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Carnegie Mellon University

Research Showcase @ CMU Computer Science Department

School of Computer Science

1983

A robust sorting network Larry Rudolph Carnegie Mellon University

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CMU-CS-84-104

University l . » f c r ^ * ?

Pittsburgh

G

PA 1 5 2 1 3 - 3 8 9 0

A Robust Sorting Network Larry Rudoiph Department of C o m p u t e r S c i e n c e C a r n e g i e - M e l l o n University A u g u s t 1983

Abstract B e g i n n i n g with the r e c e n t l y i n t r o d u c e d ' b a l a n c e d sorting network* that sorts an input v e c t o r of s i z e N=-2 a n d c o n s i s t s of n identical b l o c k s w h e r e e a c h block is c o m p o s e d of n p h a s e s of N/2 c o m p a r a t o r s per p h a s e , w e p r o p o s e a s h u f f l e - e x c h a n g e t y p e layout consisting of a s i n g l e block with t h e output r e c i r c u l a t e d b a c k as input until sorting is a c h i e v e d . T h e main a d v a n t a g e of the p r o p o s e d d e s i g n is that n o c o m p a r a t o r in the network is critical in the s e n s e that a n y faulty c o m p a r a t o r c a n be b y p a s s e d w i t h o u t d i s t u r b i n g the functionality of the n e t w o r k (just its s p e e d ) . T h e novelty of the d e s i g n is that the r o b u s t n e s s is d e r i v e d from the u n d e r l y i n g algorithm. T h e network will sort in Lhe p r e s e n c e of m a n y faulty c o m p a r a t o r s . M o r e o v e r , of the NlogN/2 c o m p a r a t o r s , only A' pairs of c o m p a r a t o r s a r e critical. T h a t is, the n e t w o r k fails only w h e n b o t h c o m p a r a t o r s in a n y of these pairs fail. T h e s e results e n a b l e o n e to build large sorting n e t w o r k s on a single wafer s o that a high p e r c e n t a g e of the f a b r i c a t e d w a f e r s c a n b e u s e d ; s o m e of the w a f e r s wiil sort v e r y quickly (the o n e s with no faulty c o m p o n e n t s ) , most will sort at s o m e w h a t s l o w e r than optimal s p e e d s , but only a few will fail to b e useful as s o r t i n g n e t w o r k s ( d u e to too many, b a d l y p l a c e d faults). n

1

T h e a d v e n t of V L S I t e c h n o l o g y is impacting almost all a s p e c t s of s o c i e t y . Unfortunately, o n e of the major p r o b l e m s facing V L S I t e c h n o l o g y increasing manufacturing c o s t s is t h e low yield in mass p r o d u c t i o n of c h i p s a n d wafers. T h a t is, a l t h o u g h it is c h e a p to p r o d u c e large quantities of a single c h i p o r wafer, only a small fraction of them function c o r r e c t l y . T h e rest are r e n d e r e d useless d u e to r a n d o m flaws i n t r o d u c e d during the fabrication p r o c e s s . T h e flaws tend to h a v e the most a d v e r s e effect on the active elements (e.g., gates, transistors) a n d less effect o n the wires. O n e w a y of increasing t h e yield is b y employing d e s i g n s that function d e s p i t e fabrication flaws. W e p r e s e n t a layout for a sorting network that c a n withstand many faulty c o m p o n e n t s . A l t h o u g h a small network c a n usually fit o n a single c h i p , a m u c h larger network c o u l d b e made to fit o n a single wafer. P r e v i o u s l y k n o w n sorting networks s h a r e d the p r o p e r t y that almost all the c o m p o n e n t s (comparators) are c r u c i a l , a n d s o large networks p r o d u c e d o n a single wafer w o u l d be e x p e n s i v e d u e to the resulting low yield. T h i s is not the c a s e with o u r layout; most faulty c o m p a r a t o r s c a n b e b y p a s s e d a n d still allow the n e t w o r k to sort, albeit s o m e w h a t s l o w e r . Related w o r k c a n b e classified into t w o different areas, sorting n e t w o r k s and fault-tolerant systems. T h e r e has b e e n m u c h r e s e a r c h in sorting n e t w o r k s a n d the related area of parallel sorting algorithms. B a t c h e r [ B a t c h e r 68] i n t r o d u c e d the Bitonic network as well as the O d d - E v e n network (see also [ K n u t h 68]) b o t h requiring 0 ( [ log A ] ) s t e p s to sort input v e c t o r s of s i z e N (see also, H o n g and S e d g e w i c k [ H o n g a n d S e d g e w i c k 82] and Perl [Perl 83] for additional insights into s u c h networks). T h e r e has also b e e n m u c h r e s e a r c h in sorting algorithms for parallel p r o c e s s o r s , s o m e parts of w h i c h are relevant to sorting networks, for e x a m p l e , Valiant [Valiant 75], B o r o d i n a n d H o p c r o f t [ B o r o d i n a n d H o p c r o f t 82], a n d Kruskal [Kruskal 83]. Rief a n d Valiant [Reif a n d Valiant 83] g i v e an algorithm that requires 0 ( log A O e x p e c t e d time to sort o n a particular t y p e of network, w h e r e a s , Ajtai et. al [Ajtai el al 83] recently s h o w e d that there a r e 0(N\ogN) s i z e d networks that c a n sort in 0 ( log AO steps, a l t h o u g h the large c o n s t a n t s make their network impractical. W i n s l o w a n d C h o w [ W i n s l o w a n d C h o w 83] review a n d c o m p a r e v a r i o u s sorting m a c h i n e s that make different a s s u m p t i o n s a b o u t h o w the input a n d the output are c o n n e c t e d to the m a c h i n e . 2

T h e s t r u c t u r e of the p a p e r is as follows: W e first d e s c r i b e the ' b a l a n c e d sorting network', w h i c h has recently b e e n i n t r o d u c e d [ D o w d et al 83a, D o w d et al 83b]. T h e 'crucial c o m p a r a t o r s ' are then identified a n d their effect a n a l y z e d . T h e third s e c t i o n first r e v i e w s the layout p r o p o s e d in the i n t r o d u c t o r y p a p e r a n d then modifies the layout in t w o w a y s : first to r e d u c e the n u m b e r of critical c o m p a r a t o r s a n d t h e n to eliminate all of t h e m . A n analysis of the i n c r e a s e d yield is also p r e s e n t e d .

1. The Balanced Sorting Network In this s e c t i o n w e review the d e s i g n a n d layout of the " b a l a n c e d sorting n e t w o r k " i n t r o d u c e d b y D o w d et. al [ D o w d et al 83a]. T h e network requires [ log N\ s t a g e s of N/2 c o m p a r a t o r s to s o r t N items a n d c o n s i s t s of a s e q u e n c e of log N identical merging b l o c k s , w h e r e e a c h block p o s s e s s e s a highly regular, r e c u r s i v e d e s i g n (see F i g u r e 1-1 for a merging network of s i z e 16). A n o v e l a s p e c t of the network is that t h e b l o c k s a r e identical not smaller r e c u r s i v e v e r s i o n s as in t h o s e of B a t c h e r [ B a t c h e r 68]. 2

Specifically, a b a s i c unit of the n e t w o r k is a t w o input, t w o o u t p u t c o m p a r a t o r transforming the arbitrary o r d e r of the t w o input elements into n o n d e c r e a s i n g o r d e r . E a c h phase of t h e b a l a n c e d merging n e t w o r k is c o m p o s e d of N/2 of these c o m p a r a t o r s with the first p h a s e c o m p a r i n g elements x ( 0 ) with x ( N - l ) , x ( l ) with x(N-2), • • • , x ( A V 2 - l ) with x(N/2), w h e r e x is the input v e c t o r . T a k i n g the a p p r o a c h of an 'oblivious* algorithm in that e v e n t h o u g h the first p h a s e d o e s not g u a r a n t e e a

2

0 -* 1 2 3 4— 1 5 6 7 Input 8 9 10 — 11 12 13 14

Phase 3 Phase 4

Phase 2

Phase 1

, 1. 1

ul

J

i

. I —

J

.—

1

i i — a —

4

—a

• —

Figure 1 -1:

Output

I —

J

.—

1 1 i . 1i t — p - ^

A B a l a n c e d M e r g i n g B l o c k of S i z e 16

partition of the input into t w o halves, w e p r e t e n d that it d o e s a n d s o c o n t i n u e to apply the s a m e p r o c e d u r e to both halves of the output of the first p h a s e r e c u r s i v e l y . T h u s , log N p h a s e s c o m p r i s e a merging network. W e n u m b e r the p h a s e s from 1 to log N. F i g u r e 1-1 d e p i c t s a b a l a n c e d merging network for N-16

elements using K n u t h ' s [ K n u t h 68]

c o m p a r a t o r - n e t w o r k representation w h e r e horizontal lines r e p r e s e n t the input lines x ( z ) , 0 < / < Af, a n d vertical lines r e p r e s e n t c o m p a r i s o n s b e t w e e n the elements o n the c o r r e s p o n d i n g input lines.

Since

the output of a merging network (from n o w o n w e call s u c h a merging network a b l o c k ) may not b e s o r t e d , w e c o n t i n u e to apply t h e s e b l o c k s until sorting is o b t a i n e d .

( F i g u r e 1-2 s h o w s the full

b a l a n c e d sorting network for s i z e 8.) E a c h b l o c k , as its name implies, is a merging network, h o w e v e r , it is not easily o b s e r v e d e x a c t l y w h a t is being m e r g e d . T h e first p h a s e of the merging network a p p l i e d to a recursively balanced v e c t o r partitions the elements s o that t h e N/2 smallest elements a r e in t h e first half of t h e v e c t o r a n d t h e N/2 largest elements are in the s e c o n d half of t h e v e c t o r . M o r e o v e r , e a c h half is r e c u r s i v e l y b a l a n c e d s o that e a c h s u b s e q u e n t p h a s e acts r e c u r s i v e l y to sort the input. T h e b a l a n c e d sorting network is v e r y similar to the bitonic a n d t h e o d d - e v e n sorting n e t w o r k s i n t r o d u c e d b y B a t c h e r [ B a t c h e r 68]. c o m p o s e d of N/2 c o m p a r a t o r s .

T h e s e n e t w o r k s also c o n s i s t of [ log N]

1

s t a g e s with a s t a g e

M o r e o v e r , they are b o t h build u p o n merging n e t w o r k s , h o w e v e r ,

d e s p i t e their similarity, there is n o permutation b e t w e e n the input lines of either of B a t c h e r ' s t w o n e t w o r k s a n d the b a l a n c e d sorting network.

T h e d i f f e r e n c e s b e t w e e n the b a l a n c e d network a n d

t h o s e of B a t c h e r a r e evident from t h e following lemma w h i c h is satisfied b y neither the o d d - e v e n n o r bitonic m e r g e n e t w o r k s .

3

Block 1

Block 2

Block 3

Input

Sorted Output

Figure 1 -2:

A B a l a n c e d S o r t Network of S i z e 8

L e m m a 1: i) If n o e x c h a n g e o c c u r s d u r i n g a block, t h e n the input of the b l o c k is s o r t e d . ii) A network w h i c h s o r t s a n y input c a n b e c o n s t r u c t e d by serially c o m p o s i n g a finite n u m b e r of b l o c k s . Proof: i) A s i n g l e b l o c k performs all c o m p a r i s o n s x ( / - 1) with x(/) for 1 < i< n (among others) a n d t h u s if n o e x c h a n g e o c c u r s the input must b e s o r t e d . ii) E a c h e x c h a n g e d e c r e a s e s the n u m b e r of inversions (that is, pair of elements w h i c h are out of o r d e r ) . S i n c e a permutation has at most ( ? ) inversions, part (i) implies that ) b l o c k s suffice to sort. • In addition to differentiating b e t w e e n the sorting networks, this lemma is important in two respects. It d e m o n s t r a t e s that only s o m e of the c o m p a r i s o n s a r e n e e d e d to sort. It also s u g g e s t a p r o c e d u r e for d e c i d i n g w h e n the o u t p u t is s o r t e d . T h e following implementation strategy, w h i c h is a s s u m e d t h r o u g h o u t the rest of the paper, arises from these o b s e r v a t i o n s . S i n c e a s u c c e s s i o n of identical b l o c k s a r e r e q u i r e d , only o n e block is actually n e e d e d . T h e o u t p u t of the b l o c k is recirculated b a c k as input (see F i g u r e 1-3). M o r e o v e r , by the first part of t h e lemma, the d e c i s i o n to recirculate c a n b e b a s e d o n w h e t h e r a n y e x c h a n g e s o c c u r r e d within a block. Not only d o e s this allow a faster c o m p l e t i o n time for certain input v e c t o r s and the elimination of a l o g i V c o u n t e r , but it also e n a b l e s a m o r e fault tolerant network, as will b e s h o w n later.

2. Critical Comparators In this s e c t i o n w e identify the 'critical c o m p a r a t o r s ' of the recirculating network (see F i g u r e 1-3). S i n c e m a n y fabricated c h i p s , o r more significantly, wafers, are likely to contain faulty c o m p a r a t o r s , it is d e s i r a b l e that the fabricated p r o d u c t still sort o n c e the faulty c o m p a r a t o r s a r e b y p a s s e d . Recall that w e are c o n s i d e r i n g a recirculating network consisting of o n e b l o c k of c o m p a r a t o r s with the o u t p u t r e c i r c u l a t e d b a c k w h e n e v e r there is at least o n e e x c h a n g e o c c u r r i n g in the block. We c o m p a r e a c o m p l e t e b a l a n c e d sorting network with o n e missing s o m e of its c o m p a r a t o r s . T h e term

4

Output

Input

Figure 1 -3:

iteration

A S i n g l e B l o c k B a l a n c e d S o r t N e t w o r k of S i z e 8

is u s e d to indicate the m o v e m e n t of d a t a t h r o u g h o n e block of the c o m p l e t e network a n d t h e

term pass refers to the m o v e m e n t of d a t a t h r o u g h o n e b l o c k of the incomplete network. W h e n s o m e c o m p a r a t o r s are missing, there will usually b e m o r e p a s s e s than iterations to p r o d u c e the s o r t e d output for a g i v e n input. C o n s i d e r a size 8 network with an input v e c t o r x consisting of all z e r o s e x c e p t for a o n e in t h e fourth position, i.e. x ( 3 ) = 1 o r x = [0,0,0,1,0,0,0,0] (the result after the s o r t s h o u l d b e [0,0,0,0,0,0,0,1]). It is e a s y to s e e that t h e x ( 3 ) : x ( 4 ) c o m p a r i s o n (a first p h a s e c o m p a r i s o n ) is c r u c i a l . W i t h o u t it, the 1 will n e v e r c h a n g e its p o s i t i o n . O n the o t h e r h a n d , the x [ 4 ] : x [ 7 ] c o m p a r a t o r (a s e c o n d p h a s e c o m p a r a t o r ) is not c r u c i a l .

In the first b l o c k , the first p h a s e e x c h a n g e s x [ 3 ] with x [ 4 ] , the next p h a s e d o e s

nothing, the third p h a s e e x c h a n g e s x [ 4 ] with x [ 5 ] . A s e c o n d pass t h r o u g h the block will p r o d u c e t h e s o r t e d o u t p u t using t h e x [ 5 ] : x [ 6 ] c o m p a r a t o r in the s e c o n d p h a s e a n d the x [ 6 ] : x [ 7 ] c o m p a r a t o r in t h e third p h a s e . B e f o r e presenting t h e main t h e o r e m , w e i n t r o d u c e s o m e notation a n d q u o t e a few lemma's p r o v e d in [ D o w d et al 83a]. • G r e e k letters r e p r e s e n t a string of bits with a s u p e r s c r i p t to indicate the n u m b e r of bits. F o r e x a m p l e 1, lrV(a ^"" 00)= ln>( a -JfV- 10) = w

2

n

2

Une> (pJ- 0a -JQ) Line> (pi~ 1 a 'J\) 2

l r V U " ~ ^ ~ 0 1 ) = Line> (fii' 0a ^l) lrV( a 'Jfi^ 11) = Line> (pj~ 1 a^O).

n

2

2

n

n

2

2

n

2

A natural question to ask is w h i c h c o m p a r a t o r s in the layout c o r r e s p o n d to the critical c o m p a r a t o r s of the network. It is c l e a r that in t h e last s t a g e all the c o m p a r a t o r s are critical. T h e following t h e o r e m s u p p l i e s the general a n s w e r . T h e o r e m 1 3 : In the t stage, the critical c o m p a r a t o r s are those of the form: th

(t=l) ( l < / < logAO (/ = log AO

10"- 0:01"- 1 lO"-'- ^-^: I0 " " p^ l p»~ 0:fi -n 2

2

1

n

l

(

l

l

n

P r o o f : W e s h o w that the 'ln()' values of the a b o v e indices c o r r e s p o n d to the ' L i n e O ' v a l u e s w h o s e i n d i c e s are t h o s e of the n e t w o r k s critical c o m p a r a t o r s . L e m m a 12 a n d Definition 2 are u s e d in the following equalities. Ca§g(/=1) ln (1 O ^ O ) = L i n e H l 0 0 ) InHl 0 " - 1 ) = L i n e H O I " ! ) x

H

2

1 1

2

C a s e (1 < / < log AO T h e r e a r e t w o c a s e s arising from t h e l o w o r d e r bit of ft*" . 1

Subcase

(p'-^P'-iQ) ln'(1 O ^ - ^ j S ^ O O ) ln'(1 C - ' - ^ ' ^ O I )

Subcase

(p'- =p• P v e c t o r that c a u s e s all c o m p a r a t o r s to s w a p during the first pass is 2

T

h

e

n

t

h

e

i n

u t

17

1 2 3 4

Layout (i) Layout Layout (iii) Layout (iv)

5000

Size of Network (N)

probability

of a comparator

F i g u re 4 - 4 :

fault (p) a .01

Probability of Network W o r k i n g

18

probability

of a comparator

Fig u re 4 - 5 :

fault (p) - .001

Probability of N e t w o r k W o r k i n g

19

20

defined as follows: x(i) = i n

v

i„_ , i _ , i . , • • • 2

n

3

w

4

In other w o r d s , every other bit is c o m p l e m e n t e d . Let y be the output v e c t o r after o n e pass t h o u g h the block. If y ( i ) =

• • • , i ^ , i^, • • •, then it is easy to identify the p h a s e k c o m p a r a t o r that failed to + 1

swap. W h e n actually c o n s t r u c t i n g a s i n g l e block recirculating sorting network as outlined a b o v e , o n e n e e d not initially permute the input b y T or r s i n c e the initial input is u n o r d e r e d . If this is the c a s e then the test input v e c t o r just p r e s e n t e d must first b e p e r m u t e d b y T. 3

G i v e n an input that f o r c e s e v e r y c o m p a r a t o r to e x e c u t e a s w a p , it is then possible to d e s i g n a self modifying circuit. A single input bit indicates that the circuit is in d e b u g m o d e . While in this m o d e e a c h c o m p a r a t o r tests to s e e that it e x e c u t e s a s w a p . If a c o m p a r a t o r d o e s not s w a p then the c o m p a r a t o r c a n automatically disable itself (i.e. put itself into b y p a s s m o d e ) .

6. Conclusion A l t h o u g h the b a l a n c e d sorting network has the s a m e time a n d s p a c e requirements as that of the bitonic sorting network, w e h a v e s h o w n that its a d v a n t a g e s are more than c o s m e t i c . T h e network c a n be realized as a highly robust s h u f f l e - e x c h a n g e layout. It is t h u s p o s s i b l e to p r o d u c e a fairly large sorting network on a single wafer s o that most of the wafers fabricated c a n be u s e d . T h e major 'trick' u s e d in o u r d e s i g n w a s the alternation of roles p l a y e d by e a c h c o m p a r a t o r . It w o u l d b e interesting to find other algorithms that c a n exploit this trick. A s the requirements for l a r g e - s c a l e parallel p r o c e s s i n g g r o w s , s o d o e s the n e e d to d e v e l o p 'robust* algorithms that c a n function in the p r e s e n c e of many failures. A n o t h e r requirement a p p e a r s to b e a simply c o m p u t e d function that indicates termination as well as the requirement for p r o g r e s s to o c c u r at e a c h iteration. A n immediate application of the results of this p a p e r may be routing networks. O u r sorting network c a n be c o n s i d e r e d to be a routing network with e a c h c o m p a r a t o r s e n d i n g an input value to the output port c o r r e s p o n d i n g to a certain bit in the destination a d d r e s s (instead of as the result of a c o m p a r i s o n ) . A l t h o u g h a single s h u f f l e - e x c h a n g e b l o c k c a n route an single input to any single output, s o m e permutations of N inputs require more than o n e pass. P e r h a p s o u r method c a n be u s e d to build a r o b u s t routing network?

^ h i s may not be the case if the input is assumed to be 'almost* sorted.

21

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[ B a t c h e r 68]

[ B e n e s 65]

Ajtai, M., J . K o m l o s , a n d E. S z e m e r e d i . A n C ( n log n) soritng Network. In 15th Annual ACM Symposium on Theory B a t c h e r , K. E. Sorting n e t w o r k s a n d their applications. AFIPS Spring Joint Computer Conference B e n e s , V . E. Mathematical theory of connecting A c a d e m i c P r e s s , 1965.

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32:307-314,1968.

networks

and telephone

traffic.

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