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,O,, Introductionq+@0Learning a hidden graph with adaptive algorithms1100  i Hung-Lin Fu Department of Applied Mathematics National Chiao Tung University Hsin Chu, Taiwanjj3@@      (Motivated by bioinformatics applications))#( ) P 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/$$=$0&;  8     K A PCR (Polymerase Chain Reaction) reaction reveals if two primers are proximate (adjacent to the same gap). Multiplex PCR can treat multiple primers simultaneously and outputs if there is a pair of adjacent primers in the input set and even sometimes the number of such pairs. /3K 30&"  )    Two primers of each contig are  mixed together Find a Hamiltonian cycle by PCRs! LY"/*"h   *     "  DPrimers are treated independently. Find a perfect matching by PCRs. L#!# N"        Goal6  Our goal is to provide an experimental protocol that identifies all pairs of adjacent primers with as few PCRs (queries) (or multiplex PCRs respectively) as possible.  Nj      &Mathematical Models(   Hidden Graphs (Reconstructed) ty W$$$3 @e      3SDescribed into Math Part II &   Hidden Graph R      679;=Z!Known Results (Matching)6 $$ $3   The information-theoretic lower bound for matching is (1+o(1))nlgn bound can be reached by an adaptive algorithm. [Bouvel, et al. 05 ]. Proof. Nonadaptive algorithms require queries. [Alon, Beigel, Kasif, Rudich, Sudakov 02 ]. Proof 9AAAAAAA/ 6$w       ,              vX 0X 0,& Strategy: first to find one vertex &'& &  Theorem: [Angluin 06 ] A vertex in a hidden graph on n vertices can be reconstructed with at most queries. Proof. Tn2 I a &   t 1#Results Example of Find-One-Vertex2$  $  f$Known Results on Other Graphs(3  + Hamiltonian[lower][upper] Star. P!P ! h          >X 0 X 0X 0$X 0'+(Hamiltonian cycle ~ adap.&   DO(nlgn) bound can be reached by an adaptive algorithm. [Grebinski, Kucherov 1997]. Proof. To process all vertices one-by-one by storing them in the independent set of chains. case I: no/no case II: yes/no case III: yes/yes at most 2nlgn queries. BACKS]CCCCCC+f 2        X 0@DAHow about more general graphs?,   BCT  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                X 0DE: Algorithm 3 ,  b It is left to find all the edges between independent sets. 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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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WiýDLGǤF5V|I*R򕔱b\9O S:veQHÀꁺj,e<d]'LMRaٽHqݶ,A\om01QPjPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!i *drs/shapexml.xmlPK-!GNdrs/downrev.xmlPK b$ g/  H X 0޽h ? 6GzQ___PPT10i._p+D=' = @B +1 0   `z (  `& `  p'0e0eP? jL 1 `}  ' nConcluding remarks e  D ` # '0e0eP?gQ[HrbMn@S 2 ` 'L___PPT10,$z___PPT9\T Reduce the rounds of Algorithm 1 (i.e., obtain an efficient algorithm to find a maximal matching). Learning a hidden graph in Quantitative k-multi-vertex model. c  )  a aCaaaa a a   ` J'?>\HrbMn@S 4"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!s6d drs/shapexml.xmlUMo1#,_QGEm!qx$8 !.H++R_0co zBUmg{37bsi]at[I-LVq48stht+U*%U'WID஛Jj<ˍ-֎Nji6%Uz>V|`YǙC{r|ӷO8cڞ\_31uuz'e!2cܲM!45f_Sw$ܖ M(L3t-Ζ)D.<hٺ$vee%ͥbtQʭ> D:PDz#UBfK}(x,kL]qf7*!hR.,+~e:j"l?䋋{a@3lE d~*7. bFzNYn N#p3ݪctc fZ`jJ+AYJ gsk-Ş=3CnaWf.pu<@盻8QVH{pi@tVyXDRqʥnbi(դq"".G2KƐ45m108JK364,a%B0MR+y;lIU 1yXg֕~_IK[&(} oQEF=H-h܉.WF:I9N3Rg}]>P:J ˮ-@qV^L,ͬ.ή X'C$\p7/O?ߧsrV Jc~1" A}XYԢA]H^{WPK!!drs/downrev.xmlDMK@Eax;;1HH@ƈg5`4I.tys9bG1Qd㝂U\q'tGHˋ5VnOSz!+T0*)"| S:veQKud,%d]8MNuZA$Zxj?_z @^_=r el rPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!s6d *drs/shapexml.xmlPK-!!drs/downrev.xmlPK x24$ g/   H ` 0޽h ? 6GzQy___PPT10Y+D=' m= @B +#1 0 "" hO"(  h h  py0e0eP? jL 1 `}   f References     " h # k0e0eP?gQ[HrbMn@S 2" PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!3pmdrs/shapexml.xml\n7;lYi 9ѠV\k]K9OA!K/[}#ȩo]PnwIo)? 3GۜIfX]ٳC #H$|Wy2|lLmp,SQFה{/x2 o8{Y8Zr|t:hA=C$=b&A *xj \!n0gWB猂2=cK zqԩq)~21Ye B`HmMBǠQّH 30׉+Y-+bXzǩ=0z(X ߪ0Dr&dt&Wl)zꩾ"b=YRAU FPDqRWa?4JӉȝx*EyId*\Sr"WBҍi8L !p2-\2!+MY,+\#n .(H3v9I4!v.fipd$l8hߓ+L4Q8|%$!-InP }7;B&1 ,lxk9EjN)P趕DQU,NRdI<%6ӊҟ`(4驳 S#7JL9- I(oN{( È +10o+gW-U(y J4=W(^Nl oBa^XhR::kf y< e}ϱA{9i( "T+ ܋b7#\ (RB 8>8o[gOL<;gwz~?a;~aa8wpw&V3͝fS .Ow'"iÙ͌腨q.{g{jW{m h.0|#05Ri^&H;4v|# ׺]E~l"[<'{b◱ r+T/*s΄U̺*w, ;<\ؔRS"ɽHhACݏO1oauUK ܔ]< ryCeˤ3}\t2Ͱ߽$ s<~TpUu A׳eҶ¤wU㗽`g5]ZI*h&6$Ik1+CBE/\,G|bٚ{_/mWo0W,N,ؤT&TX]28l'в>nUaDr|仯0VJdngdBC=x2!hAސέt/sHi?˔9SKXpVSi'uҤ5J2;[nhAl#-|\ЈVʽ^3*"V"yi.HZGCUFI}'QߖJFMU{7b-+y,{r; %B 7]fm|Vgyhs9?)sLͥV cXxhV{ .V'e;Iߕ] x/ǷC(Bq;ɐ|^&uoy(7{[d 蘄ɍgm+RvΌ$pX?j#lgЎ^*-*qU76 쌸hx;:\KyCZVXB 5uG vx]h]u.\tnhf 2hˆʍ )ﯖ+<"8"ڈ#⺠>uuۅ$]kySBNt5aa5'=3_͆˚AֵTmSR^҉abn;7PK!ߴdrs/downrev.xmlDn0Dk+V@P`AmNE@?`7q؎lC{z;N\هY4r q "D:gݚ {)6"Al(ARJ0u=)F*OCNγ,Z4\} >仳ZSYϒ^caw/T [u#$d,An~PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!3pm*drs/shapexml.xmlPK-!ߴ drs/downrev.xmlPK }`Z}  Z $[1] N. Alon, R. Beigel, S. Kasif, S. Rudich,and B. Sudakov. Learning a hidden matching, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 197 206, 2002. [2] D. Angluin and J. Chen. Learning a hidden graph using O(log n) queries per edge. Manuscript, 2006. [3] D. Angluin and J. Chen. Learning a hidden hypergraph of Machine Learning Research 7, 2215-2236, 2007. [4] R. Beigel, N. Alon, S. Kasif, M. S. Apaydin and L. Fortnow. An optimal procedure for gap closing in whole genome shotgun sequencing, In RECOMB, 22 30, 2001. [5] V. Grebinski and G. Kucherov. Optimal query bounds for reconstructing a Hamiltonian cycle in complete graphs, In fifth Israel symposium on the Theory of Computing Systems, 166-173, 1997. [6] V. Grebinski and G. Kucherov. Reconstructing a Hamiltonian cycle by querying the graph: Application to DNA physical mapping. Discrete Applied Math., 88(1-3): 147 165, 1998.  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Lower bound Main Ideas Reference0Example (Algorithm A(V): Finding an edge on V) Algorithm 2 Algorithm 3 Dont Stop! Complexity v 33Concluding remarks References v 36  ϥΦr ²]pdO OLE A{ vD$0(T\ _PID_HLINKS _TemplateIDA6288,19,Reference274,21,Appendix I291,25,Appendix~Matching297,26,Appendix~Matching II-1292,17,Results Boolean model274,23,Appendix~Count.!273,20,Hamiltonian cycle ~ adap."272,12,Described into Math Part I274,22,Appendix~Count.TC011408051028Root EntrydO)^D@PicturesCurrent User,SummaryInformation(_sQ useruser  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesCurrent UserSummaryInformation(PowerPoint Document(|Q DocumentSummaryInformation8