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Referring to the information of  belonging to H , we wish to identify G by edge-detecting queries, each of which tells whether a subset (with some constraints) of [n] induces an edge (or more info.) of G. Who is hiding there?63I3l 6 NClassical Group Testing Problem  Given a set of items, each of which is either positive (abnormal) or negative (normal). Identify positive items by group test which determines whether a subset of items contains a positive item. Blood testing for Aids in World War II was an example and we got more in recent years.`.& 3FW3  O Complex Model  RAn extension of classical group testing problem. A set of complexes, each of which is a subset of basic items, is given and the property (positive or negative) of each complex is waiting to be determined. The corresponding group test determines whether a subset contains at least one positive complex (instead of just one positive item). hSZ: 3G33E! S QCompetitive Group Testing  }In classical group testing, the number of positive substances is usually assumed bounded while for competitive group testing, no information on the number of positive substances is given. A learning graph problem can be viewed as a competitive group testing on complex model with each complex of size two. If we are learning hypergraphs, then the size of complex is getting larger.~ZQ 332T .&E   . 1Motivated by applications in DNA physical mapping22#( 2 P R Whose Idea?  gIn 1996, A. Sorokin et al. proposed the following idea in Genome Research: For physical mapping, some cloned fragments are produced to cover the whole DNA molecule with some gaps and then some of them are assembled to form longer continuous fragments (contigs). However, the information about the mutual placement of the contigs on the DNA sequence is lost. jh !3` x.@  ?   SContinued &   Multiplex PCR (Polymerase chain reaction) is a tool to tell whether there are two neighboring contigs among a set of contigs so as to identify the relative replacements of the contigs. The relative replacements of the contigs correspond to a  graph !>3t^    4  #   1   Whole-genome shotgun sequencing Short reads are obtained and covering the genome with redundancy and possible gaps. Circular genome  X9  X*   $ u Reads are assembled into contigs with unknown relative placement. fZBZ3ZB%)&  U   y Primers : (short) fragments of DNA characterizing ends of contigs. D/$$=$0&;  8    K A PCR (Polymerase Chain Reaction) reaction reveals if two primers are proximate (adjacent to the same gap). Multiplex PCR can treat multiple primers simultaneously and outputs if there is a pair of adjacent primers in the input set and even sometimes the number of such pairs. /3K 30&"  )    TTwo primers of each contig are  mixed together Find a Hamiltonian cycle by Multiplex PCRs if circular genomes are concerned! dYN/*3$h   * &  $  "  NPrimers are treated independently. Find a perfect matching by Multiplex PCRs. d#+#3N"  %      Goal6  Our goal is to provide an experimental protocol that identifies all pairs of adjacent primers with as few PCRs (queries) (or multiplex PCRs respectively) as possible. 8z   Nj      &Mathematical Models(   Hidden Graphs (Reconstructed) ty W$$$3 @e      3SDescribed into Math Part II &   Hidden Graph R      679;=TThen, &   We can delete the edge and start it over again to find another edge until all the edges are found. Clearly, if the hidden graph we are looking for is of large size compare to n, then this algorithm is not going to be a good one. We may simply ask all the n(n-1)/2 2-subsets of [n].,c53  f$Known Results on Other Graphs(3   . PP     T  Reference    Reconstructing a Hamiltonian cycle by querying the graph: Application to DNA physical mapping [Grebinski and Kucherov 98 ] Learning a hidden Matching [ N. Alon et al, 04 ] Learning a hidden graph using O(lgn) queries per edge. [Angluin and Chen 04 ] Learning a hidden subgraph [Alon and Asodi, 05 ] Combinatorial search on graphs motivated by bioinformatics applications: a brief survey [Bouvel, Grebinski and Kucherov, 05 ] Learning a hidden hypergraph [Angluin and Chen, 06 ] Math h|CC# _     (  -    #      b                 0BMBetter Lower Bound!  In reconstructing a graph with n vertices and m edges, all graphs of order n and size less than m must be considered as graphs to be reconstructed. (Using computation trees.) Theorem. log(S0 i m(n(n-1)/2 chooses i)) edge-detecting queries are required to identify a graph drawn from the family of all graphs with n vertices and m edges. W 3     z 3 3 W CU5 Steps Algorithm  )You may ask my coauthors for the details!**3 * VUsing Designs (Affine Planes)  An affine plane of order q is a balanced incomplete block design, 2 - (q2, q, 1) design, such that the set of blocks can be partitioned into parallel classes. A Steiner triple system of order 9, STS(9), is an example of an affine plane of order 3. It is well-known that the existence of an affine plane of order q is equivalent to the existence of a complete set of mutually orthogonal Latin squares of order q which exists when q is a prime power.`H33V3c  WContinued &   Theorem. For each positive integer n there exists a prime p such that 2n p2 n. There exists a pairwise balanced design (PBD), 2  (n, K, 1) design, such that K contains positive integers not greater than p and the design has at most p(p+1) blocks. 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For any , edge-detecting queries are required to identify a graph drawn from the class of all graphs with vertices and edges. 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