i . Call the truncated equation Fi.
7.
Find the l e a s t c o s t blocking s e t s R i , l , R i , 2 L i = c(R
8. 9.
i,1 ) and U i + l
=
,... , R i , k .
in Rf and set 1
Ui.
For j = 1 , 2 ,... , k i do compute Q i , j = a minimal f i r e protection set for (n-R.
.,T),
1 ,J
c ( Q . , ) = c o s t of Q i , j , 1
,J
nSi = an associated c o l l e c t i o n of s - t n e t s in M-R. t h a t would i, j be protected by Q . ., one s - t net f o r each a r c and vertex 1 ,J i n Q. : 1 ,J’
10.
set Ui+l
11.
if c ( Q ~ , =~ 0, ) t h e n set Z =
12.
i f Z # 0, then go t o 14;
13.
i f Ui+l
= rnin(Ui+l,Li
=
+
c(Q.1 , J. ) ) ;
Zu R.1 ,J.,
and go t o 14;
L i , then go t o 14, e l s e go t o 15.
14. Continue. 1 5 . I f Z = 0, then go t o 4.
16. Stop. Z i s the c o l l e c t i o n of minimum c o s t f i r e protection s e t s , and L i = U . 1+1 i s t h e minimum c o s t .
A boolean algebraic analysis of fire protection
223
4. AN EXAMPLE A f i r e spread network n i s shown i n F i g u r e 1 w i t h t h e j o i n t a r c w e i g h t s enclosed i n parentheses.
F i g u r e 1.
A F i r e Spread Network n
L e t t h e c o l l e c t i o n o f t a r g e t s e t s be
T = IT1,Tzl T1 = I2,31
T2 = {3,81.
There a r e many f i r e p r o t e c t i o n s e t s R f o r ( n , ~ ) ,t h e f o l l o w i n g ones b e i n g easy t o f i n d by i n s p e c t i o n
43
R {2,3 ,83 {31
50.0
I ( 2 , 1 ) ;(3,4),(4,3); (5,6),(6,5) ;(7,8),(8,7)l { ( I 1 3 ) $4 );(3,4) ,(433);(3,5),(5,3)1
27.0
26.0
.
F i n d i n g an optimum s o l u t i o n by i n s p e c t i o n , however, i s d i f f i c u l t even f o r such a small problem as t h i s . Our Boolean a l g e b r a i c a l g o r i t h m f o u n d 11 s - t n e t s b e f o r e c o n v e r g i n g i n t h r e e i t e r a t i o n s t o t h e s e t o f two optimum s o l u t i o n s h a v i n g c o s t 2 2 . 0 .
We summarize t h e
i n t e r m e d i a t e and f i n a l r e s u l t s by d i s p l a y i n g t h e l e a s t c o s t b l o c k i n g s e t s Ri,j, t h e c u t s Qi,j,
t h e bounds Li and Ui,
p r o d u c t o v e r t h e new s - t n e t s ASi-l,
as w e l l as Boolean f o r m u l a s f o r Pi-l, and Fiy
the
t h e t r u n c a t e d d i s j u n c t i v e normal f o r m
Si which have c o s t s 6 Ui, 1 6 i 6 3. As Boolean v a r i a b l e s we use t h e v e r t e x numbers I , t h e i n d i v i d u a l a r c names I-J, and t h e j o i n t
whose terms a r e b l o c k i n g s e t s o f a r c names I--J,
1 6 I , J 6 8.
z = g Fo = 1
(Boolean)
Lo = 0.0
Qo
= I ( 2 , 1 ) ;( 2 A ),(4,2);(3,5),(5,3);(4,6),(6,4)1
U1 = c ( Q 0 ) = 23.0
224
B. L. Hulme et a[. 7 v 8 v 3 - 5 v 5-7 v 7-8)A v 3 v 2-1 v 1-3)A ( 5 v 6 v 4 v 3 v 7 V 8 v 5--6 v 6-4 v V 5-7 v 7-8) A (5v 3 v 7 v 8 v 5-3 v 5-7 v 7-8)A ( 2 v 4 v 3 v 2--4 v 3 - - 4 ) A (3 v 4 v 2 v 3--4 v 2--4)
Po = ( 3
v
5 (2v 1
F,
= Fo
A
5-7
A
Po ( t r u n c a t e d on U1 )
1-3
A 2--4v
5 - 7 A 2-1
A
2--4V
A
A
2--4v
7-8
1-3
7-8 A 2-1 A 2--4V 5-7
A
1-3
A A
3--4
term value 13.0 14.0 14.0 15.0
3--4V
18.0
5 - 7 A 2-1 3--4V 7-8A 1 - 3 A 3--4V
19.0
7-8A 2-1 A 3 - - 4 v 3-5 A 1-3 A 5--6 A 5-3 A 2 - - 4 V 3-5 A 1-3 A 3--4 A 5-3 V
20.0
A 3-5 A 1-3 A 3-5 A 2-1 A 3-5 A 2-1 A
3 - 5 A 2-1
A 5-3 A 2--4V 6-4 A 5-3 A 2--4 v 3 - 4 A 5-.3V 6-4 A 5-3 A 2--4 5-6
19.0 21 .o
22.0 22.0 22.0
23.0 23.0
term value
18.0 19.0 19.0 20.0 21 .o
A boolean algebraic analysis of fire protection
225 22.0 22.0
22 .o
22.0 22.0
22.0 23.0 23.0 23.0 23.0 23.0 23.0 23.0 23.0
term value 22.0 22.0 23.0
23.0 23.0
B. L. Hulme et al.
226
Thus, Z i s a complete s e t o f optimum s o l u t i o n s having c o s t 22.0.
5.
CODE PERFORMANCE
A procedure f i l e , PROTECT, w r i t t e n i n t h e Cyber Control Language, implements o u r a l g o r i t h m on the CDC 7600 [ti]. Although FIRE f i n d s minimum c u t s i n polynomial time w i t h D i n i c ' s O(n4 ) algorithm, where n i s t h e number o f v e r t i c e s , t h e Boolean expansion o f products i n t o d i s j u n c t i v e normal form can r e q u i r e an amount o f time t h a t i s exponential i n t h e number o f s - t n e t s found, even w i t h t r u n c a t i o n . Table I shows t h a t r e a l i s t i c problems have been completely solved i n 60 t o 108 seconds.
We a l s o stopped t h e i t e r a t i o n a f t e r 890 seconds on a r e a l i s t i c problem
having v e r t e x weights which were l a r g e r e l a t i v e t o t h e a r c weights.
In t h i s case
we accepted t h e one f e a s i b l e s o l u t i o n , Ri ,1
as t h e best
U Qi,l,
having c o s t Ui+l
s o l u t i o n o b t a i n a b l e i n t h i s amount o f time.
No. No. oof f No. oof f No. No. oof f No. oof f Minimum Minimum CDC7600 CDC7600 No. No. No. oof f Target Target Target Target No. No. oof f Cost Run Time Time No. oof f No. Prob. No. ss- -t t nneet st s Cost Run Prob. Arcs VVeer tritci ceess Sets Sets I It teer raat ti oi onnss Found Found Solutions S o l u t i o n s (Seconds) (Seconds) No. VVertices e r t i c e s Arcs No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 3 5 4 4 8 8 9 9 20 20 20 20 20 20 20 20 a5 85 85
2 6 12 10 10 20 20 32 32 62 62
62
60 60
60 60 60 512 512 512
2 3 2 4 3 3 4 5 5 5 4 4 4 6 4 5 8 15 1 5. 20
1 2 1 2 1
2
2 4 3 2 3 3 2 3 4 6 7 ia ia 22
2 2 2 3 4 3 4 5 3 9
3
6 5 5 3 3 3 3 6 16
2 6 7 5 4 11 9 10 6 46 12 22 5
24 12 24 12 12 22 20
1 13 10
13.5 13.5 14.8 17.4 20.7 21.2 23.1 35.6 24.9
1 1 1 1 2 56 15 4
20.9 43.1 25.0 38.0 19.9 24.1 21.8 75.5 60.1 107.6
85.6
227
A boolean algebraic analysis of'fire protection
6.
CONCLUSION
As i n d i c a t e d i n S e c t i o n 2, t h e f i r e a l g o r i t h m o f S e c t i o n 3.3 g e n e r a l i z e s t o an a l g o r i t h m f o r f i n d i n g a l l o f t h e minimum w e i g h t cobases o f any independence system whose c i r c u i t s a r e p o s i t i v e l y weighted.
I n a d d i t i o n t o t h e t r a n s l a t i o n o f s - t nets
t o c i r c u i t s and minimal f i r e p r o t e c t i o n s e t s t o cobases, one o t h e r correspondence i s needed.
We assume t h e e x i s t e n c e o f an independence a l g o r i t h m A which, g i v e n
any subset XE E , decides whether o r n o t X i s independent.
We f u r t h e r assume
t h a t , i f X i s dependent, t h e n A produces one c i r c u i t f r o m X.
The a l g o r i t h m A i s
needed i n t h e greedy a l g o r i t h m , which r e p l a c e s t h e FIRE s u b r o u t i n e a t steps 2 and 9.
u Qi,j
T h i s greedy a l g o r i t h m f i n d s n o t o n l y a s u p e r s e t Ri,j
a l s o new c i r c u i t s aSi a t each stage.
o f a cobase b u t
The p r o o f o f c o r r e c t n e s s f o r t h i s general
a l g o r i t h m i s s i m p l y a t r a n s l a t i o n o f t h e argument g i v e n i n S e c t i o n 3.1.
REFERENCES Barlow, R.E., F u s s e l l , J.B. and S i n g p u r w a l l a , N.D., e d i t o r s , R e l i a b i l i t y and F a u l t Tree A n a l y s i s , Proceedings o f t h e Conference on R e l i a b i l i t y and F a u l t Tree A n a l y s i s , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , September 3-7, 1974, SIAM, P h i l a d e l p h i a , 1975.
-
B e r r y , O.L., Nuclear power p l a n t f i r e p r o t e c t i o n p h i l o s o p h y and a n a l y s i s , SAND80-0334, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, May 1980. O i n i c , E.A., A l g o r i t h m f o r s o l u t i o n o f a problem o f maximum f l o w i n a network w i t h power e s t i m a t i o n , S o v i e t Math. Dokl., 11 (1970), 1277-1280. Edmonds, J . , and Fulkerson, D.R., 8 (1970) 299-306.
B o t t l e n e c k extrema, J. C o m b i n a t o r i a l Theory,
Hulme, B.L., S h i v e r , A.W. and S l a t e r , P.J., FIRE: a s u b r o u t i n e f o r f i r e p r o t e c t i o n network a n a l y s i s , SAND81-1261, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, December 1981. Hulme, B.L., Shiver, A.W. and S l a t e r , P.J., Computing minimum c o s t f i r e p r o t e c t i o n , SAND82-0809, Sandia N a t i o n a l Laboratoires, Albuquerque, NM, June 1 9 8 2 Welsh, O.J.A.,
M a t r o i d Theory (Academic Press, London, 1976).
W o r r e l l , R.B., S e t e q u a t i o n t r a n s f o r m a t i o n system (SETS), SLA-73-0028A, N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, January 1975.
Sandia
W o r r e l l , R.B. and Stack, D.W., A SETS u s e r ' s manual f o r t h e f a u l t t r e e analyst, SANO77-2051, Sandia N a t i o n a l L a b o r a t o r i e s , A1 buquerque, NM, November 1978. Zimmermann, U., L i n e a r and C o m b i n a t o r i a l O p t i m i z a t i o n i n Ordered A l g e b r a i c S t r u c t u r e s , v o l . 10, Annals of D i s c r e t e Mathematics (North-Holland, Amsterdam, 1981).
A BOOLEAN ALGEBRAIC ANALYSIS
B.L. Hulme and A.W.
Shiver
Sandia N a t i o n a l L a b o r a t o r i e s A1 buquerque, NM 87185 U.S.A.
OF FIRE PROTECTION P.J. S l a t e r U n i v e r s i t y o f Alabama H u n t s v i l l e , AL 35899 U.S.A.
I n a comolex f a c i l i t y , such as a n u c l e a r power p l a n t , t h e d e s t r u c t i o n by f i r e o f c e r t a i n c r i t i c a l combinations o f equipment can have p o t e n t i a l l y s e r i o u s consequences. T h i s paper d e s c r i b e s a computational procedure which can be used t o f i n d minimum c o s t ways t o p r o t e c t t h e c r i t i c a l combinat i o n s o f equipment f r o m a s i n g l e - s o u r c e f i r e by p r o t e c t i n g c e r t a i n areas and s t r e n g t h e n i n g c e r t a i n b a r r i e r s a g a i n s t f i r e . The procedure y i e l d s a complete s e t o f optimum s o l u t i o n s by i t e r a t i v e l y computing upper and l o w e r bounds on t h e minimum c o s t . The f i r e p r o t e c t i o n s e t s e v o l v e f r o m Boolean a l g e b r a i c computations which o b t a i n minimum c o s t b l o c k i n g s e t s a s s o c i a t e d w i t h t h e l o w e r bounds w h i l e t h e upper bounds a r e produced by maxflow-mincut c a l c u l a t i o n s i n a network. The problem can be viewed as one o f m i n i m i z a t i o n o v e r t h e cobases o f an independence system.
1.
THE FIRE PROTECTION PROBLEM
The f i r e spread p o s s i b i l i t i e s i n a f a c i l i t y a r e modeled by a f i r e spread network n (V,A),
a weighted d i r e c t e d graph c o n s i s t i n g o f a s e t V o f v e r t i c e s , a s e t A o f
d i r e c t e d a r c s , d i s j o i n t f r o m V b u t j o i n i n g one v e r t e x o f V t o another, and nonn e g a t i v e w e i g h t s a s s o c i a t e d w i t h t h e a r c s and v e r t i c e s .
The v e r t e x s e t V i s
determined by p a r t i t i o n i n g t h e f a c i l i t y l a y o u t i n t o s u i t a b l e areas and a s s i g n i n g one v e r t e x t o each area.
The a r c s e t A i s determined by f i r s t i d e n t i f y i n g which
f i r e areas a r e p h y s i c a l l y a d j a c e n t and t h e n d o i n g combustion c a l c u l a t i o n s [2]
for
each f i r e area t o see whether t h e f u e l load, v e n t i l a t i o n r a t e , room c o n f i g u r a t i o n , e t c . , can s u p p o r t a f i r e s u f f i c i e n t t o p e n e t r a t e t h e s u r r o u n d i n g w a l l s , c e i l i n g o r f l o o r i n t o an a d j a c e n t area.
The a r c s a r e d e f i n e d i n p a i r s .
I f f i r e can
spread i n one o r b o t h d i r e c t i o n s between t h e v e r t i c e s i and j , t h e n b o t h t h e arcs ( i , j )
and ( j , i )
E
A a l t h o u g h t h e y may have d i f f e r e n t weights.
V e r t i c e s where v i t a l components o f s a f e t y systems a r e l o c a t e d a r e c a l l e d t a r g e t vertices.
A minimal s e t o f t a r g e t v e r t i c e s whose d e s t r u c t i o n c o u l d have unaccep-
t a b l e consequences i s c a l l e d a t a r g e t s e t .
The c o l l e c t i o n T = IT1,T 2,...,Tk3
of
t a r g e t s e t s t o be p r o t e c t e d i s t y p i c a l l y determined by f a u l t t r e e a n a l y s i s [1,9]. Being minimal, no t a r g e t s e t Ti c o m p l e t e l y c o n t a i n s a n o t h e r .
Since a l l v e r t i c e s
6.L. Hulme et al.
216
i n a t a r g e t s e t must be d e s t r o y e d i n o r d e r t o i n i t i a t e an u n a c c e p t a b l e e v e n t , p r o t e c t i n g one t a r g e t v e r t e x i s s u f f i c i e n t t o p r o t e c t any t a r g e t s e t t o which i t belongs. The a r c s and v e r t i c e s o f
11
a r e weighted w i t h f i r e p r o t e c t i o n costs.
The v e r t e x
weight, c ( i ) , i s t h e c o s t o f p r e v e n t i n g f i r e f r o m p r o p a g a t i n g t h r o u g h area i as
For example,
w e l l as p r e v e n t i n g f i r e damage t o any v i t a l equipment l o c a t e d t h e r e . t h i s m i g h t be t h e c o s t o f a n adequate s p r i n k l e r system. i n d i v i d u a l arc weight, c ( i , j ) , through the b a r r i e r ( i , j ) ,
Each a r c has an
w h i c h i s t h e c o s t o f p r e v e n t i n g t h e spread o f f i r e
f o r example by i m p r o v i n g t h e f i r e r a t i n g o f t h e
b a r r i e r o r by r e d u c i n g t h e f u e l l o a d i n a r e a i. The a r c s come i n p a i r s , as mentioned above, even though one a r c m i g h t have a z e r o w e i g h t i n d i c a t i n g t h a t i t i s already protected against f i r e . j o i n t arc weight, b ( i , j ) ,
Each p a i r of a r c s , ( i , j ) and ( j , i )
t h e c o s t o f p r o t e c t i n g b o t h ( i , j ) and ( j , i ) .
has a Since
t h e r e may be economies p o s s i b l e i n p r o t e c t i n g both, we have max { c ( i , j ) ,
A network n ' ( V ' , A ' ) in
17'
c(j,i)l 4 b(i,j) 6 c(i,j)
i s a subnetwork o f n ( V , A )
a r e simply r e s t r i c t i o n s from
sequence vOalvla
11 t
o n'.
+
.
c(j,i)
i f V ' s V, A '
C_
(11
A, and t h e w e i g h t s
A path i n n i s an a l t e r n a t i n g
?... akvk o f d i s t i n c t v e r t i c e s vi and d i s t i n c t a r c s ai such t h a t
t o vi, 1 6 i 4 k. D e l e t i o n o f a v e r t e x v f r o m M means t h e removal ai j o i n s v . 1-1 n o t o n l y o f v b u t a l s o o f a l l a r c s i n c i d e n t t o o r f r o m v, w h i l e d e l e t i o n o f a n a r c means removal o f o n l y t h e a r c . The most l i k e l y a c c i d e n t a l f i r e s a r e s i n g l e - s o u r c e f i r e s .
They may s t a r t a t any
v e r t e x and f l o w a l o n g p a t h s o f p o s i t i v e l y w e i g h t e d a r c s and v e r t i c e s .
I f every
v e r t e x i n some t a r g e t s e t has a common predecessor v e r t e x a l o n g such paths, t h e n t h e f a c i l i t y i s vulnerable t o a f i r e propagating from t h a t vertex.
This motivates
the following definitions.
An s - t n e t f o r (Iz,T)
i s a minimal, p o s i t i v e l y weighted, subnetwork a' i n M con-
s i s t i n g o f a source v e r t e x s, a t a r g e t s e t Ti II
such t h a t f o r e v e r y t a Ti
set f o r
(II,T)
E
T, and o t h e r v e r t i c e s and a r c s o f
t h e r e i s a p a t h i n M' f r o m s t o t .
A f i r e protection
i s a s u b s e t R o f a r c s and v e r t i c e s whose d e l e t i o n ( p r o t e c t i o n )
y i e l d s a subnetwork
11-R
n o t c o n t a i n i n g any s - t n e t .
The c o s t o f a f i r e p r o t e c t -
e c t i o n s e t R i s t h e sum o f t h e w e i g h t s o f i t s elements, u s i n g t h e j o i n t w e i g h t b(i,j)
instead o f c ( i , j )
+
c ( j , i ) when b o t h ( i , j )
and ( j , i ) E R.
The minimum c o s t f i r e p r o t e c t i o n problem i s t o f i n d a l l o f t h e minimum c o s t f i r e protection sets f o r
(M,r).
211
A boolean algebraic analysis of fire protection
2.
A BLOCKING SYSTEM
F o l l o w i n g t h e t e r m i n o l o g y o f Edmonds and F u l k e r s o n [4],
we d e f i n e a c l u t t e r s on a
f i n i t e s e t E t o be a c o l l e c t i o n o f subsets o f E, no one o f which c o n t a i n s a n o t h e r . The c l u t t e r R on E c o n s i s t i n g o f t h e minimal subsets o f E having nonempty i n t e r s e c t i o n w i t h e v e r y member o f S i s c a l l e d t h e b l o c k i n g c l u t t e r , o r t h e b l o c k e r , of S.
We s h a l l c a l l t h e members o f R b l o c k i n g s e t s a n d t h e p a i r (S,R) a b l o c k i n g
sys tern
.
F o r p r e s e n t purposes we l e t E = V
t h e s e t o f a r c s and v e r t i c e s i n n.
An s - t
The c o l l e c t i o n S o f a l l t h e s - t n e t s f o r ( n , T ) i s a c l u t t e r
n e t i s a subset o f E.
on E .
u A,
The b l o c k i n g c l u t t e r R o f S i s a c o l l e c t i o n o f minimal subsets o f a r c s and
v e r t i c e s such t h a t each member o f R c o n t a i n s a t l e a s t one element o f e v e r y s - t net. O b v i o u s l y t h e b l o c k i n g s e t s i n R a r e minimal f i r e p r o t e c t i o n s e t s .
Our t a s k i s t o
f i n d t h e minimum c o s t b l o c k i n g s e t s i n R , and we propose t o do t h i s w i t h o u t constructing either
s
o r R s i n c e t h e s e c o l l e c t i o n s can be q u i t e l a r g e .
We a r e i n d e b t e d t o t h e r e f e r e e who p o i n t e d o u t t h a t t h i s problem can a l s o be p l a c e d i n t h e framework o f o p t i m i z a t i o n o v e r independence systems
P O , p.191.
An
independence system S = ( E , I ) i s a f i n i t e s e t E t o g e t h e r w i t h a c o l l e c t i o n I o f if I
subsets o f E c l o s e d under i n c l u s i o n , i . e . , elements o f I a r e c a l l e d independent s e t s .
E
I and J
G
I , then J
E
I.
A maximal independent s e t i s a
The
&.
A subset o f E n o t i n 1 i s c a l l e d a dependent s e t , and a minimal dependent s e t i s
a circuit. E\Bi
The independence system S* = (E,I*)
whose bases a r e t h e complements
o f t h e bases Bi o f S i s c a l l e d t h e dual independence system o f S.
S* i s a cobase o f S, and a c i r c u i t o f S* i s a c o c i r c u i t o f S. system S i s c a l l e d a m a t r o i d i f I and J
E
A base o f
An independence
I w i t h IJI=II 1+1 i m p l i e s t h a t t h e r e
e x i s t s an element j e J \ I such t h a t I IJ { j }e I .
Among t h e many p r o p e r t i e s o f
m a t r o i d s a r e t h e f a c t s t h a t a l l bases have t h e same c a r d i n a l i t y and t h a t t h e greedy algorithm
[lo,
p.3061 w i l l always f i n d a maximum w e i g h t independent s e t
g i v e n any nonnegative w e i g h t f u n c t i o n on E. I t i s known f o r m a t r o i d s [7,
pp.36-371 and more g e n e r a l l y f o r independence
systems t h a t t h e c i r c u i t s and t h e cobases f o r m a b l o c k i n g system. problem, where E =
VU
independence system.
I n the f i r e
A, l e t S, t h e c o l l e c t i o n o f s - t nets, be t h e c i r c u i t s o f an The cobases, which a r e b l o c k i n g s e t s o f S, a r e minimal f i r e
p r o t e c t i o n s e t s , whose complements, t h e maximal p r o t e c t e d subnstworks, a r e bases. A subset (subnetwork)
X
C
E i s independent ( p r o t e c t e d ) i f X does n o t c o n t a i n any X i s dependent ( u n p r o t e c t e d ) . T h i s independence
c i r c u i t ( s - t n e t ) , otherwise
system i s n o t a m a t r o i d because t h e bases can have d i f f e r e n t c a r d i n a l i t i e s . Therefore, t h e greedy a l g o r i t h m w i l l n o t always s o l v e t h e f i r e problem.
Our
i t e r a t i v e a l g o r i t h m , which uses Boolean a l g e b r a t o produce t h e b l o c k i n g s e t s , i s
B. L. Hulme et al
218
d e s c r i b e d i n t h e language o f t h e f i r e problem t o m i n i m i z e o v e r t h e m i n i m a l f i r e p r o t e c t i o n s e t s o b t a i n e d as b l o c k i n g s e t s o f t h e s - t n e t s .
It i s e a s i l y t r a n s -
l a t e d t o an a l g o r i t h m f o r m i n i m i z i n g o v e r t h e cobases o f any independence system, where t h e cobases a r e o b t a i n e d a l g e b r a i c a l l y as b l o c k i n g s e t s o f t h e c i r c u i t s .
3.
3.1.
A SOLUTION ALGORITHM
The B a s i c Prdcedure
The b a s i s f o r o u r a l g o r i t h m can be e x p l a i n e d by a b r i e f d e s c r i p t i o n o f a t h e o r e t i c a l a l g o r i t h m w h i c h o m i t s a l l t h e r e f i n e m e n t s needed f o r a p r a c t i c a l computation. The t h e o r e t i c a l a l g o r i t h m c o n s i s t s o f g e n e r a t i n g a n e s t e d sequence o f subc l u t t e r s Si
in
s,
sl= s* '=... c s i ==... c-s,
(2)
Every b l o c k i n g s e t R e Ri i n t e r s e c t s e v e r y
and t h e i r c o r r e s p o n d i n g b l o c k e r s Ri.
From ( 2 ) i t f o l l o w s t h a t R a l s o i n t e r s e c t s e v e r y member o f
member o f Si.
T h i s means t h a t e v e r y b l o c k i n g s e t R E Ri i s a s u p e r s e t o f Si-,,Si-2,...,S,. some b l o c k i n g s e t f o r e v e r y p r e v i o u s Sj, 1 Q j Q i - 1 , so t h a t min c ( R ) 6 m i n c ( R ) 4 RER R6R
... .c min
R6Ri
c ( R ) .c m i n c ( R )
R e
I n each b l o c k e r Ri t h e l e a s t c o s t b l o c k i n g s e t s Ri,j c ( R ~ , ~=) m i n c ( R ) z Li R=Ri
.
(3)
and t h e i r c o s t ,
,
(4)
Thus, ( 3 ) shows t h a t Li i s a l o w e r bound on t h e minimum c o s t o f
are i d e n t i f i e d . f i r e protection.
Every l e a s t c o s t b l o c k i n g s e t Ri,j
E
Ri
i s t e s t e d t o see i f i t b l o c k s n o t o n l y
e v e r y s - t n e t i n Si b u t a l s o e v e r y s - t n e t i n S.
T h i s means a s k i n g i f M-R
i,j c o n t a i n s any s - t n e t s , a q u e s t i o n answerable by s t a n d a r d p a t h f i n d i n g t e c h n i q u e s . If
11-R.
1 ,j
c o n t a i n s s - t n e t s , some o f them s h o u l d be saved t o u n i t e w i t h Si a t t h e
n e x t stage. Moreover, Ri,j
If
11-R. . c o n t a i n s no s - t n e t s , t h e n Ri,j i s a b l o c k i n g s e t i n R. 1 ,J i s a minimum c o s t member o f R because o f ( 4 ) and ( 3 ) , and hence
R . . i s one o f t h e minimum c o s t f i r e p r o t e c t i o n s e t s which we seek. 1 ,J I f t h e r e a r e any o t h e r minimum c o s t members o f R , t h e y w i l l a l s o b e l o n g t o R i and be i d e n t i f i e d a l o n g w i t h Ri,j.
T h i s can be seen as f o l l o w s .
I f R i s another
member o f R h a v i n g t h e same minimum c o s t , c ( R ) = c ( R ~ , ~ ) ,t h e n R i n t e r s e c t s e v e r y
R i s minimal w i t h respect t o t h e p r o p e r t y o f i n t e r s e c t i n g e v e r y member o f Si. I f t h i s were n o t so, t h e n R would c o n t a i n an element i n a d d i t i o n t o a b l o c k i n g s e t o f Si. From t h e p o s i t i v i t y o f
member o f S and hence e v e r y member o f Si.
A boolean algebraic analysis of fire protection
219
t h e w e i g h t s o f t h e elements i n s - t n e t s , R would have a c o s t c(R) > c ( R ~ , ~ ) a, Being a minimal subset o f E which i n t e r s e c t s every member o f S .
contradiction.
R
E
Ri.
1’
T h e r e f o r e , whenever one optimum s o l u t i o n i n R i s found among t h e l e a s t a l l t h e o t h e r s w i l l be found t h e r e a l s o .
c o s t members o f Ri,
The t h e o r e t i c a l a l g o r i t h m must converge t o a complete s e t o f optimum s o l u t i o n s because S i s f i n i t e and hence t h e number o f i t e r a t i o n s i s f i n i t e . l a r g e t h i s process can r e q u i r e enormous amounts o f t i m e .
When S i s
B e f o r e d i s c u s s i n g ways
t o make t h i s a l g o r i t h m more p r a c t i c a l , we summarize i t s s t e p s as f o l l o w s . Theoretical Algorithm
8, Z
8.
1.
Set i = 0, So =
2.
I f n has s - t n e t s , then save some i n AS
=
a1 ready p r o t e c t e d .
e l s e stop, h a v i n g shown t h a t n i s
3.
Set i
4.
C o n s t r u c t t h e b l o c k e r Ri o f Si.
5.
O b t a i n a l l o f t h e l e a s t c o s t b l o c k i n g s e t s R i,j i n R 1. .
6.
F o r e v e r y Ri,j
=
i+l, Si = S i - l u
0
i f M-R.
ASi.
7.
3.2
If Z =
0,
1 ,j
ASi-,.
o b t a i n e d i n 5, do has no s - t nets, t h e n save R .
1
,j
i n Z e l s e save some s - t n e t s i n
t h e n go t o 3 e l s e stop, having a l l t h e optimum s o l u t i o n s i n Z .
Aspects o f a P r a c t i c a l A l g o r i t h m
The f i r s t p r a c t i c a l m a t t e r t o be c o n s i d e r e d i s t h e presence o f s i n g l e t o n s i n T. Any t a r g e t v e r t e x t which forms a s i n g l e t o n Ti = I t } i n T must be p a r t o f any f i r e p r o t e c t i o n set.
I n p a r t i c u l a r , a l l s i n g l e t o n t a r g e t v e r t i c e s must belong t o
a l l minimum c o s t f i r e p r o t e c t i o n s e t s .
I n o r d e r t o s i m p l i f y t h e computations we
assume t h a t a l l s i n g l e t o n s have been d e l e t e d from T , t h a t t h e c o r r e s p o n d i n g t a r g e t v e r t i c e s have been e i t h e r d e l e t e d f r o m n o r g i v e n z e r o weights, and t h a t these v e r t i c e s and t h e i r c o s t s w i l l be combined w i t h t h e a l g o r i t h m i c r e s u l t s t o f o r m complete s o l u t i o n s t o t h e f i r e p r o t e c t i o n problem. The procedure s t a t e d above i s r e a l l y n o t an a l g o r i t h m u n t i l we s t a t e s p e c i f i c a l l y (a)
how t o s e l e c t t h e new s - t n e t s &‘i-l
(b)
how t o c o n s t r u c t t h e b l o c k e r Ri o f Si.
i n f o r m i n g Si and
These i s s u e s a r e r e l a t e d by t h e f a c t t h a t t h e more s - t n e t s t h a t a r e produced a t On t h e
each stage, t h e l a r g e r i s I S i I and t h e more d i f f i c u l t i t i s t o o b t a i n R i .
B. L. Hulme et al.
220 o t h e r hand, i f ISi/
grows t o o s l o w l y t h e method w i l l be slow t o converge.
Our
approach t o i s s u e ( b ) d e t e r m i n e s o u r approach t o ( a ) , so we s h a l l d e a l f i r s t w i t h (b). T h e o r e t i c a l l y t h e b l o c k e r Ri o f a c l u t t e r Si can be c o n s t r u c t e d a l g e b r a i c a l l y by f o r m i n g t h e Boolean p r o d u c t o v e r t h e members o f Si o f t h e Boolean sum of t h e elements i n t h e members, expanding t h e p r o d u c t by t h e d i s t r i b u t i v e l a w ( a ( b = ab
v ac),
v
c)
and s i m p l i f y i n g t h e sum-of-products by idempotence (aa = a, a V a = a )
and a b s o r p t i o n ( a
v
ab = a ) .
The terms i n t h e r e s u l t i n g d i s j u n c t i v e normal f o r m
correspond t o t h e b l o c k i n g s e t s i n Ri. Both t h e t i m e and space r e q u i r e m e n t s o f t h i s a l g e b r a i c t e c h n i q u e can grow exponent i a l l y w i t h ISi o f Ri,
1.
We have chosen t o combat t h i s g r o w t h by n o t c o n s t r u c t i n g a l l
b u t i n s t e a d o n l y a s u b s e t R i G Ri of b l o c k i n g s e t s h a v i n g a c o s t no g r e a t -
e r t h a n a c u r r e n t upper bound, Ui,
a s s o c i a t e d w i t h some f e a s i b l e s o l u t i o n .
The Boolean a l g e b r a i c m a n i p u l a t i o n language SETS [8] has t h e c a p a b i l i t y n o t o n l y o f expanding and s i m p l i f y i n g Boolean f o r m u l a s b u t a l s o o f t r u n c a t i n g sums-ofp r o d u c t s on any g i v e n n u m e r i c a l t h r e s h o l d f o r t e r m v a l u e s , where t e r m v a l u e s can be computed as sums o f v a r i a b l e v a l u e s .
We a s s o c i a t e f i r e p r o t e c t i o n c o s t s w i t h
t h e Boolean v a r i a b l e s r e p r e s e n t i n g t h e v e r t i c e s and a r c s o f t h e s - t n e t s .
I n any
Boolean sum c o n t a i n i n g t h e v a r i a b l e I - J f o r a r c ( i , j ) we i n c l u d e 1 - 4 r e p r e s e n t i n g t h e j o i n t a r c s ( i , j ) and ( j , i ) , ensures t h a t , i f ( i , j ) and ( j , i )
provided t h a t b ( i , j )
c c(i,j)
c(j3i).
t
This
a r e a t t r a c t i v e choices f o r a b l o c k i n g set, then
t h e c o r r e s p o n d i n g Boolean t e r m w i l l c o n t a i n t h e v a r i a b l e 1 - 4 h a v i n g c o s t b ( i , j ) rather than the product ( I - J ) ( J - I )
having c o s t c ( i , j )
t
c(j,i).
Also, i t i s
e f f i c i e n t t o o m i t f r o m a Boolean sum t h e v a r i a b l e I - J f o r an a r c ( i , j ) , l e a v i n g 1 - 4 f o r t h e a r c p a i r , whenever c ( i , j )
= b(i,j).
a r c s can be p r o t e c t e d f o r t h e same c o s t as t h e s i n g l e a r c . c(j,i) c(j,i),
while
T h i s i s because b o t h
O f course, i f
= 0 so t h a t I - - J has been o m i t t e d f r o m t h e sum because b ( i , j )
= c(i,j)
t
t h e n I - J i s l e f t i n t h e sum.
Thus, we need an upper bound Ui on which t o t r u n c a t e terms so t h a t t h e r e s u l t i n g Boolean f o r m u l a F i , whose terms a r e t h e b l o c k i n g s e t s i n R t , m i g h t be more manageable i n s i z e .
N o t i c e t h a t t h e s e t o f s - t n e t s Si does n o t need t o be saved
f o r t h e n e x t stage.
I t i s enough t o save t h e t r u n c a t e d f o r m u l a Fi.
A t the
i + l - s t stage a Boolean p r o d u c t o v e r t h e new s - t n e t s ASi can be m u l t i p l i e d by Fi, expanded, t r u n c a t e d on Uitl
Q
Ui,
and s i m p l i f i e d t o f o r m Ryt1.
In t h i s way o n l y
b l o c k i n g s e t s w i t h c o s t s l o w enough t o p o s s i b l y produce minimum c o s t s o l u t i o n s a r e c o n s t r u c t e d and saved a t each stage. I t i s i m p o r t a n t i n p r a c t i c e t o n o t i c e t h a t t h e c o m p u t a t i o n o f RY+,
i s much f a s t e r
A boolean algebraic analysis of fire protection
i f , b e f o r e t h e m u l t i p l i c a t i o n by Fi, expanded, t r u n c a t e d on Uitl,
221
t h e Boolean product-of-sums c v e r aSi i s
and s i m p l i f i e d w h i l e repeated use i s made o f t h e
d i s t r i b u t i v e law ( a v b ) ( a V c ) = a V bc. T h i s r e p e a t e d l y saves p r o d u c i n g t h e c r o s s p r o d u c t s ac by e i t h e r t r u n c a t i o n o r a b s o r p t i o n .
v ab and t h e n
d e l e t i n g them
Our need d u r i n g t h e Boolean computation f o r an upper bound c o s t Ui a s s o c i a t e d w i t h a f e a s i b l e s o l u t i o n determines how we deal w i t h i t e m ( a ) .
Constructing s - t
n e t s s i m p l y by p a t h f i n d i n g i n M-R. . does n o t produce a f e a s i b l e s o l u t i o n . We use 1 ,J a s u b r o u t i n e FIRE [5] which computes b o t h f e a s i b l e s o l u t i o n s and s - t n e t s . By u s i n g D i n i c ' s a l g o r i t h m [3] t o f i n d a minimum c u t Q . . s e p a r a t i n g two b l o c k i n g 133
s e t s o f T i n a network r e l a t e d t o n-Ri,j
and weighted w i t h v e r t e x and j o i n t a r c
(M,J)
Qi,j
i s a f i r e protection set f o r
Qi,j
i s m i n i m a l , i t may n o t be a minimum c o s t supplement t o RiYj.
c(Ri , j )
+
u
f o r (M-R. . , T ) . Hence, R . . 1 ,J 133 h a v i n g c o s t c(R. .) + c ( Q ~ , ~ ) A . lthough
weights, FIRE produces a f i r e p r o t e c t i o n s e t Qi,j
1 ,J
However,
c(Qi , j ) i s an upper bound which may be used t o o b t a i n Ui+l.
See [6]
f o r t h e d e t a i l s o f t h e minimum c u t c a l c u l a t i o n s . FIRE produces t h e supplementary s - t n e t s
. 1 ,J
Q.
f o r minimality.
as an immediate byproduct o f t e s t i n g
I n t h e network G = M - R ~ , -~ Qi,j
-
{ z e r o weighted a r c s and
v e r t i c e s } , which c o n t a i n s no s - t nets, each element e e Q . . i s p u t back i n , one 1 ,J a t a time, and a p a t h f i n d e r t e s t s G + e f o r t h e e x i s t e n c e o f an s - t n e t . I f G + e has no s - t n e t , e i s n o t e s s e n t i a l t o Qi,j,
so e i s d e l e t e d f r o m QiYj.
Otherwise,
e i s essential t o Q
because t h e s - t n e t j u s t f o u n d c o n t a i n s e and w i l l be uni, j p r o t e c t e d i f e i s n o t p r o t e c t e d . Thus, FIRE produces o n l y one s - t n e t f o r each
a r c and v e r t e x i n Q. ., h e l p i n g t o keep I A S ~ fI r o m b e i n g t o o l a r g e , and each s - t 1 ,J n e t c o n t a i n s i t s corresponding a r c o r v e r t e x . Consequently, p r o t e c t i n g Q . . p r o t e c t s a l l t h e new s - t n e t s , ASi,
and p r o t e c t i n g Ri,j
i n Si,
so t h a t Ri,jU Qi,j
cost.
The Boolean computation o f Ri+l
c o s t p r o t e c t i o n s e t s Ritl
p r o t e c t s Si+l
,j
= Si
u ASi,
1,J
protects a l l o f the s - t nets b u t perhaps n o t w i t h minimum
a t t h e n e x t stage w i l l produce minimum
f o r Sitl.
The f i n a l p r a c t i c a l m a t t e r concerns Step 6 o f t h e t h e o r e t i c a l a l g o r i t h m . sometimes i n e f f i c i e n t t o t e s t
fl o f
3.3
., p a r t i c u 7 1J I t i s always s u f f i c i e n t t o t e s t o n l y
t h e l e a s t c o s t blocking sets R.
l a r l y when t h e r e i s a l a r g e number o f them. t h e f i r s t one, Ri,,,
It i s
u n t i l t h e f i n a l s t a g e m, and o n l y t h e n t o t e s t a l l Rm,j.
THE ALGORITHM
With t h e above d e t a i l s i n mind we can now s t a t e a much more p r a c t i c a l a l g o r i t h m .
22 2
1. 2.
B. L. Hulme et al,
Set i = 0 , S 0 = 0, Z = 0, L0 = 0 , Fo = 1 (Boolean). Compute Qo = a minimal f i r e protection s e t f o r ( n , T ) , U1 = c ( Q 0 ) = c o s t of Qo, aS0 = an associated c o l l e c t i o n of s - t nets i n by
3.
If U, = 0 , then go t o 16.
4.
Set i = i + l .
5.
Form the Boolean products =
11
t h a t would be protected
Q,, one s-t net f o r each a r c a n d vertex in Qo.
A Ve SenSi-, eeS
,
Fi-1 A Pi-1
Fi
si
( i n e f f e c t constructing
=
si-l U AS^-^).
6.
Expand F i , truncating terms on t h e value U i , and simplify t o form R Y , the family of blocking s e t s R e R i having c(R) 6 U i . Call the truncated equation Fi.
7.
Find the l e a s t c o s t blocking s e t s R i , l , R i , 2 L i = c(R
8. 9.
i,1 ) and U i + l
=
,... , R i , k .
in Rf and set 1
Ui.
For j = 1 , 2 ,... , k i do compute Q i , j = a minimal f i r e protection set for (n-R.
.,T),
1 ,J
c ( Q . , ) = c o s t of Q i , j , 1
,J
nSi = an associated c o l l e c t i o n of s - t n e t s in M-R. t h a t would i, j be protected by Q . ., one s - t net f o r each a r c and vertex 1 ,J i n Q. : 1 ,J’
10.
set Ui+l
11.
if c ( Q ~ , =~ 0, ) t h e n set Z =
12.
i f Z # 0, then go t o 14;
13.
i f Ui+l
= rnin(Ui+l,Li
=
+
c(Q.1 , J. ) ) ;
Zu R.1 ,J.,
and go t o 14;
L i , then go t o 14, e l s e go t o 15.
14. Continue. 1 5 . I f Z = 0, then go t o 4.
16. Stop. Z i s the c o l l e c t i o n of minimum c o s t f i r e protection s e t s , and L i = U . 1+1 i s t h e minimum c o s t .
A boolean algebraic analysis of fire protection
223
4. AN EXAMPLE A f i r e spread network n i s shown i n F i g u r e 1 w i t h t h e j o i n t a r c w e i g h t s enclosed i n parentheses.
F i g u r e 1.
A F i r e Spread Network n
L e t t h e c o l l e c t i o n o f t a r g e t s e t s be
T = IT1,Tzl T1 = I2,31
T2 = {3,81.
There a r e many f i r e p r o t e c t i o n s e t s R f o r ( n , ~ ) ,t h e f o l l o w i n g ones b e i n g easy t o f i n d by i n s p e c t i o n
43
R {2,3 ,83 {31
50.0
I ( 2 , 1 ) ;(3,4),(4,3); (5,6),(6,5) ;(7,8),(8,7)l { ( I 1 3 ) $4 );(3,4) ,(433);(3,5),(5,3)1
27.0
26.0
.
F i n d i n g an optimum s o l u t i o n by i n s p e c t i o n , however, i s d i f f i c u l t even f o r such a small problem as t h i s . Our Boolean a l g e b r a i c a l g o r i t h m f o u n d 11 s - t n e t s b e f o r e c o n v e r g i n g i n t h r e e i t e r a t i o n s t o t h e s e t o f two optimum s o l u t i o n s h a v i n g c o s t 2 2 . 0 .
We summarize t h e
i n t e r m e d i a t e and f i n a l r e s u l t s by d i s p l a y i n g t h e l e a s t c o s t b l o c k i n g s e t s Ri,j, t h e c u t s Qi,j,
t h e bounds Li and Ui,
p r o d u c t o v e r t h e new s - t n e t s ASi-l,
as w e l l as Boolean f o r m u l a s f o r Pi-l, and Fiy
the
t h e t r u n c a t e d d i s j u n c t i v e normal f o r m
Si which have c o s t s 6 Ui, 1 6 i 6 3. As Boolean v a r i a b l e s we use t h e v e r t e x numbers I , t h e i n d i v i d u a l a r c names I-J, and t h e j o i n t
whose terms a r e b l o c k i n g s e t s o f a r c names I--J,
1 6 I , J 6 8.
z = g Fo = 1
(Boolean)
Lo = 0.0
Qo
= I ( 2 , 1 ) ;( 2 A ),(4,2);(3,5),(5,3);(4,6),(6,4)1
U1 = c ( Q 0 ) = 23.0
224
B. L. Hulme et a[. 7 v 8 v 3 - 5 v 5-7 v 7-8)A v 3 v 2-1 v 1-3)A ( 5 v 6 v 4 v 3 v 7 V 8 v 5--6 v 6-4 v V 5-7 v 7-8) A (5v 3 v 7 v 8 v 5-3 v 5-7 v 7-8)A ( 2 v 4 v 3 v 2--4 v 3 - - 4 ) A (3 v 4 v 2 v 3--4 v 2--4)
Po = ( 3
v
5 (2v 1
F,
= Fo
A
5-7
A
Po ( t r u n c a t e d on U1 )
1-3
A 2--4v
5 - 7 A 2-1
A
2--4V
A
A
2--4v
7-8
1-3
7-8 A 2-1 A 2--4V 5-7
A
1-3
A A
3--4
term value 13.0 14.0 14.0 15.0
3--4V
18.0
5 - 7 A 2-1 3--4V 7-8A 1 - 3 A 3--4V
19.0
7-8A 2-1 A 3 - - 4 v 3-5 A 1-3 A 5--6 A 5-3 A 2 - - 4 V 3-5 A 1-3 A 3--4 A 5-3 V
20.0
A 3-5 A 1-3 A 3-5 A 2-1 A 3-5 A 2-1 A
3 - 5 A 2-1
A 5-3 A 2--4V 6-4 A 5-3 A 2--4 v 3 - 4 A 5-.3V 6-4 A 5-3 A 2--4 5-6
19.0 21 .o
22.0 22.0 22.0
23.0 23.0
term value
18.0 19.0 19.0 20.0 21 .o
A boolean algebraic analysis of fire protection
225 22.0 22.0
22 .o
22.0 22.0
22.0 23.0 23.0 23.0 23.0 23.0 23.0 23.0 23.0
term value 22.0 22.0 23.0
23.0 23.0
B. L. Hulme et al.
226
Thus, Z i s a complete s e t o f optimum s o l u t i o n s having c o s t 22.0.
5.
CODE PERFORMANCE
A procedure f i l e , PROTECT, w r i t t e n i n t h e Cyber Control Language, implements o u r a l g o r i t h m on the CDC 7600 [ti]. Although FIRE f i n d s minimum c u t s i n polynomial time w i t h D i n i c ' s O(n4 ) algorithm, where n i s t h e number o f v e r t i c e s , t h e Boolean expansion o f products i n t o d i s j u n c t i v e normal form can r e q u i r e an amount o f time t h a t i s exponential i n t h e number o f s - t n e t s found, even w i t h t r u n c a t i o n . Table I shows t h a t r e a l i s t i c problems have been completely solved i n 60 t o 108 seconds.
We a l s o stopped t h e i t e r a t i o n a f t e r 890 seconds on a r e a l i s t i c problem
having v e r t e x weights which were l a r g e r e l a t i v e t o t h e a r c weights.
In t h i s case
we accepted t h e one f e a s i b l e s o l u t i o n , Ri ,1
as t h e best
U Qi,l,
having c o s t Ui+l
s o l u t i o n o b t a i n a b l e i n t h i s amount o f time.
No. No. oof f No. oof f No. No. oof f No. oof f Minimum Minimum CDC7600 CDC7600 No. No. No. oof f Target Target Target Target No. No. oof f Cost Run Time Time No. oof f No. Prob. No. ss- -t t nneet st s Cost Run Prob. Arcs VVeer tritci ceess Sets Sets I It teer raat ti oi onnss Found Found Solutions S o l u t i o n s (Seconds) (Seconds) No. VVertices e r t i c e s Arcs No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 3 5 4 4 8 8 9 9 20 20 20 20 20 20 20 20 a5 85 85
2 6 12 10 10 20 20 32 32 62 62
62
60 60
60 60 60 512 512 512
2 3 2 4 3 3 4 5 5 5 4 4 4 6 4 5 8 15 1 5. 20
1 2 1 2 1
2
2 4 3 2 3 3 2 3 4 6 7 ia ia 22
2 2 2 3 4 3 4 5 3 9
3
6 5 5 3 3 3 3 6 16
2 6 7 5 4 11 9 10 6 46 12 22 5
24 12 24 12 12 22 20
1 13 10
13.5 13.5 14.8 17.4 20.7 21.2 23.1 35.6 24.9
1 1 1 1 2 56 15 4
20.9 43.1 25.0 38.0 19.9 24.1 21.8 75.5 60.1 107.6
85.6
227
A boolean algebraic analysis of'fire protection
6.
CONCLUSION
As i n d i c a t e d i n S e c t i o n 2, t h e f i r e a l g o r i t h m o f S e c t i o n 3.3 g e n e r a l i z e s t o an a l g o r i t h m f o r f i n d i n g a l l o f t h e minimum w e i g h t cobases o f any independence system whose c i r c u i t s a r e p o s i t i v e l y weighted.
I n a d d i t i o n t o t h e t r a n s l a t i o n o f s - t nets
t o c i r c u i t s and minimal f i r e p r o t e c t i o n s e t s t o cobases, one o t h e r correspondence i s needed.
We assume t h e e x i s t e n c e o f an independence a l g o r i t h m A which, g i v e n
any subset XE E , decides whether o r n o t X i s independent.
We f u r t h e r assume
t h a t , i f X i s dependent, t h e n A produces one c i r c u i t f r o m X.
The a l g o r i t h m A i s
needed i n t h e greedy a l g o r i t h m , which r e p l a c e s t h e FIRE s u b r o u t i n e a t steps 2 and 9.
u Qi,j
T h i s greedy a l g o r i t h m f i n d s n o t o n l y a s u p e r s e t Ri,j
a l s o new c i r c u i t s aSi a t each stage.
o f a cobase b u t
The p r o o f o f c o r r e c t n e s s f o r t h i s general
a l g o r i t h m i s s i m p l y a t r a n s l a t i o n o f t h e argument g i v e n i n S e c t i o n 3.1.
REFERENCES Barlow, R.E., F u s s e l l , J.B. and S i n g p u r w a l l a , N.D., e d i t o r s , R e l i a b i l i t y and F a u l t Tree A n a l y s i s , Proceedings o f t h e Conference on R e l i a b i l i t y and F a u l t Tree A n a l y s i s , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , September 3-7, 1974, SIAM, P h i l a d e l p h i a , 1975.
-
B e r r y , O.L., Nuclear power p l a n t f i r e p r o t e c t i o n p h i l o s o p h y and a n a l y s i s , SAND80-0334, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, May 1980. O i n i c , E.A., A l g o r i t h m f o r s o l u t i o n o f a problem o f maximum f l o w i n a network w i t h power e s t i m a t i o n , S o v i e t Math. Dokl., 11 (1970), 1277-1280. Edmonds, J . , and Fulkerson, D.R., 8 (1970) 299-306.
B o t t l e n e c k extrema, J. C o m b i n a t o r i a l Theory,
Hulme, B.L., S h i v e r , A.W. and S l a t e r , P.J., FIRE: a s u b r o u t i n e f o r f i r e p r o t e c t i o n network a n a l y s i s , SAND81-1261, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, December 1981. Hulme, B.L., Shiver, A.W. and S l a t e r , P.J., Computing minimum c o s t f i r e p r o t e c t i o n , SAND82-0809, Sandia N a t i o n a l Laboratoires, Albuquerque, NM, June 1 9 8 2 Welsh, O.J.A.,
M a t r o i d Theory (Academic Press, London, 1976).
W o r r e l l , R.B., S e t e q u a t i o n t r a n s f o r m a t i o n system (SETS), SLA-73-0028A, N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, January 1975.
Sandia
W o r r e l l , R.B. and Stack, D.W., A SETS u s e r ' s manual f o r t h e f a u l t t r e e analyst, SANO77-2051, Sandia N a t i o n a l L a b o r a t o r i e s , A1 buquerque, NM, November 1978. Zimmermann, U., L i n e a r and C o m b i n a t o r i a l O p t i m i z a t i o n i n Ordered A l g e b r a i c S t r u c t u r e s , v o l . 10, Annals of D i s c r e t e Mathematics (North-Holland, Amsterdam, 1981).