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1 The researeh in this report 7JJas supported in part by a National, Science Foundation Eeseareh Grant and lJaB lJritten under the direction of Professor I. M. Chakravarti. fhti,tten JU'{,y~ 19'10~ printed October~ 19'10. N07JJ at the Department of MathematiCB~ EUsabeth Ci.ty State Elizabeth City~ North CaroZina 2'1909 ON THE CONSTRUCTION OF SYSTEMS AND DESIGNS USEFUL IN THE THEORY OF RANDOM SEARCH1 V. P. Mang1ik * Department of Statistics University of North Carolina at Chapel, Hilt Institute of Statistics Mimeo Series No. 717 October, 1970 2 lJniversity~ ON THE CONSTRI1crION OF SYSTEMS AND DESIGNS USEFUL IN THE THEORY OF RANDOM SEARCH by V. P. Manglik University of North CattoZina at Chapel Bill ABsTRACT This paper presents some methods of constructing systems and designs useful in random search theory. A brief introduction to basic concepts useful in simple search and relation of search systems with important combinatorial configurations is given in Section 1. Section 2 gives some new pairwise balanced (PWB) designs. Construction of the systems is discussed in Sections 3 and 4. In Section 5, a criteria for comparing search systems is given together with some methods for constructing systems optimal in this sense. The results are given without proof because proofs will appear elsewhere [1]. 1. I NTR>DLCTION Problems of search occur in almost every field. such as medical diagnosis, the parameter of a probability distribution, search for an unknown or hidden object, etc. The general search problem is to determine an unknown object x e S, the basia set of all possible objects of search, by selecting one after the other certain test funations f ,f , •• from a family l 2 F defined on S and observing The research in this report tUas supported in part by a National Scienae Foundation Research Grant and was wPitten undeP the direotion of ppofessop I. M. ChakpavaPti. 2 their values at x till we have enough information . finite consis ting of, say, a j ~ S n elements a ( i · 1,2, ••• ,m) (j • 1,2, ••• ,n) O,l,.,q-l to find a ,., an; 2 l f ~ i and taking only associate an mx n matrix at the point E F A system a ~ j S. depending on q distinct values M are all different. of values of b jt f r Lt=l b jt = n. M correspond to the flmctions and S. is called sepazaating on S A separating system F is a separating system on Y be the finite set different points of Renyi [2] introduced corresponding to the value of The rows of F of functions on proper subset of at is To a simple search problem, we can M III «fi(a ») j the columns to the elements of of S such that at each stage the choice of a function depends to some ex- important notions of simple search theory. i When F tent on chance, the search is called simple ztemdom seazaoh. f x. f S, j III F S. S on e: F, j III S f E F such that 1,2, ••• ,m; Associating with each element of iff the columns is called minimaZ if no For every {Yl'Y2' ••• 'Yr} S t· let the set fj(a). Yt 1,2, ••• ,r, a probability of P(fj(x) = Yt) = bjt/n III Pjt' entropy of f j i$ r H(fj ) = Lt=l Pjt·tagU/Pjt). Then for any separating system lin, we have t;lIIlH(fj) ~ logn, and an F on S, optimaZ separating system is one for which the equality holds. For a choice of R (ai ,ai , ••• ,ai ) -It 12k III k distinct elements of S, k elements. k if j if, R.. -It (ai ,8t , ••• ,ai ) = R.., 12k -It Further, let us choose a sequence of ,Yt .••• ,yp of Y also and let 1 2'K l{f: f(aij)=YR ; j=1,2, ••• ,k; fEF}I. k let 12k Yt O'l"de'l" 8t,8t , ... ,ai; 12k I{f: f(ai )=f(ai ) •••• =f(ai ), fEF}I. System F is weakZy homogeneous of orde'l" choice of say, ~(Yt ,Yt ' ••• 'YI ; ai' 1 2 lc 1 k k elements ai ) = k Then the system is strongZy homogeneoua of ~(Ytl,Yt2,••• ,Ytk; ail,ai2, ••• ,aik) dependent of the choice of independent of the elements of S. III ~(Ytl'Ylz,••• ,Ytk) in- 3 A system noted by F F(m,n) of m functions defined on and its m x n systems will thus be denoted by MW S matrix by M. SHS(m,n) and of n elements will be de- Strongly and weakly homogeneous WHS(m,n) with matrices M S and respectively. Following result is an easy consequence of above definitions and of partial- ly balanced arrays ([ 3]) : A partially balanced array of t N columns, and parameters A(Y1'Y2"" 'Yt) SHS(N,m) of order t in s m rows, s symbols, strength is equivalent to transposed matrix of a symbols with parameters Rt (Y1'Y2""'Yt) = A(Y1'Y2""'Y t )' PWB designs ([ 4]) are found to be closely related to search systems. posed matrix of PWB design. SHS(m,n) of order 2 in 2 symbols gives incidence matrix of However, converse is not ture, in fact there are examples when the system is not even WHS. In Section 3, a stronger result is given: Incidence matrix of a PWB design is related to the matrix of a SHS 2 Trans- in 2 symbols iff each treatment has equal number of replications. of order (These PWB designs have been termed as SYmmetrical Unequal Block (SUB) arrangements by [5].) 2. Sat£ New PAB I£slGNS 2.1. PWB designs based on finite geometrical structures: Using the results on finite geometrical structures ([6]) following PWB designs can be obtained by considering points not lying on the structure as treatmen ts and the lines as blocks. (a) PWB«q+1)(q+1-n); k {(n-1)q+n, n} - Me. 1 = q+1, in k 2 PG(2,q) = q+1-n'; with q A. 1), = O(modn), from an n' = q/n. 4 (b) PWB(v. q2(q2_q+l); PWB(v = q(q 2+1); quadric in k 2 k k 2 • q, k = q-l; 3 and lines as blocks. k • q(q+l), l from = in • q+l" l PG(3,q) PWB(v • q(q +1); • q(q-l); ~. 1) 2 PG(2"q2), q pm. • q2" l =0 x q+ l + yq+l + zq+l (c) k k = 1) from elliptric Also 2 2 ~ ~:II • q; q+l) i, obtainable by taking planes as blocks. 2.2. PWB designs based on finite Baer subplanes: A sUbplane of a projective (affine) plane and lines L such that (ii) V V p' E: P L' E: P 3 3 A proper sub-plane is a subset Q of points p Q is itself a projective (affine) plane, relative to incidence relation given in (i) P P. A Baer sub-plane is a sub-plane such that a unique line L of is incident with p EQ ~ L' a unique point Q(2"m) Q 3 p' E P(2,s)" is incident with s . pn L p. is a Baer sub-plane iff 2 s. m ([1]). Considering points not in as treatments and lines as blocks following Q PWB designs are obtained. (a) 3 PWB(v· m(m -1); k • s-m, l k = s; 2 A. 1) from Baer projective sub- plane. (b) 2 2 PWB(v. m (m _l); sub-plane. k l .. s-m" k l • s-l, k 3 • s; ~. 1) from Baer affine 5 3. STRaG.Y HoMxiENEous Consider" system F(m,n) SYsTEMS OF ORJER 2 IN 2 SVM30LS and its matrix M having entries, say, 0 and For i-th and j-th columns of M let Aij • Number of rows having 1 in both the columns lPij .. Number of rows having 0 in both the columns Yij .. Number of rows having 1 in i-th and 0 in j -th column ~ij 0 in i-th and 1 in j-th column. We then have A+ ~ For a ~ +r +A A• (ii) r .. A, matrices. mJ, n II: SHS(m,n) A and y. ~ mx n NN' J - N l x n matrix with all entries 1. 6 • ~Jn .. (m"'A-2~)J n • of order 2 in 2 symbols is specified by only two parameters .. N.. n n If we denote C is n r .. yJ , H' .. yI by n + N, then AJ • n corresponds to parameters (vi) J of order 2 we shall have nxn (v) where Jl.Jn , Thus a r .. A'. are symmetric and SHS(m,n) (i) (iv) n x n 4 A and Matrices of rows having .. Number m - A - 2y submatrix M l of SHS(m,n) and M mxn of order 2 in 2 symbols with y. will correspond to SHS(m,n) of same 1. 6 order and with same pa~ameters for leting columns from the matrix of u ~ l SHS n. This means that de- does not affect the prop- erty of strong homogeneity. Above properties of SHS(m,n) of order 2 in 2 symbols lead us to following theorem. THEOf£M 3.1. with entries A necessary and sufficient condition for an and 0 to be the matrix of a 1 bols with parameters A and (A+Y)1 = r (i) 1'M • (i1) C · H'M III x n matrix M of order 2 in 2 sym- V is that 1 VI + AJ i.e. each column of H have same number of l's i.e. diagonal entries of off-diagonal entry is CoRou.ARv 3.1. SHS(m,n) m Care r and every . A. A necessary condition for SBS(m,n) of order 2 in 2 sym- m ~ n. bols is In view of this theorem, we conclude that incidence matrix of a SUB arrangement is closely related to a V = (r-A). A and with k i =k SHS(b,v) Remembering that of order 2 in 2 symbols with parameters BIB designs are a special case of these this includes all BIB designs. Therefore, for constructing SHS(m,n) of order 2 in 2 symbols, the whole force of BIB designs is available together with other SUB arrangements. 4. WeAKLy HOMOOENEOus SYSTOO OF ORIER 2 IN 2 SVf4nS In the notations of Section 3, for a WHS(m,n) of order 2 in 2 symbols, we have A+ ~ + r + /). = mJ, A+ ~ • aJ 7 where a ij "ij+41ij • Number of rows having i-th and j-th columns same and for III WHS(m,n) of order 2 a ij • a for all i and j. These properties give follow- ing theorems. THEOREM 4.1. Let A be a AA' then A'· M represents a Let AA' (m-a)I III n THECREM 4.3. AI + BJ, n • 1 2 = ••• such that ' 'J ~ " i. .. .. rn i.. of order 2 only if r l n x m matrix with entries M to represent where A be a Let or l A ••• = A. r.2 •••• A WHS(m,n) A be a a necessary condition for that 0 • r III r , n was is a SUS. Furthermore, in this case THEOREM 4.2. n)( m matrix with entries 1 and -1, then was (m,n) with parameter a is a = 2a - m, n)( m matrix with entires 1 and -1 such that + aJ (i) AA' III (ii) -"31 :S i!.m :S (ii1) m and a are both even (or odd), then (m-B)! 1 A' .. M represents a WHS(v,v) WHS(m,n) with parameter a III (B+m)/2. can be obtained from the 2 and 3-class association schemes. In this direction following results are obtained. THEOREM Bl 4.4. A sufficient condition for the in a 2-class association scheme to represent a is that a III v x v association matrix WHS(v,v) with parameter a 8 Tt£OREM 4.5. represent a Association matrix WHS(v,v) with parameter B 1 a in a 3-c1ass association scheme will if 3 r = a pii + 2P23 i-I THEOREM 4.6. v x v 1 + Consider a 2-c1ass association scheme in association matrices with entries 2 3 \' 3 L Pii i=l and BO,Bl'B 2 - Then -1 will represent a v x v v 3 + 2P23 objects with 2 N .. Ii=oCiB i matrix with parameter a rI v WHS (v, v) if either (i) . (1,1,-1) , t 2 P12 - P12 .£ = (1,-1,1) , pt .£ or (ii) It can be noted that THECREM 4.7. v x v eter (a) (b) (c) (d) Nand a ~ v - p -N 2 12 1, a .. V - 2pi2 . -1, a • 2 V - 2P12- represent same WHS_ Consider a 3-class association scheme in association matrices 3 N • t i=oCiBi 12 .. with entries B , i 1 i . 0,1,2,3_ and Then the -1 will represent a v v x v objects with matrix WHS(v, v) with param- if one of the following holds. 1 1 2 2 3 3 P13 + P23 • P13 + P23 • 1 + P13 + P23' 1 1 a v - 2(P13+P23) 2 2 3 3 1 1 e = (1,1,-1,1) , P12 + P23 • 1 + P12 + P2 3 • P12 + P23' 1 1 a • v - 2(P 2+P23) 1 1 1 2 2 3 3 .£ = (1,-1,1,1) , 1 + P12 + PI3 • P12 + P13 P12 + P13' 2 2 a V - 2(P +P12) 13 1 1 2 2 3 3 (1,1,-1,-1), P12 c 1 + P1 2 + PI3 • 1 + P12 + P1 3' + P1 3 1 1 a ... v - 2(P12+P13) .£ = .. (1,1,1,-1) , - .. .. .. 9 (e) £ .. (1,-1,1,-1), a (f) £ • • v - 1 1 1 + PIJ + P2J • (1,-1,-1,1), J 3 v - 2(P 1 3+P23) (1,-1,-1,-1), a .. £ • .. 2(pi2+P~J) a • (g) 1 + Pi2 + pi3 V - 2. In view of the theorems 4.4 to 4.7, we have to look for the association schemes which satisfy conditions of anyone theorem. Among the 2-c1ass assa- ciation schemes condition hold good for the following. (a) Triangular Association schemes with (b) L (n) (c) NLg(n) (d) SLB (e) Pseudo Geometric Association schemes with schemes with g schemes with schemes with t .. k-r+l For a a = R 2 2 for 2 v. n , k· 21:'-1 or 5. v .. n n" 2g. and k . 21+1. PI(n,N,F,x) R2/RI fine a tk [(r-l)(k-l)+t], for RANooM SEARcH of order 2 in 2 symbols 0 SYSTEMS and I [:~t ' with parameter following result is given in [2]. using family of v .. t.. k-r-l. P1(n,N,p,x) where v .. 6. n . 2g. ca.PARISOi OF SIKJL.E WHS(RI,n) v .. 15, WHS F 1 - (n-l) is probability of determining of functions. for fixed ~ n. But to be optimal WHS this bound is attained. x ~ S uniquely in N steps Maximizing this probability means minimization ~/R1 ~ (n-2)/2(n-l), for fixed n if R 2 (for proof see [2]). and R I De- are such that 10 For a y we have SHS(Rl'n) R /R 2 III l of order 2 in symbols R - 2y. l 0 and 1, Thus we look for these with parameter SUB " and arrangements for which R1-2y THEOREM • 2n-2 (n-l) 5.1. Only 4y(n-l) , or PWB or n b 4(r-") (v-I) • v design with above parameters is the one for which all blocks are of same size. THEOREM 5.2. k • t, "III t-l BIB design gives the optimal WHS (4t-2 ,2t) • THEOREM 5.3. of an optimal WHS ments. SHS D with parameters (in fact of an optimal 4t-2, SHS) of order 2 on a set S of 4t n = 2t 4t III 2t, r III 2t-l, was is SHS. imply existence S with n III 2t ele- elements, there exists a Pl(n,N,F,x) is maximised exists. Existence of Hadamard matrix of order WHS(4t-l,4t) v In fact this S such that the lowest bound for if Hadamard ma trix of order THEOREM 5.4. III Existence of a Hadamard matrix of order In other words, for every set of order 2 on b 4t imply the existence of order 2 in 2 symbo,ls with parameter This research was done mder the direction of r. M. Chakravarti. a III 2t-l. 11 ·w REfERENCES [1] Manglik, V.P. (1970), "Construction of Designs and Systems Useful in the Theory of Random Search," Ph.D. Dissertation under preparation at University of North Carolina at Chapel Hill. [2) B4nyi, A. (1965), [31 Chakravarti, I.M. (1963), "Orthogonal and Partially Balanced Arrays and their Application in Design of Experiments," Butt. Amer. "On the Theory of Random Search," Math. Soc... 71, p. 808. Mem'ka.. Heft 3, p.231. [4] Bose, R.C. (1968), ''Notes on the application of combinatorial theory to design of experiments." [5] Kishen, K. (1940), "Symmetrical Unequal Block Arrangements, rr Sankhya, 5, p. 329. [6) Barlotti, A. (1965), "Some topics in finite geometrical structures," lust. of Stat. Mimeo Series No. 439, University of North Carolina at Chapel Hill. [7} Dembowaski, P. (1968), [8] Blackwelder, W.C. (1966), "Construction of BIB designs from association schemes," Inst. of Stat. Mimeo Series No. 481, University of North Carolina at Chapel Hill. [9] Hall, M. (1967), Finite Geometries. CorrhinatoPia1. Theory.. Blaisdell Pub. Co.