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LIBRARY OF THE 
 
 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAICN 
 
 510.84 
 IA6r 
 
 no .679-681 
 
 cop. 2. 
 
The person charging this material is re- 
 sponsible for its return to the library from 
 which it was withdrawn on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, and underlining of books 
 are reasons for disciplinary action and may 
 result in dismissal from the University. 
 
 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
 
 NOV 5 
 
 OCT 13 
 
 197? 
 
 RECO 
 
 JAN 2 3 
 
 L161 — O-1096 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/lowerboundsonmer680yaoa 
 
j~. , yijjCDCS-R-T^-68o 
 
 s*f?VJL^l, 
 
 LOWER BOUNDS ON MERGING NETWORKS 
 
 by 
 
 Andrew Chi- Chin Yao 
 Foong Frances Yao 
 
 
 October, I97U 
 
 The Librarv "' 
 
 JAN lMST5 
 
 ..viS 
 
uiucdcs-r-tH-680 
 
 LOWER BOUNDS ON MERGING NETWORKS 
 
 Andrew Chi-Chih Yao 
 Foong Frances Yao 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 Research is supported by NSF GJ-U1538 
 
ABSTRACT 
 
 Let M(m,n) be the minimum number of comparators needed in an (m,n)- 
 
 merging network. It is shown that M(m,n) 2 n(£g(m+l)) /2 , which implies 
 
 that Batcher's merging networks are optimal up to a factor of 2 + e for 
 
 almost all values of m and n. It is also proved that M(2,n) = [3n/2], and 
 
 that r = lim M(m,n)/n exists, r is determined to within 1. 
 m m 
 
 n-x» 
 
1. Introduction 
 
 An (m,n) -merging network [2] is a network in which the input consists 
 
 of two sorted sets {x n £ x_ £ •• • £ x } and {y n <: y_ <: •• • $ y }, and the 
 
 id m id n 
 
 output is the sorted set {z_ S z_ £ ••• £ z } with {z.'s} = {x.'s, y 's}. 
 
 ^ 12 m+n i j k 
 
 The network is built of comparators which are themselves (l ,l) -merging 
 networks. A comparator is usually drawn as in Figure 1, and a merging 
 network is shown in Figure 2. One can choose any input convention that 
 specifies how the sequence {x.'s} are to be interspersed in the {y 's}. By 
 standard technique [3], it can be shown that a transformation exists between 
 any two such conventions which preserves the number of comparators used. 
 
 Let M(m,n) be the minimum number of comparators needed in an (m,n)- 
 merging network. The famous "odd-even merge" by Batcher [l] readily gives 
 the following upper bound for M(m,n), which is also the best upper bound 
 currently known. 
 
 M(m,n) <? (n + m)(Ugm]/2 + m/2 Ugm1 ) + 0(m) 
 
 for m ^ n (l) 
 
 On the other hand, Floyd [2] proved a lower bound on M(m,n) for the special 
 
 case m = n. These efforts determined M(n,n) asymptotically to within a 
 
 factor of 2. The behavior of M(m,n) for general m and n, however, is not 
 
 well understood. 
 
 In this paper we shall derive a lower bound on M(m,n) by a different 
 
 approach. This bound, together with (l), determines M(m,n) up to a factor of 
 
 2 + e for almost all m and n. We shall also show that M(2,n) = f3n/2l, and 
 
 that r = lim M(m,n)/n exists. Main results are in Theorems 1, 8, 9 and 12. 
 m 
 
 n-*» 
 
 t lg is logarithm to the base 2. 
 
min (x,y) 
 
 max (x,y) 
 
 Figure 1. A comparator. 
 
 ^3 
 
 * <» 
 
 — ^ I - I. ■! ■ . ■ ■■■III! I 
 
 1 ^ 
 
 f 1 
 
 ,, 
 
 Figure 2. A ( 3,^) -merging network. 
 
2. ( 2, n) -Merging Networks 
 
 It has been conjectured that Batcher's merging networks are optimal 
 in terms of the number of comparators used. In this section we shall lend 
 support to this conjecture by showing that the odd-even merge yields optimal 
 (2 ,n) -merging networks. Indeed, Batcher's network uses [3n/2] comparators 
 (Figure 3), and we will prove the following theorem: 
 
 Theorem 1 M(2,n) = [3n/2l 
 
 To prove the theorem, it suffices to show that [3n/2] is a lower 
 bound for M(2,n). We shall need some notations and lemmas. 
 
 The lines in a network can be numbered from the top down as first, 
 second, ..., etc. A comparator is called an "( i,j ) -comparator" if its two 
 input lines are numbered the i and the j with i < j. In Figure 3, a 
 is a ( 3, 8) -comparator. A comparator is said to be of the form (i,*) if it 
 is an (i, j ) -comparator for some j. Comparator of the form (*,j) is defined 
 similarly. 
 
 By the remark made in Section 1, without loss of generality we can 
 consider only those (2 ,n)-merging networks where x, and x are inputs to 
 the top and the bottom line respectively. Furthermore, assume the input 
 (n + 2) -tuple (x , y , y , •••, y , x ) to be always a permutation of 
 (l, 2, •••,n+2). Now, if we fix x = n + 2, then for every possible 
 value of x_^ between 1 and n + 1, there is a unique path that takes x to 
 the appropriate output line. (Figure U(a)). Moreover, in these paths x 
 only moves downward through comparators. That is, if an (i,j ) -comparator 
 is contained in one of these paths , then it is used to take x from line i 
 down to line j (clearly the value of x has to be no less than j). We 
 define A to be the set of all comparators contained in these paths. Similarly, 
 
y l — ' — " 
 
 y 2 • 
 
 y 3 • J « 
 
 y k „ 
 
 y 5 1 
 
 y 6 . 
 
 * — 1 — i 
 
 Figure 3. Batcher's (2 ,n)-merging network where n = 6, 
 
 y U 
 y 5 
 y 6 
 
 x^ 
 
 m f 
 
 Figure Ma). The A set of comparators 
 
 *L 
 
 y. 
 
 y 2 
 
 
 y 3 
 
 
 
 J k 
 
 
 
 
 '5 
 
 
 
 
 y 6 
 1 
 
 — 
 
 b— 
 
 i 
 
 Figure Mb). The B set of comparators, 
 
while fixing x = 1, we consider all the possible paths traversed by x and 
 let B be the set of all comparators contained in them. Here, the comparators 
 in B are only used to move x upward (Figure Mb)). 
 
 Lemma 1 For each j, 2 < j < n + 1, the following are true: 
 
 (i) There is a unique comparator in A which is of the form (*,j). 
 
 (ii) There is a unique comparator in B which is of the form (j,*). 
 
 Proof We will only prove (i), since (i) and (ii) are symmetrical. 
 
 As we fix x = n + 2 and let x = j , a descending path will take x to the 
 j-th output line. The last comparator crossed by x in this path must be of 
 the form (*,j). We shall now show that for any fixed j,2<j<n+l, there 
 is only one comparator in A which is of the form (*,j). 
 
 Let w be the maximum value of x that will cause x_ to cross a comparator 
 of the form (*,j). Clearly j $ w <: n + 2. Denote by a the ( i , j )-comparator 
 crossed by x = w. The following two statements are true by the definition of 
 w: 
 
 (1) All the values u of x where j < u < w, will cause x to follow 
 the same path until past comparator a. 
 
 (2) When x assumes any value v such that w<v£n+2,x does not 
 cross any comparator of the form (*,j). 
 
 As a consequence of (l) and (2), a must be the only comparator of the 
 form (*,j) in A. Q 
 
 Because of Lemma 1, we can introduce the following mappings T and T'. 
 
 Definition 2 For any ( j ,k) -comparator a in ADB (note that 
 
 2$j<ksn+l),we will let T(a) be the unique comparator in A of the 
 form (*,j); and let T'(a) be the unique comparator in B of the form (k,*). 
 
 It is easy to see that both T(a) and T'(a) must lie to the left of a. 
 
Lemma 2 T is a mapping from AOB into A-B. 
 
 Proof We will show that for aeAOB, T(a)fB. Suppose this is not 
 true, then T(a)eAOB. Assume a is of the form (j,*), then T(a) is of the form 
 (*,j). Since T(a)eAflB, by the definition of T ' we have T'(T(a)) = a. However, 
 T(o) must lie to the left of a; and T"(T(a)), or a, must lie to the left of 
 T(a) . This is a contradiction. 
 
 Lemma 3 The mapping T is one to one. 
 
 Proof Let a and a' "be two separate comparators in AfiB. Assume 
 a is of the form (j,*) and a 1 is of the form (j',*). By Lemma 1 we must 
 
 have J 4 i l * Therefore it is impossible to have T(a) = T(a'). I I 
 
 Proof of Theorem 1 Lemma 2 and Lemma 3 together imply 
 
 | A-B | :> | AOB | . 
 It follows that 
 
 | A-B | $ \g |A| 1. 
 However it is easy to see that |a| = n and |b| = n. (There are n internal nodes 
 in a binary tree with n + 1 leaves.) Therefore 
 
 |A-B| :> [n/2]. 
 This leads to 
 
 | aub| = | A-B | + |b| 
 
 * [n/2l + n 
 
 = f3n/2l. □ 
 
3. Lover Bound for M(m,n) 
 
 The upper bound on M(m,n) as given in equation (l) can "be rewritten as 
 follows : 
 
 M(m,n) £ (n + m)Ug(m + l) + constant)/2 for m <: n. (2) 
 
 In this section we shall derive a lower bound, 
 
 M(m,n) 5 nUg(m + l))/2 (3) 
 
 By comparing (2) and (3), we see that for any e > 0, M(m,n) is determined to 
 within a factor of 2 + e for all sufficiently large m and n. The rest of this 
 section is devoted to a prcof of (3). 
 
 In an (m,n)-merging network, we can look at the input set as a column 
 vector with m + n components, and each comparator as a function which, given 
 a vector, either transposes two of its components or does not change it. We 
 shall assume that the input vector is a permutation of (l, 2, •••, m + n) and 
 that the x.'s are inputs to the first m lines. Thus, if a network contains I 
 
 comparators, appearing in sequence from left to right as a , a , '", a , then 
 
 T 
 any input vector V = (x , • • • , x , y , • • • , y ) is transformed into 
 
 T 
 (l, 2, •••, m + n) through a chain of vectors: 
 
 a a a 
 
 1 2 ^ I 
 
 T 
 where V = (l, 2, ••*, m + n) . 
 
 Now, a set of r different input vectors can be viewed as forming an 
 
 (m + n) x r matrix. The a. 's can then be regarded, by their columnwise 
 
 actions, as transformations on (m + n) x r matrices. We will choose r = m + 1, 
 
 and consider the effect of the a.'s when the following input matrix A is 
 
 given : 
 
A o = 
 
 n + 1 \ 1 1 
 
 n + 2 n+2 ^ 2 
 
 n+3 n+3 n + 3 
 
 n + m 
 1 
 2 
 
 n + m 
 2 
 3 
 
 n + m 
 3 
 
 1+ 
 
 1 
 2 
 3 
 
 1 
 2 
 3 
 
 • 
 
 m 
 m + 1 
 
 m + 2 
 
 n + 1 n + 2 
 
 m + n 
 
 In the i-th column of A , the upper part is the ordered list of length m, 
 (l, 2, '•', i - 1, n + i, n + i + 1, • • * , n + m) , and the lower part is the 
 ordered list of length n, (i, i+1, •••, n + i - l). 
 Let 
 
 a a 
 
 A — ^ A l ~ ^ A 2 
 
 Z 
 
 » A. 
 
 (10 
 
 be the sequence of transformations that A undergoes in an (m,n) -merging 
 
 network , where 
 
 \ • 
 
 \ 
 
 m + n 
 
 m 
 
 *. 
 
 To derive a lower "bound on I , the number of comparators in the network, 
 we define an entropy function: 
 
Definition h Given a vector v = (a_ 5 a,_ , •••, a J _.) where 
 
 1 d m+1 
 
 1 <: a <: m + n for all k, we define 
 
 p. = |{k|a = i}| for 1 S i < m + n, 
 
 and 
 
 m+n 
 E(v) = i | 1 p. Jig p. 
 
 (if p. = 0, then p. lg p. is taken to be 0.) 
 
 Definition 5 For the matrices A, (0 £ h <: i) defined in (h) , let 
 
 m+n 
 
 E( V " J £ 1 E < v j ) 
 
 where v. is the j-th row vector of A, . 
 
 m 
 Lemma 6 E ( A Q ) = 2 -^ J ^ J (5) 
 
 and E(A ) = (m + n) ■ (m + l)£g(m + l) (6) 
 
 Proof In A , only the first m row vectors have non-zero 
 
 entropies. For 1 $ j £ m, the j-th row vector of A has entropy 
 
 j Jig j + (m + 1 - j )Jlg(m + 1 - j ) , hence (5) follows. Equation (6) is true 
 
 because every row vector of A has entropy (m + l)Jlg(m + l). \_j 
 
 For each h , A n and A n _ differ in at most two rows , thus the difference 
 h h-1 
 
 in their entropies is bounded as implied by the following lemma. 
 
 Lemma 7 Let v and v' be two vectors satisfying the conditions of 
 Definition k, and let w and w' be obtained from v and v' by exchanging certain 
 corresponding components. Then 
 
 E(w) + E(w') £ E(v) + E(v') + 2(m + l) 
 
 Proof Let p., p.', q. , q. ' denote the number of i's appearing in 
 
 i l ' i l 
 
 v, v', w, w' respectively. Then 
 
 p. + p. ' =q. + Q. ' i = 1, 2, • • • , m + n 
 
 l ill 
 
10 
 
 Hence 
 
 E(w) + E(w') - E(v) - E(v' ) 
 
 = 2 q ± ig q i + 2 q i » Hg q. 1 - Z ? ± Ig v ± - Z v ± ' ig p.' 
 i i i i 
 
 v< 2 (q t + a i f )Ag(q i + q i ') - Z P; . £g Pi - E p ± ' Jig <g ± x 
 
 i i i 
 
 = 2 (P- + P i ')Ag(p i + p.') - 2 p. Jig p. - I p.' ig p.' (T) 
 
 i i 
 
 p. + p. ' p. + p. ' 
 
 Since it is easy to show that p. Jig + p. ' Jig ; assumes its 
 
 l p. l p. 
 l l 
 
 maximum value p. + p. ' when p. = p. ' , it follows from (T) that 
 i i i i 
 
 E(w) + E(w') - E(v) - E(v' ) 
 
 * 2 (p. + p.') 
 i 
 
 = 2 p. + 2 p. ' 
 .1 .1 
 l l 
 
 = (m + l) + (m + l) 
 
 = 2(m + 1) 
 
 This proves Lemma 7« 
 
 Theorem 8 M(m,n) > n(Jlg(m + l))/2 
 
 Proof By Lemma 6, 
 
 m 
 E(A Q ) = 2 ^ J JLg j 
 
 <: m(m + l) Jig m. 
 Thus 
 
 E(A ) - E(A ) 5 (m + n)(m + l)Jlg(m + l) - m(m + l) Jig m 
 $ n(m + l)£g(m + l) . 
 On the other hand, Lemma 7 implies that 
 
 E(A h ) - E(A _ ) £ 2(m +1) for 1 $ h $ I. 
 
11 
 
 It follows that 
 
 , „ n(m + l)&g(m + l) _ n&g(m + l) 
 * S 2(m + 1) " 2 
 
 where il is the number of comparators in any (m,n) -merging network. 
 
12 
 
 h. Asymptotic Behavior of M(m,n) 
 
 In this section, we study the asymptotic behavior of M(m,n) for fixed 
 
 m. 
 
 Theorem 9 For any fixed m, lim M(m,n)/n exists. 
 
 n-*° 
 
 To prove Theorem 9> we shall make use of the following lemma. 
 
 Lemma 10 M(m,n)/n < M(m,n )/n + mn /n for n < n. 
 
 Proof Divide the inputs {y £ y £ ••• $ y } into groups 
 
 S., , S^, •••, S r , i where each group, except maybe the last one, contains 
 1 2 I n/n n I 
 
 n n elements (Figure 5). Let S contain the largest n elements, S contain 
 the next largest n elements, etc. We can then construct an (m,n)-merging 
 network by first merging the m elements on top with S . After that the largest 
 n elements in the inputs are picked out at the bottom n lines. We then 
 merge the top m elements with S~ , and so on. This implies 
 
 M(m,n) < rn/n "lM(m, n Q ) 
 
 $ (n/n Q )M(m, n Q ) + M(m, n Q ) 
 
 Since it is clear that M(m, n ) < m n , the lemma is proved. 
 
 As a consequence of Lemma 10, we obtain the following corollary. 
 Corollary 11 M(m,n)/n <: M(m,n)/n + e(n) for fixed m, n , where 
 
 e(n) -> as n -*■ °°. 
 
 def 
 Proof of Theorem 9 For fixed m, let us write R (n) = M(m,n)/n. 
 
 m 
 
 Since R (n) is bounded from below (say by 0), we can consider L = lim inf R (n). 
 m m 
 
 n-*» 
 
 We will show that R (n) actually converges to L. 
 
 m 
 
 For any 6 > 0, choose an n such that 
 
 R (n n ) * L + 6 
 m 
 
13 
 
 m 
 
 '[n/n Q l 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n-n Q +l 
 
 
 
 
 n 
 
 
 
 
 Figure 5. An (m,n) -merging network built 
 
 of (m,n) -merging networks. 
 
lU 
 
 By Corollary 11, 
 
 R (n) $ R (n_) + e(n) (8) 
 
 m m 
 
 <: L + 6 + e(n) for all n > n , 
 
 where e(n) -*■ as n ■*■ °°. 
 
 But (8) implies that 
 
 lim sup R (n) £ L. 
 
 m 
 n-x» 
 
 Therefore L = lim R (n). This proves the theorem. 
 
 m 
 n-H» 
 
 Theorem 12 Let r = lim M(m,n)/m. Then 
 m 
 
 Ug(m + l))/2 S r m < Ug ml/2 + m/2^ g ml (9) 
 
 Proof This follows from Theorem 8 and (l). 
 
 Straightforward calculation will show that the difference between 
 
 the upper bound and the lower bound for r in (9) is less than 1. 
 
 m 
 
15 
 
 CONCLUSION 
 
 We have shown that Batcher's (m,n) -merging network is in general 
 optimal up to a constant factor. And at least in one non-trivial case (m = 2) , 
 we have shown that Batcher's merging network is in fact the best possible. It 
 will be interesting to see whether Batcher's merging network is optimal for 
 more cases; in particular, when m = 3. 
 
16 
 
 REFERENCES 
 
 [l] Batcher, K. E. , Sorting Networks and Their Applications, Proc. AFIPS Spring 
 Joint Comp. Conf . , 32, 307-31 1 *. 
 
 [2] Knuth, D. E. , The Art of Computer Programming, Vol. 3, pp. 220-233. 
 
 [3] Knuth, D. E., The Art of Computer Programming, Vol. 3, p. 239, Exercise l6, 
 
JIBLIOGRAPHIC DATA 
 iHEET 
 
 1. Report No. 
 
 uiucdcs-r-T 1 +-68o 
 
 Title and Subtitle 
 
 LOWER BOUNDS ON MERGING NETWORKS 
 
 Author(s) 
 
 Andrew Chi-Chih Yao, Foong Frances Yao 
 
 Performing Organization Name and Address 
 
 University of Illinois 
 Department of Computer Science 
 Urbana, Illinois 61801 
 
 2. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 1800 G Street, N.W. 
 Washington, D.C. 20550 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 October, 197^ 
 
 8. Performing Organization Rept. 
 No. 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 NSF GJ-U1538 
 
 13. TyP e °f Report & Period 
 Covered 
 
 14. 
 
 15. Supplementary Notes 
 
 6. Abstracts 
 
 Let M(m,n) be the minimum number of comparators needed in an (m,n)- 
 merging network. It is shown that M(m,n) 5 nUg(m+l))/2, which implies 
 that Batcher's merging networks are optimal up to a factor of 2 + e for 
 almost all values of m and n. It is also proved that M(2,n) = [3n/2], and 
 
 that r = lim M(m,n)/n exists, r is determined to within 1. 
 
 m m 
 
 n -x» 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 Merging network, comparator 
 
 17b. Identifiers /Open-Ended Terms 
 
 17e. COSAT1 Field/Group 
 18. Availability Statement 
 
 Unlimited 
 
 FORM NTIS-35 (10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 20 
 
 22. Price 
 
 USCOMM-DC 40329-P7I 
 
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