Applications of combinatorial designs in computer science

The theory of combinatorial designs has been used in widely different areas of computation concerned with the design and analysis of both algorithms and hardware. Combinatorial designs capture a subtle balancing property that is inherent in many difficult problems and hence can provide a sophisticated tool for addressing these problems. The role of combinatorial designs in solving many problems that are basic to the field of computing is explored in this paper. Case studies of many applications of designs to computation are given; these constitute a first survey, which provides a representative sample of uses of designs. More importantly, they suggest paradigms in which designs can be used profitably in algorithm design and analysis.

[1]  Gerald Berman.  The Application of Difference Sets to the Design of a Balanced Multiple-Valued Filing Scheme , 1976, Inf. Control..

[2]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

[3]  Luc Teirlinck.  Non-trivial t-designs without repeated blocks exist for all t , 1987, Discret. Math..

[4]  Peter Frankl,et al.  Complexity classes in communication complexity theory , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[5]  J. A. M. Schreuder,et al.  Constructing timetables for sport competitions , 1980 .

[6]  Walter A. Burkhard.  Partial match retrieval , 1976 .

[7]  Marlies Gerber,et al.  A zero-entropy mixing transformation whose product with itself is loosely Bernoulli , 1981 .

[8]  P. Erdös,et al.  On Pairwise Balanced Block Designs with the Sizes of Blocks as Undtorm as Possible , 1982 .

[9]  A. Rosa,et al.  Tables of Parameters of BIBDs with r⩽41 including Existence, Enumeration, and Resolvability Results , 1985 .

[10]  Alexander Rosa,et al.  An Updated Bibliography and Survey of Steiner Systems , 1980 .

[11]  共立出版株式会社.  コンピュータ・サイエンス : ACM computing surveys , 1978 .

[12]  Ronald L. Rivest.  On Hash-Coding Algorithms for Partial-Match Retrieval (Extended Abstract) , 1974, SWAT.

[13]  Hanfried Lenz,et al.  Design theory , 1985 .

[14]  Paul Camion.  A Deterministic Algorithm for Factorizing Polynomials of Fq [X] , 1983 .

[15]  Fan Chung Graham,et al.  Optical orthogonal codes: Design, analysis, and applications , 1989, IEEE Trans. Inf. Theory.

[16]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1970, SYMSAC '71.

[17]  Pavol Hell,et al.  Parallel Sorting with Constant Time for Comparisons , 1981, SIAM J. Comput..

[18]  Richard M. Wilson,et al.  An Existence Theory for Pairwise Balanced Designs II. The Structure of PBD-Closed Sets and the Existence Conjectures , 1972, J. Comb. Theory, Ser. A.

[19]  János Demetrovics,et al.  Contribution to the theory of data base relations , 1979, Discret. Math..

[20]  F. Chung.  On concentrators, superconcentrators, generalizers, and nonblocking networks , 1979, The Bell System Technical Journal.

[21]  Robert C. Minnick,et al.  Magnetic Core Access Switches , 1962, IRE Trans. Electron. Comput..

[22]  D. Ray-Chaudhuri.  Combinatorial Information Retrieval Systems for Files , 1968 .

[23]  Vojtech Rödl,et al.  The number of submatrices of a given type in a Hadamard matrix and related results , 1988, J. Comb. Theory, Ser. B.

[24]  Andries E. Brouwer.  On associative block designs , 1976 .

[25]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[26]  Damaraju Raghavarao,et al.  Constructions and Combinatorial Problems in Design of Experiments , 1972 .

[27]  Richard M. Wilson,et al.  An Existence Theory for Pairwise Balanced Designs, III: Proof of the Existence Conjectures , 1975, J. Comb. Theory, Ser. A.

[28]  Michael O. Rabin,et al.  Probabilistic Algorithms in Finite Fields , 1980, SIAM J. Comput..

[29]  John R. Beaumont,et al.  Studies on Graphs and Discrete Programming , 1982 .

[30]  Lisheng Wu,et al.  On minimum matrix representation of closure operations , 1989, Discret. Appl. Math..

[31]  G. Koch.  A class of covers for finite projective geometries which are related to the design of combinatorial filing systems , 1969 .

[32]  Neal Zierler.  A conversion algorithm for logarithms on GF(2n) , 1974 .

[33]  Douglas R. Stinson,et al.  A Combinatorial Approach to Threshold Schemes , 1987, CRYPTO.

[34]  Richard M. Wilson,et al.  An Existence Theory for Pairwise Balanced Designs II. The Structure of PBD-Closed Sets and the Existence Conjectures , 1972, J. Comb. Theory A.

[35]  G. Constantine.  A Load-Sharing Matrix Switch , 1958, IBM J. Res. Dev..

[36]  G. Constantine,et al.  New developments in load-sharing matrix switches , 1960 .

[37]  D. R. Fulkerson,et al.  Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems , 1974 .

[38]  Mitchell P. Marcus,et al.  Doubling the Efficiency of the Load-Sharing Matrix Switch , 1959, IBM J. Res. Dev..

[39]  Douglas R. Stinson.  Some Constructions and Bounds for authentication Codes , 1986, CRYPTO.

[40]  Robert T. Chien,et al.  Orthogonal Matrices, Error-Correcting Codes and Load-Sharing Matrix Switches , 1959, IRE Trans. Electron. Comput..

[41]  M. Colbourn,et al.  On Cyclic Steiner 2-Designs , 1980 .

[42]  Richard C. Singleton.  Load-Sharing Core Switches Based on Block Designs , 1962, IRE Trans. Electron. Comput..

[43]  G. R. Blakley,et al.  Safeguarding cryptographic keys , 1899, 1979 International Workshop on Managing Requirements Knowledge (MARK).

[44]  Zoltán Füredi,et al.  Minimum matrix representation of closure operations , 1985, Discret. Appl. Math..

[45]  Sakti P. Ghosh,et al.  File Organization Schemes Based on Finite Geometries , 1968, Inf. Control..

[46]  Nicholas Pippenger,et al.  Sorting and Selecting in Rounds , 1987, SIAM J. Comput..

[47]  David K. Chow.  New Balanced-File Organization Schemes , 1969, Inf. Control..

[48]  Paul C. van Oorschot,et al.  A geometric approach to root finding in GF(qm) , 1989, IEEE Trans. Inf. Theory.

[49]  Albrecht Beutelspacher,et al.  Geometric Structures as Threshold Schemes , 1986, EUROCRYPT.

[50]  Joe Kilian,et al.  The organization of permutation architectures with bussed interconnections , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[51]  Ronald L. Rivest,et al.  On the Optimality of Elia's Algorithm for Performing Best-Match Searches , 1974, IFIP Congress.

[52]  Walter A. Burkhard,et al.  Hashing and trie algorithms for partial match retrieval , 1976, TODS.

[53]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1985, JACM.

[54]  V. Rödl,et al.  The number of sub-matrics of a given type in a Hadamard matrix and related results , 1987 .

[55]  Nelson M. Blachman,et al.  On the Wiring of Two-Dimensional Multiple-Coincidence Magnetic Memories , 1956, IRE Trans. Electron. Comput..

[56]  C. Moler,et al.  Advances in Cryptology , 2000, Lecture Notes in Computer Science.

[57]  Davis Avis.  A note on some computationally difficult set covering problems , 1980, Math. Program..

[58]  Yossi Azar,et al.  Tight Comparison Bounds on the Complexity of Parallel Sorting , 2018, SIAM J. Comput..

[59]  D. Stinson,et al.  Threshold schemes from combinatorial de - signs , 1989 .

[60]  B. Bollobás,et al.  Sorting in one round , 1981 .

[61]  Richard M. Wilson,et al.  An Existence Theory for Pairwise Balanced Designs I. Composition Theorems and Morphisms , 1972, J. Comb. Theory, Ser. A.

[62]  P. Erdös,et al.  Representation of Group Elements as Short Products , 1982 .

[63]  S. P. Ghosh,et al.  Application of finite geometry in file organization for records with multiple-valued attributes , 1968 .

[64]  Paul C. van Oorschot,et al.  On splitting sets in block designs and finding roots of polynomials , 1990, Discret. Math..

[65]  Eli Upfal,et al.  The complexity of parallel computation on matroids , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[66]  R. Häggkvist,et al.  Sorting and Merging in Rounds , 1982 .

[67]  Martenc Jones Colbourn.  Algorithmic Aspects of Combinatorial Designs: A Survey , 1985 .

[68]  Verzekeren Naar Sparen,et al.  Cambridge , 1969, Humphrey Burton: In My Own Time.

[69]  Werner Buchholz,et al.  File Organization and Addressing , 1963, IBM Syst. J..

[70]  Ronald L. Rivest,et al.  Partial-Match Retrieval Algorithms , 1976, SIAM J. Comput..

[71]  Noga Alon,et al.  Expanders, sorting in rounds and superconcentrators of limited depth , 1985, STOC '85.

[72]  T. V. Lakshman,et al.  Efficient decentralized consensus protocols , 1986, IEEE Transactions on Software Engineering.

[73]  Charles J. Colbourn,et al.  The complexity of completing partial Latin squares , 1984, Discret. Appl. Math..

[74]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[75]  Charles J. Colbourn,et al.  Embedding Partial Steiner Triple Systems Is NP-Complete , 1983, J. Comb. Theory, Ser. A.

[76]  A. Street,et al.  Combinatorics of experimental design , 1987 .

[77]  Richard M. Wilson.  Some partitions of all triples into steiner triple systems , 1974 .

[78]  Adi Shamir,et al.  How to share a secret , 1979, CACM.

[79]  J. Bond,et al.  Combinatorial Designs and Hypergraphs of Diameter One a , 1989 .

[80]  Shmuel Schreiber.  Covering All Triples on n Marks by Disjoint Steiner Systems , 1973, J. Comb. Theory, Ser. A.

[81]  A. J. W. Hilton.  The reconstruction of latin squares with applications to school timetabling and to experimental design , 1980 .

[82]  Robert T. Chien,et al.  A Class of Optimal Noiseless Load-Sharing Matrix Switches , 1960, IBM J. Res. Dev..

[83]  G. R. BLAKLEY.  Safeguarding cryptographic keys , 1979, 1979 International Workshop on Managing Requirements Knowledge (MARK).

[84]  R. C. Rose,et al.  The Design of Combinatorial Information Retrieval Systems for Files with Multiple-Valued Attributes , 1969 .

[85]  Andrew M. Odlyzko,et al.  Discrete Logarithms in Finite Fields and Their Cryptographic Significance , 1985, EUROCRYPT.