Composition problem is considered for partition constrained vertex subsets of  n  dimensional unit cube 
E^n . Generating numerical characteristics of E^n  subsets partitions is considered by means of the same 
characteristics in  1 − n  dimensional unit cube, and construction of corresponding subsets is given for a special 
particular case. Using pairs of lower layer characteristic vectors for  E^(1-n)  more characteristic vectors for 
E^n are composed which are boundary from one side, and which take part in practical recognition of validness of a given 
candidate vector of partitions

Aslanyan, Levon

Sahakyan, Hasmik

Bulgarian Digital Mathematics Library at IMI-BAS

International Book Series "Information Science and Computing"   139GENERATING MORE BOUNDARY ELEMENTS OF SUBSET PROJECTIONS Hasmik Sahakyan, Levon Aslanyan Abstract: Composition problem is considered for partition constrained vertex subsets of n  dimensional unit cube nE . Generating numerical characteristics of nE  subsets partitions is considered by means of the same characteristics in 1−n  dimensional unit cube, and construction of corresponding subsets is given for a special particular case. Using pairs of lower layer characteristic vectors for 1−nE  more characteristic vectors for nE are composed which are boundary from one side, and which take part in practical recognition of validness of a given candidate vector of partitions.  Keywords:  monotone Boolean functions, (0,1)-matrices ACM Classification Keywords: G.2.1 Discrete mathematics: Combinatorics Conference: The paper is selected from Seventh International Conference on Information Research and Applications – i.Tech 2009, Varna, Bulgaria, June-July 2009 Introduction Set systems are considered represented as subsets of vertices of n  dimensional unit cube nE . Given a subset of nE , its numerical characterization is composed by partitions and their sizes, which are the coordinates of the corresponding associated vector of partitions. General numerical characterization of vertex subsets of nE  through their partitions is considered in [SA, 2006], [S, 2009] where a complete and simple structural description of the set of all integer-valued vectors, which serve as associated vectors of partitions is given, The description is through the set of its boundary elements. Elements of boundary set with minimum and maximum weight are known by [SA 2001], [SA, 2006]. Current research focuses on generating more boundary elements via the known ones for smaller dimensions, and constructs a set of corresponding subsets of vertices.  Basic Structural Description Let nE  be the set of vertices of n -dimensional unit cube, },,1},1,0{/),,{( 1 nixxxE inn "" =∈= . For an arbitrary variable ix , consider partition of the cube into two subcubes according to the value of nixi ≤≤1, . Denote these subcubes by 111−=nxE  and 101−=nxE  correspondingly. Similar to this, each subset of vertices nEM ⊆ can be partitioned into 11=xM  and 01=xM . For a given m , nm 20 ≤≤  consider an m -element subset M  of nE . Vector  ),,( 1 nssS "=  is called associated vector of partitions of M  if || 11== xi Ms  for all ni ≤≤1 . Let )(nmψ  denote the set of all associated vectors of partitions of m -subsets of nE .  Let nm 1+Ξ  denote the n  dimensional )1( +m -valued grid, i.e., the set of all integer-valued vectors ),,,( 21 nsssS "=  with msi ≤≤0 , ni ,,1"= . Obviously nmm n 1)( +Ξ⊆ψ .  9 – Intelligent Processing   140 Vector )(nS mψ∈  is called an upper (lower) boundary vector for )(nmψ  if no vector nmR 1+Ξ∈  with SR >  ( RS > ) belongs to )(nmψ . By )(nmψ  and )(nmψ , respectively, denote the sets of all upper and lower boundary vectors of )(nmψ . The sets )(nmψ  and )(nmψ  contain equal numbers of elements [S, 2009 ]. Let r  denote this number of elements and let },,{)( 1 rm SSn" =ψ   and  },,{)( 1 rm SSn " =ψ  so that ( jS, jS) is a pair of complementary vertices: jiji sms −= , for ni ≤≤1 . For each vector jS  all coordinates are greater than or equal to 2/m  and for each vector jS all coordinates are less than or equal to 2/m . For a pair jS, jS, from )(nmψ  and )(nmψ  respectively, let )( jSI   (the notion )( jSI   also may be used) denotes the vector set of the sub-cube of nm 1+Ξ  spanned by these vectors: }/{)( 1 jjnmj SQSQSI ≤≤Ξ∈= + . proves that the collection of all sub-cubes )}(/)({ nSSI mjj ψ ∈  composes the basic set )(nmψ :  Theorem 1. ∪ rjjm SIn1)()(==ψ  Boundary Cases Basic object of analysis in this section is the set of all monotone Boolean functions in nE , with exactly m  ones. Let )(1 nMm  be the set of all associated vectors of partitions of m -subsets corresponding to the ones of these monotone Boolean functions. Similarly )(0 nMm  is the set of all associated vectors of partitions of m -subsets corresponding to the zeros of such monotone Boolean functions. It is easy to check that )()( 1 nMn mm ⊆ψ  and that )()( 0 nMn mm ⊆ψ . Thus for description of )(nmψ  suffice to find those monotone Boolean functions which  correspond to )(nmψ  (and/or )(nmψ ). Suppose the weights (sum of all coordinates) of vectors of )(nmψ  belong to some interval ],[ maxmin LL . A specific set of monotone Boolean functions is constructed in [AS, 2001], for which the corresponding associated vectors of partitions belong to )(nmψ  and have the weight minL . The value of minL  and the coordinates of associated vectors are also analyzed in detail. Let niiD ,,1 "  be the set of vertices of nE  arranged in decreasing order of numeric value of the binary vectors ><niixx ,,1" . It is easy to check that the first 12 −n  elements of niiD ,,1 "  form the set of vertices of 111−=nxiE  and the reminder 12 −n  elements form 101−=nxiE  being arranged in the same decreasing order of numeric value of ><niixx ,,2" . Denote these sets by niiixD,121"=  and niiixD,021"=  respectively. Similarly the first 22 −n  elements of niiixD,121"=  form 21,1 21−==nxx iiE  and the rest 22 −n  elements form 2 0,1 21−==nxx iiE  - arranged in decreasing order of numeric value of the vectors ><niixx ,,3" , etc. It follows that each initial part (subset) of niiD ,,1 "  (respectively niiixD,121"= , niiixD,021"= , etc.) of arbitrary size serves as the set of ones of some monotone Boolean function. Let )(,,1 mD nii "  denote them -length initial segment of niiD ,,1 " . Denote by nD  the overall set of all enumerations niiD ,,1 " , and let )(mDn  be the m -length initial segments of enumerations of )(mDn . International Book Series "Information Science and Computing"   141The class of monotone Boolean functions, which have sets of ones belonging to )(mDn  we denote by )(mDnf   and let )(mDnS  be the corresponding set of associated vectors of partitions.  Theorem 2. )(nS mψ∈  has weight  minL  if and only if )(mDnSS∈ .  The proof of theorem which is not the target to bring here is by induction on n.   Note. Let us consider vector ),,( 1 nssS "=  of )(mDnS , which is the associated vector of partitions for some )(,,1 mD nii " . It is easy to check that niiisss ≥≥≥ "21. Further, vector S  obeys the following very useful property: any vector ,*),*,1,,( "" +ji iiss  for arbitrary njj ≤≤1, , does not belong to )(nmψ .  Generating New Boundary Elements Thus, we consider the problem of description of all monotone Boolean functions, which correspond to )(nmψ .  Let we have the solution of this problem for 1−n : given 1m , 11 20 −≤≤ nm , and we may describe the monotone Boolean functions corresponding to )1(1−nmψ . Consider the way of composing the target construction through composing the ones which we have in dimensions n-1: consider partition of nE  according to some variable ix , and consider the pairs of monotone Boolean functions in 111−=nxE  and 101−=nxE  respectively, which correspond to the upper boundary vectors and find such pairs, that in nE  (their union in nE ) correspond to )(nmψ . Consider partition of nE  according to the value of 1x  and consider monotone Boolean functions in 111−=nxE  and 101−=nxE  with 1m  and 2m  ones respectively, where 21,mm  are arbitrary partition of m  with the only requirement that 21 mm ≥ . We intend to find feasible pairs of monotone functions from 111−=nxE  and 101−=nxE  to get at first monotone functions in nE , and then - monotone functions which correspond to )(nmψ . Consider the following particular case. In 111−=nxE , as well as in 101−=nxE  consider monotone functions for which corresponding associated vectors belong to sets )1(1−nmψ  and )1(2 −nmψ  respectively, and have the minimum possible weights. By theorem 2 these are functions of the form )( 11 mDnf− and )( 21 mDnf− respectively. Consider the case of the same order of variables, let it be )( 1,2 mD n"  and )( 2,2 mD n" . Denote by ),,( ''2'nssS "=  and ),,( ''''2'' nssS "=  - the corresponding associated vectors. Obviously this pair is feasible for getting a monotone function in nE . Now we prove that it is feasible for getting an upper boundary vector as well, that is the vector ),,,(),,,( 21'''''2'21 nnn sssssssmS "" =++=  belongs to )(nmψ .  The choice of this particular case is caused by theorem 2; that is that we have the description of monotone Boolean functions which correspond to the upper boundary vectors of minimum weight.  Theorem 3.    )(),,,(),,,( 21'''''2'21 nsssssssmS mnnn ψ"" ∈=++=  9 – Intelligent Processing   142 Proof. Suppose this is not the case and the vector )~,,~,~(~ 21 nsssS "= , SS >~  serves as the associated vector of partitions for some monotone Boolean function (component wise comparison is used).  Consider cases: a) 11~ ss = . Consider partition of nE  according to the value of 1x . We get monotone functions in 111−=nxE  and 101−=nxE  with 1s  and )1( 1 +− sm  ones respectively. Denote by ),,( **2 nss "  and ),,( ****2 nss "  the corresponding associated vectors. Let 1122 ~,,~ −− == ii ssss "  and ii ss >~ . We confirm that 2*2 'ss =  and ''2**2 ss = , 1* 1 ', −− = ii ss"  and 1** 1 '' −− = ii ss , otherwise 2*2 'ss >  (similarly 3*3 'ss > , 1* 1 ', −− = ii ss" ) implies )1(),,(1''2' −∉= nssS mn ψ"  and 2**2 ''ss >  (similarly 3**3 ''ss > , 1** 1 '', −− = ii ss" ) implies )1(),,(2''''2'' −∉= nssS mn ψ"  due to the note above. ii ss >~  implies either ii ss '* >  or ''** ii ss > . By the same reasoning each of them leads to contradiction. b) 11~ ss > . Consider partition of nE  according to the value of 1x . We get monotone functions in 111−=nxE  and 101−=nxE  with 1~s  and 1~sm −  ones respectively. There exist )~(~ 11 sms −−  vertices in 111−=nxE  belonging to the set of ones of the function, for which the corresponding vertices in 101−=nxE  does not belong to the set of ones of the function. We move these vertices from 111−=nxE  to 101−=nxE . It follows from the case a) that nn ssss == ~,,~ 22 " . To provide 2s  in nE  we need to provide 2's  in 111−=nxE  and ''2s  in 101−=nxE , otherwise either )1(),,(1''2' −∉= nssS mn ψ"  or )1(),,( 2''''2'' −∉= nssS mn ψ" , etc. By theorem 2 the obtained sets are )( 1,2 mD n"  and )( 2,2 mD n"  respectively, which is contradiction since 21 mm ≥ .  As a corollary we get that for any m~ , mmm ≤≤ ~2/ , a vector of )(nmψ  exists with a coordinate which equal to m~ . We may also mention the topic of construction of 1m -subsets of vertices with the given associated vector of partitions ),,,(),,,( 21'''''2'21 nnn sssssssmS "" =++= .  Given that the sets )( 1,2 mD n"  and )( 2,2 mD n"  can be constructed by the “interval bisection” method [SA, 2006], then a construction of the required set can be provided in the following way: construct first column putting 1m  consecutive ones and ( 1mm − ) consecutive zeros; continue the construction in this 2 sections by the same “interval bisection” method.  The weight of S  and the values of coordinates can be calculated. Conclusion Generating numerical characteristics of partitions by means of the same characteristics in 1−n  dimensional unit cube for a particular case, leads to more boundary elements. The corresponding subsets can be easily constructed. International Book Series "Information Science and Computing"   143Acknowledgement The paper is partially financed by the project ITHEA XXI of the Institute of Information Theories and Applications FOI ITHEA and the Consortium FOI Bulgaria (www.ithea.org, www.foibg.com). Bibliography [R, 1963] H. J. Ryser. Combinatorial Mathematics, The Mathematical Association of America, 1963, p. 150. [A, 1979] L. Aslanyan. Isoperimetry problem and related extremal problems of discrete spaces, Problemy Kibernetiki, 36, pp. 85-126 (1976). [GGP 1997] R.J.Gardner, P. Gritzmann, D. Prangenberg. On the computational complexity of reconstructing lattice sets from the X-rays. Technical Report (970-05012), Techn.Univ. Munchen, 1997. [AS, 2001] L. Aslanyan, H. Sahakyan. On the boundary cases of partitioning of subsets of the n-dimensional unit cube, Computer Science & Information Technologies Conference, Yerevan, September 17-20 (2001), 164-166. [SA, 2006] H. Sahakyan, L. Aslanyan. The boundary descriptors of the n-dimensional unit cube subset partitioning, Journal “Information Theories and Applications”, ISSN 1310-0513, Volume 13 (2006), Number 3. [S, 2009] H. Sahakyan. Numeral characterization of n-cube subset partitioning, Discrete Applied Mathematics, vol.157 (2009), issue 9, 2191-2197. Authors' Information Hasmik Sahakyan – Leading Researcher, Institute for Informatics and Automation Problems, P.Sevak St. 1, Yerevan 14, Armenia, e-mail: hasmik@ipia.sci.am  Levon Aslanyan – Head of Department, Institute for Informatics and Automation Problems, P.Sevak St. 1, Yerevan 14, Armenia, e-mail: lasl@sci.am  

Generating More Boundary Elements of Subset Projections

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Generating More Boundary Elements of Subset Projections

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