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(@ s *  ___PPT10ph    ___PPT9    .___PPTMac11  (namd Symbol    hnamd` Arial&Monotype Typography    (namd Symbol   Given a set M of sequences, what is the minimum number Rmin(M) of recombinations needed for constructing evolutionary histories that explain M? 0 n{Z0 n{ {  [[ { {  {$$[=((,,{00,7@bB2z )@ 0Dj   >  r *@ S UR  < ^ "@ s *P*   `___PPT10@8___PPT9xpH___PPT2001$GJ___PPTMac11$  (namd Symbol     hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol   LxKreitman s data from the adh locus of D. Melanogaster (1983)z= 2    6   +@ <W ? #" @t ;M = 2${ /@ N`ۆ Ԕ?#" @`,$,P___PPT100(v___PPT9XP___PPTMac11  (namd Symbol&   7Minimization is NP-hard. (Wang et al 2000, Semple 2004)X88 n{Z  +fF6~H @ 0޽h ? 3f3FKf___PPT10.qPLY+$DO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/@%(+8+0+/@ +O 3 5=-=` L<( PB#̽# Ll L C    <  L s */h  @'@___PPT10 V___PPT980___PPTMac11  (namd Symbol    hnamd` Arial&Monotype Typography    (namd Symbol   mRmin(M)0 2    [ L s *`d  )$,p@___PPT10 V___PPT980___PPTMac11  (namd Symbol    (namd Symbol    (namd Symbol   UL(M) <2 2 C  C L 6 @ 0___PPT106___PPT9 ___PPTMac11  (namd Symbol   EM, a set of sequences] L s * c p0$,|8___PPT10F___PPT9( ___PPTMac11  (namd Symbol    (namd Symbol   K< U(M)( 2C  N L s * nUe W xp@___PPT10 V___PPT980___PPTMac11  (namd Symbol    (namd Symbol    (namd Symbol   tMinimum No efficient method.< 2022  L s *`I `P $D,H___PPT10( f___PPT9H@___PPTMac11  (namd Symbol    (namd Symbol     hnamd` Arial&Monotype Typography   (namd Symbol   w#Lower bound There are many methods.8$ 2  b  L s * e$D,p@___PPT10 V___PPT980___PPTMac11  (namd Symbol    (namd Symbol    (namd Symbol   \Upper Bound Novel.. 2  L s *`0    $@, p___PPT10PH___PPT9~ ___PPTMac11X P    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol    hnamd` Arial&Monotype Typography    (namd Symbol     hnamd` Arial&Monotype Typography  FdOur Contribution: Efficient, practical algorithms for computing lower and upper bounds on Rmin(M).e 2{{/  [ [[[ \eE5}b L s *`Qb  PF8$D,___PPT10x     ___PPT9     6___PPTMac11  (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    hnamd` Arial&Monotype Typography    (namd Symbol    (namd Symbol    (namd Symbol     hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol   Key idea: If L(M) = U(M), then we know Rmin(M). Empirical observation: L(M) = U(M) frequently for a surprisingly large range of data. 2 {     $$(({,, 0064488(_llD D H L 0޽h ? 3f3FKfJB___PPT10".qӯJ+Ia6D ' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Le%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L2%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L2%(++0+L ++0+L ++0+ L ++0+ L +z  3 pZhZp(I`Y( @e)@a) `l ` C  P`  < B ` NDԔ? 3"`@C hC   ` Hۑ ? 3"`@< l 0___PPT106___PPT9B___PPTMac11   hnamd` Arial&Monotype Typography  ;M( 2$/ ` B  ? 3"`@@ $,8___PPT10F___PPT9( V___PPTMac110(   hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography  {!1. Given a set of intervals, and(" 2!$ @`^ ` B`  ? 3"`@ $,L ___PPT10h`   ___PPT9    ___PPTMac11     hnamd` Arial&Monotype Typography$    hnamd` Arial&Monotype Typography$  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  JvComposite Problem: Find the minimum number of vertical lines so that every I intersects at least L(I) vertical lines.w 2 & {E{ [  [$$(E({,, 0AAl ` d  .`l  ,$@, ` B`  ? 3"`@i  0___PPT106___PPT9B___PPTMac11   hnamd` Arial&Monotype Typography  72( 2i ` HR ? 3"`@  0___PPT106___PPT9 ___PPTMac11  (namd Symbol{{   71( 2i ` HS ? 3"`@ qd 0___PPT106___PPT9 ___PPTMac11  (namd Symbol{{   72( 2i ` H`U ? 3"`@`  0___PPT106___PPT9 ___PPTMac11  (namd Symbol{{   72( 2i ` HU ? 3"`@P A 0___PPT106___PPT9 ___PPTMac11  (namd Symbol{{   72( 2i  ` HU ? 3"`@0 ! P 0___PPT106___PPT9 ___PPTMac11  (namd Symbol{{   73( 2i )` HU ? 3"`@ G 0___PPT106___PPT9 ___PPTMac11  (namd Symbol{{   71( 2l  P  -`ln ,$@,B  ` NDfԔ? 3"`@ 0 B  ` NDfԔ? 3"`@ B  ` NDfԔ? 3"`@F F B ` NDfԔ? 3"`@   B ` NDfԔ? 3"`@ F 0P B ` NDfԔ? 3"`@  B *` HDfԔ? 3"`@P P  0` B  ? 3"`@@( $, x___PPT10XP ___PPT9 ___PPTMac11xp ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  ,namd$ Times Roman  &2. for each interval I, a number L(I)'E{E{ E AA E {$E${ @`V 4` Bm  ? 3"`@ PP,$,___PPT10     &___PPT9     ___PPTMac11  (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{     hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol    hnamd` Arial&Monotype Typography    (namd Symbol     hnamd` Arial&Monotype Typography  4Let L(I) be a  local recombination lower bound for I. The composite recombination lower bound on Rmin(M) is given by a solution to the composite problem.7 n2d n2E{  {{{E{ !$$(( ,,00{44{88{2<<b5 @`l p  7`< ,$D, 5` <@  ? #" @  98 2( 6` NԔ?#" @p B 8` HDԔ?"0@NNN?N l B 9` HDԔ?"0@NNN?N l B :` NDԔ?"0@NNN?NP ,$D,B <` NDԔ?"0@NNN?N,,P ,$D,B =` NDԔ?"0@NNN?N, , P ,$D ,B >` NDԔ?"0@NNN?N P ,$D ,B ?` NDԔ?"0@NNN?N\\P ,$D ,B A` NDԔ?"0@NNN?NP ,$D ,B B` NDԔ?"0@NNN?N,,P ,$D ,B C` NDԔ?"0@NNN?NP ,$D,B D` TDԔ?"0@NNN?N   B E` TDԔ?"0@NNN?N B F` TDԔ?"0@NNN?N  H ` 0޽h ? 3f3FKf___PPT10.q/+!D' = @B DY' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-`%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.`%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(DT' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:`%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<`%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=`%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>`%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*?`%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A`%(D' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B`%(D' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C`%(D' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4`%(++0+` ++0+` ++0+0` ++0+4` +dn 3 R R- Q(  r  S r   < r  S A @ <   N Ԕ?#" @0 p`X`___PPT10@8___PPT9xpJ___PPTMac11$  (namd Symbol&     hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol&   H1 0 0 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1TI0 <2    B  NDԔ?#" @   N0 Ԕ?#" @0p`w0___PPT106___PPT9B___PPTMac11   hnamd` Arial&Monotype Typography   AI$( 2G{  NP Ԕ?#" @0 P `___PPT10@8___PPT9xpH___PPT2001$GJ___PPTMac11$  (namd Symbol&     hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol&   $1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 0 1T%0 <2   N` Ԕ?#" @0   h___PPT10H@___PPT9H___PPT2001$G^___PPTMac1180  (namd Symbol&     hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol&   $0 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 1`%8 <2 @B  TDԔ?"0@NNN?N@   $D,PH___PPT2001$@GB  TDԔ?"0@NNN?N@pp ,$@, # N` Ԕ?#" @0 @ $,X`___PPT10@8___PPT9xpJ___PPTMac11$  (namd Symbol&     hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol&   1 0 1 1 0 1 1 1 0 0 1 1T0 <2  $ N0 Ԕ?#" @ $ ,X`___PPT10@8___PPT9xpJ___PPTMac11$  (namd Symbol&     hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography    hnamd` Arial&Monotype Typography   (namd Symbol&   $0 0 1 1 1 0 0 1 0 0 1 1 1 0 1 1 1 1T%0 <2  ' Np Ԕ?#" @@`8P $, x___PPT10XP ___PPT9 ___PPTMac11  (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol    (namd Symbol&    (namd Symbol    (namd Symbol&    (namd Symbol&    (namd Symbol&   h(S) = 4-2-1 = 1( 2GEG  GG G $$ ) Np Ԕ?#" @04\ $ ,( p___PPT10PH___PPT9___PPTMac11  (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol    (namd Symbol&    (namd Symbol    (namd Symbol&    (namd Symbol&    (namd Symbol&   h(S) = 6-3-1 = 2( 2GG G GG  l @P  ,@P ,$D  ,B  TDԔ?"0@NNN?NP@P B * TDԔ?"0@NNN?N@ H  0޽h ? 3f3FKf___PPT10.qAR+<DG' = @B D' = @BA?%,( < +O%,( < +D\' =%(D' =%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =%(D' =%(Du' =4@BBB B%()))D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(+P+0+ ++0+ ++0+# ++0+$ ++0+' ++0+) +oA 3 ''}'( fito{5 7  B  ? 3"`@PM, $, ___PPT10ph    ___PPT9    Z ___PPTMac114 ,   (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{   wMyers & Griffiths : For every interval I, restrict the maximum size (s) of S and the maximum distance (w) between the leftmost and the rightmost columns in S. 2 E G{ $$3((,G,{00B  NDԔ?"0@NNN?N ,$@,B  NDԔ?"0@NNN?N@@,$D,B  NDԔ?"0@NNN?N,$D,B  NDԔ?"0@NNN?N,$D ,B  NDԔ?"0@NNN?N,$D,z   0    9 ,$D,B   BDԔ? #" @     <Ԕ?"0@NNN?Np 0 ;I( 2Ad  <0Ԕ?"0@NNN?N,$, |S| < s,h ( 2G G Oz      ,$@ ,B  HDԔ?"0@NNN?N 0 B  HDԔ?"0@NNN?N0  B  HDԔ?"0@NNN?N { { B  HDԔ?"0@NNN?N~ ~   <Ԕ?"0@NNN?N^   9d( 2=  <КԔ?"0@NNN?Np,$ , ed < wD( 2G  <Ԕ?"0@NNN?N  ,$ , 7Implemented in RecMin, along with the composite method.@8 2{" #  s 0e0e ? 3"`@   < l P P  PP,$D ,  <0Ԕ?"0@NNN?NP   VRecMin is a tremendous improvement over previous practical lower bound methods. But, \W0 P2H  Q "  н0e0e ? S"0e`@P` P <z___PPTMAC11TL  (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{    (namd Symbol{{   "  @H  0޽h ? 3f3FKf*"___PPT10.qO+D' = @B D' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D] ' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ ++0+ + 3 3(  O  <@Ԕ?"0@NNN?NsP  (Implemented in HapBound) R 2{{  {c  <Ԕ?"0@NNN?N0 |tl___PPT9NF ;No parameters. Much faster than RecMin. Implements additional ideas that produce lower bounds even better than the optimal haplotype bound.J0 n2 <)  [, U  @`  BN ? 3"`@ `xp___PPT10     &___PPT9     ___PPTMac11d\  hnamd` Arial&Monotype Typography   How to derive sharper bounds? In the composite method, check if each local bound L(I) is in fact equal to Rmin(I), and if not, increase L(I) by one. (-S and -M options) 2{4G{ { {{E{  {$$(G({ ,,0A0448A8 <<,D%;  NWԔ?"0e@   @ABC DEFGHIJK5%LMNOp@H@`___PPT10@8___PPT9xp2___PPTMac11   (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol&   We cast the problem as a classic set cover problem that can be formulated as an ILP problem, with 1 variable per column and 1 inequality per pair of rows. We can compute the optimal haplotype bound exactly.j x{2! {%  {[[,  @`#  N|1?"0e@   @ABC DEFGHIJK5%LMNO b0___PPT106___PPT9 ___PPTMac11  (namd Symbol&   ]'Can use either GNU ILP Solver or CPLEX.(( 2(H  0޽h ? 3f3FKf___PPT10u.u0+D=' @= @B +l  3 wo!,h(  hl h C   < Wj   ,h#"*  h <Г ?"  R1643 sec   @` h < ?"0  L78 @`  h < ?"  0   HapBound -S4  C @` h <ଜ ?"   P31 sec @`  h <@œ ?"0   L75 @`  h <͜ ?"  0  lHapBound   @`  h <؜ ?"   P5 days @`  h <ڜ ?"0   S No result   @`0  h < ?"  0  RecMin -s 48 -w 48PC C   @` h < ?"   R7944 sec   @` h < ?"0   L75 @`0 h <p  ?"  0  RecMin -s 25 -w 25PC C   @` h <  ?"  O3 sec @` h <@ ?"0   L59 @`  h <> ?" 0  :RecMin  s 8  w 12 (default) @` h <I ?" RTime{ @` h <T ?"0  Y Lower Bound  { @` h <_ ?" 0  UProgram{ @`tB h 6o ?" nB h 01 ?" nB h 01 ?"   nB h 01 ?"   nB h 01 ?"   nB h 01 ?"   tB h 6o ?" tB h 6o ?"  nB h 01 ?"0 0 nB  h 01 ?"tB !h 6o ?" &h HwԔ?"0@NNN?NP 0 p L88 Sequences, 48 sites 2H h 0޽h ? 3f3FKf___PPT10u.q| +D=' ϐ= @B +[ 3 ;;LT;(  B @ HDԔ?"0@NNN?N B ? HDԔ?"0@NNN?Npp0B  HDԔ?"0@NNN?Np 0B  HDԔ?"0@NNN?Np p B  HDԔ?"0@NNN?Np 0B  HDԔ?"0@NNN?N 0B  HDԔ?"0@NNN?N @ B  HDԔ?"0@NNN?N @ @ 0B  HDԔ?"0@NNN?N xxp B  HDԔ?"0@NNN?N  B  HDԔ?"0@NNN?N 0B  HDԔ?"0@NNN?N B  HDԔ?"0@NNN?N`@@0B  HDԔ?"0@NNN?N pp0B  HDԔ?"0@NNN?N ``0B  HDԔ?"0@NNN?N p` B @ HDԔ?"0@NNN?N@ B  HDԔ?"0@NNN?N 0B  HDԔ?"0@NNN?N@@ B  HDԔ?"0@NNN?N p B  HDԔ?"0@NNN?Np p B  HDԔ?"0@NNN?N B @ HDԔ?"0@NNN?N@ B  HDԔ?"0@NNN?Np B  HDԔ?"0@NNN?N0 0 B  HDԔ?"0@NNN?N B  HDԔ?"0@NNN?N00B  HDԔ?"0@NNN?N0B  HDԔ?"0@NNN?N@0B  HDԔ?"0@NNN?N@@B " HDԔ?"0@NNN?N0 B # HDԔ?"0@NNN?N B $ HDԔ?"0@NNN?NB % HDԔ?"0@NNN?N0@B & HDԔ?"0@NNN?N0P 0B ' HDԔ?"0@NNN?N0P P B ( HDԔ?"0@NNN?N02 ) HԔ?"6@`NNN?Np`2 * HԔ?"6@`NNN?N 2 + HXXԔ?"6@`NNN?N p` 2 , HXXԔ?"6@`NNN?N p 2 - HXXԔ?"6@`NNN?N  2 . HXXԔ?"6@`NNN?N ` 2 / HXXԔ?"6@`NNN?N 2 0 HXXԔ?"6@`NNN?NP 2 1 HXXԔ?"6@`NN      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxz{|}~N?N 2 H Ԕ?"6@`NNN?N PP XPH___PPT2001$D G 3 H Ԕ?"6@`NNN?N PXPH___PPT2001$D G 4 H Ԕ?"6@`NNN?NP P XPH___PPT2001$D G 5 H Ԕ?"6@`NNN?N` PP XPH___PPT2001$D G 6 H Ԕ?"6@`NNN?NP` XPH___PPT2001$D G ; H Ԕ?"6@`NNN?NP XPH___PPT2001$D  G2 < HXXԔ?"6@`NNN?N `  = H Ԕ?"6@`NNN?N PXPH___PPT2001$D  G 8 H Ԕ?"6@`NNN?NPXPH___PPT2001$D  GB A@ HDԔ?"0@NNN?N p B H Ԕ?"6@`NNN?N`P XPH___PPT2001$D  G C H Ԕ?"6@`NNN?N PXPH___PPT2001$D G D <Ԕ?"0@NNN?N0P 31 2 E <Ԕ?"0@NNN?N0P 39 2 F <` Ԕ?"0@NNN?N0P 37 2 G <@Ԕ?"0@NNN?N0PP 410 2 H <Ԕ?"0@NNN?N0pP 33 2 I <Ԕ?"0@NNN?N0p P 36 2 J <P Ԕ?"0@NNN?N0 P 35 2 K <%Ԕ?"0@NNN?N0 P 32 2 L <)Ԕ?"0@NNN?N00 P 38 2 M <`.Ԕ?"0@NNN?N0`P 34 22 N HXXԔ?"6@`NNN?N` P 2 O HԔ?"6@`NNN?N P  P <p6Ԕ?"0@NNN?N P  <Mutation ( 2  Q <;Ԕ?"0@NNN?N P@ A Recombination( 2 R <=   <  S <@BԔ?"0@NNN?Np  Branch and Bound construction of genealogies backwards in time (using an alternating series of coalescent, mutation, and recombination events).< 2{R  T H Ԕ?"6@`NNN?NP XPH___PPT2001$D GH  0޽h ? 3f3FKf___PPT10.v$F'+DQ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*TD' =1:Bhidden*o3>+B#style.visibility<*T%(D ' =%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*3D' =1:Bhidden*o3>+B#style.visibility<*3%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*2D' =1:Bhidden*o3>+B#style.visibility<*2%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*4D' =1:Bhidden*o3>+B#style.visibility<*4%(D' =%( D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*5D' =1:Bhidden*o3>+B#style.visibility<*5%(D' =%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*6D' =1:Bhidden*o3>+B#style.visibility<*6%(Dr' =%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*CD' =1:Bhidden*o3>+B#style.visibility<*C%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*BD' =1:Bhidden*o3>+B#style.visibility<*B%(D' =%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*8D' =1:Bhidden*o3>+B#style.visibility<*8%(Dr' =%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*=D' =1:Bhidden*o3>+B#style.visibility<*=%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*;D' =1:Bhidden*o3>+B#style.visibility<*;%(+/   3 : 2 p (  p p C `vS < 0E%}^  p C 6A14-alleles-ARG0 0A  p HP7 Ԕ?"0@NNN?NU  MSHRUB constructs genealogies that can be viewed using an open source program.(N 2I p N`LW8c?"0e@   @ABC DEFGHIJK5%LMNO( $`XP___PPT100(v___PPT9XPz___PPTMac11TL  (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol&    (namd Symbol&   'Contains U(M) recombination vertices.D( 2 [  &B p NDԔ?"0e@   @ABC DEFGHIJK5%LMNO  H p 0޽h ? 3f3FKf___PPT10u. q,+D=' n= @B +  3    ( {5 l  C `[W@  <   NA ?o?"0@NNN?NP`< d PH___PPT2001$@G  BPTԔ?"0@NNN?N@ F U11 alleles of the alcohol dehydrogenase locus of Drosophila melanogaster. (43 Sites)8V 21{ F       bB2z   BiW ? 3"`@ Eph___PPTMAC11  (namd Symbol    (namd Symbol  d___PPT10D<___PPT9      There is only one previous implemented method that computes Rmin(M) exactly. (Song and Hein, 2003) That method took about 1.5 GB of memory and 30 minutes of CPU time to find Rmin(M) = 7. c 2_ 2<[ [  [ { $$[( ([,,[0044,<s  @`H  0޽h ? 3f3FKf___PPT10u.qu#+D=' n= @B +P  3 VN ( PF?0D;   H`X 1? 3"`@P`XP___PPT100(v___PPT9XPz___PPTMac11TL  (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol   mWe tried 9 different implemented lower bound methods, aside from HapBound. They all produced either 5 or 6.Fn 2 +{+  {A%l  0S 0 S,$D,t  S 6AKreitmanARG"SB  NDԔ? 3"`@v  Hs ? C"`@`v  0___PPT106___PPT9 ___PPTMac11  (namd Symbol   >Time( 2j    H0y ? 3"`@P06H@`___PPT10@8___PPT9xp2___PPTMac11   (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol   Both HapBound (with  M option) and SHRUB produced 7 and took only a fraction of a second to analyze this data set.^s 2[  [{ e  Bj Ԕ?"0@NNN?N`f <L(M) = U(M) = 7 Rmin(M) = 7.\( 2[[[ [[ c  BPԔ?"0@NNN?N H  EAn evolutionary history, found by SHRUB, with 7 recombination events..F 2"[=,  B0PԔ?"0@NNN?N   z0It corresponds to the most parsimonious history..1 2{ H  0޽h ? 3f3FKf___PPT10z.qj+]KDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+i 3 tl ( {5 l  C p/P`@  <   6AԔ?"0@NNN?N` <  s *`3P@'0___PPT106___PPT9 ___PPTMac11  (namd Symbol   J(88 Sequences, 48 sites) 2~  0CP1 P@ TLX___PPT1080___PPT9h`V___PPTMac110(  (namd Symbol&[[    (namd Symbol&[[    (namd Symbol[[    (namd Symbol&[[    (namd Symbol[[    (namd Symbol&[[   "Our new lower bound HapBound -S -Mb# 2 [E[  [E[[ p  0XP1 @1@___PPT10 V___PPT980___PPTMac11  (namd Symbol    (namd Symbol   hUpper bound SHRUB: 2 2 [B  NDԔ?#" @ p B  NDԔ?#" @ p   s * ]P`G0___PPT106___PPT9 ___PPTMac11  (namd Symbol   N(We ignored insertion/deletion, unphased sites, and sites with missing data.)O 2O &&  H`[P1?"0@NNN?N 0 v"Composite optimal haplotype bounds# 2# B  NDԔ?#" @8  H  0޽h ? 3f3FKf___PPT10u.q+D=' Pc= @B +  3   (  r  S `P`  <   6AԔ?"0@NNN?Nx   BP ? 3"`@E` \Tz___PPTMAC11TL  (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol  L___PPT10,$v___PPT9XP pr = scaled recombination rate. q = scaled mutation rate.H90 72AA   BpPԔ?"0@NNN?N WFrequency of having L(M) = U(M) ( 2   s *НPpPTLX___PPT1080___PPT9h`V___PPTMac110(  (namd Symbol    (namd Symbol     (namd Symbol    (namd Symbol     (namd Symbol    (namd Symbol    ,Used Hudson s MS to generate1000 simulated datasets for each pair of q and r.vN 2)G  GG  s * P @ `XP___PPT100(v___PPT9XPz___PPTMac11TL  (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol    (namd Symbol   For q, r < 5, our lower and upper bounds match over 90% of the time.^E x2CC C 7 @`  N@PԔ?#" @ *p___PPT10PH___PPT9F___PPTMac11   (namd Symbol&    hnamd` Arial&Monotype Typography   {This is a significant progress, as there currently exists no other method that can find Rmin(M) for more than 9 sequences after some data reduction. x2)  [[[4 Y9 @`  HPԔ?"0@NNN?N@ ` Wn = number of sequences$( 2H  0޽h ? 3f3FKf___PPT10u.v@"+D=' ϐ= @B +} 3  (    l  C PP  < l  C 0P00 < H  0޽h ? 3f3FKf___PPT10u.q~L+D=' = @B + xP( ?? X  C  0   <x  3 5H   @  <  H  0޽hx ? 3380___PPT10.q7 x<(  <X < C  0   <x < 3 s  @  <  H < 0޽hx ? 3380___PPT10.qTn xD( 5 DX D C  0   <x D 3 s  @  <  H D 0޽hx ? 3380___PPT10.qp P( 2`# PX P C  0   < P S t  @  <@___PPT10 V___PPT980J___PPTMac11$   hnamd` Arial&Monotype Typography   @Want efficient computation of lower bound there are many methods&A*H P 0޽hx ? 3380___PPT10.qv  xd(  dX d C  0   <x d 3 t  @  <  H d 0޽hx ? 3380___PPT10.q8<  xl(  lX l C  0   <x l 3 wt  @  <  H l 0޽hx ? 3380___PPT10.qRBk  x t(  tX t C  0   <x t 3 t  @  <  H t 0޽hx ? 3380___PPT10.qN  0^( WN?bN? X  C  0   <  S t  @  < `*The most parsimonious evolutionary history#H  0޽hx ? 3380___PPT10.q!e x(  X  C  0   <x  3 t  @  <  H  0޽hx ? 3380___PPT10.qU x(  {4   X  C  0   <x  3 ps  @  <  H  0޽hx ? 3380___PPT10.q&' x(   X  C  0   <x  3 jt  @  <  H  0޽hx ? 3380___PPT10.q&' x0(  @  X  C  0   <x  3 PG   @  <  H  0޽hx ? 3380___PPT10.q&' xP( ??^  X  C  0   <x  3 pt  @  <  H  0޽hx ? 3380___PPT10.q&' x@( ??c  X  C  0   <x  3 t  @  <  H  0޽hx ? 3380___PPT10.q&' x`(  X  C  0   <x  3 I   @  <  H  0޽hx ? 3380___PPT10.qah x(  X  C  0   <x  3 ]t  @  <  H  0޽hx ? 3380___PPT10.q[ P `(  ^ S 0   < # l@$tbf1bf1#" 0e @  < jBetter bounds are produced if parameters s and w are increased. 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