Quartets and unrooted level-k networks

Phylogntis Workshop, Is Nwton Institut or Mthmtil Sins Cmrig 21/06/2011 Qurtts n unroot lvl-k ntworks Philipp Gmtt

Outlin Astrt n xpliit phylognti ntworks Lvl-k ntworks Unroot lvl-1 ntworks n irulr split systms Ronstrution rom triplts n qurtts

Astrt n xpliit phylognti ntworks Astrt phylognti ntworks: Visuliz rtiult volution t Expliit phylognti ntworks: Eh no n intrprt s urrnt or nstrl spis split ntwork SplitsTr lvl-2 ntwork Simplisti min ntwork gll ntwork Ntwork Dnrosop

Split ntworks Astrt phylognti ntworks: Visuliz rtiult volution t Split ntwork: Visuliz st o splits Split 1 Björn M. Hllström, Axl Jnk - Mmmlin volution my not stritly iurting, MBE, 2010

Split ntworks Astrt phylognti ntworks: Visuliz rtiult volution t Split ntwork: Visuliz st o splits Split 2 Björn M. Hllström, Axl Jnk - Mmmlin volution my not stritly iurting, MBE, 2010

Split ntworks Astrt phylognti ntworks: Visuliz rtiult volution t Split ntwork: Visuliz st o splits Split 3 Björn M. Hllström, Axl Jnk - Mmmlin volution my not stritly iurting, MBE, 2010

Ntwork sulss hirrhy split systm xpliit root wkly omptil k-omptil rgulr tr-siling tr-hil lvl k lvl-2 gll ntwork nst 2-omptil norml lvl-1 irulr uniyli omptil tr simpl lvl-1 split systms ontins xpliit root ntworks http://www.lirmm.r/~gmtt/rphylogntintworks.php

Outlin Astrt n xpliit phylognti ntworks Lvl-k ntworks Unroot lvl-1 ntworks n irulr split systms Ronstrution rom triplts n qurtts

Lvl-k ntworks lvl: how r is th ntwork rom tr? smll lvl tr strutur st lgorithms h 1 h 3 h 2 lvl = mximum numr o rtiultions y riglss omponnt (lo) o th unrlying unirt grph. g h i lvl-2 ntwork j k Choy, Jnsson, Skn & Sung, TCS, 2005

Lvl-k ntworks lvl: how r is th ntwork rom tr? smll lvl tr strutur st lgorithms lvl = mximum numr o rtiultions y lo. g h i lvl-2 ntwork j k lvl-1 ntwork ( gll tr ) g h i j k

Unroot lvl-k ntworks lvl: how r is th ntwork rom n unroot tr? smll lvl tr strutur st lgorithms i h g lvl = mximum numr o gs to rmov, y lo, to otin tr. unroot lvl-2 ntwork Gmtt, Brry & Pul, mnusript, 2011

Unroot lvl-k ntworks lvl: how r is th ntwork rom n unroot tr? smll lvl tr strutur st lgorithms i h g lvl = mximum numr o gs to rmov, y lo, to otin tr. = mximum ylomti numr o th los unroot lvl-2 ntwork

Unroot lvl-k ntworks lvl: how r is th ntwork rom n unroot tr? smll lvl tr strutur st lgorithms i h g lvl = mximum numr o gs to rmov, y lo, to otin tr. unroot lvl-1 ntwork (unroot gll tr) tr o yls

Unroot lvl-k ntworks i h g lvl = mximum numr o gs to rmov, y lo, to otin tr. unroot lvl-k ntwork tr o los tr o gnrtors o lvl k Unroot lvl-k gnrtors: riglss looplss 3-rgulr multigrphs with 2k- 2 vrtis lvl-2 gnrtor lvl-3 gnrtors Gmtt, Brry & Pul, mnusript, 2011

Counting ll lvl-k ntworks Unroot lvl-1 ntworks: xpliit ormul or n lvs, yls, m gs ross yls Smpl & Stl, TCBB, 2006

Counting ll lvl-k ntworks Unroot lvl-1 ntworks: xpliit ormul or n lvs, yls, m gs ross yls + symptoti vlution or n lvs: 0.207 n n-1 1.890 n Root lvl-1 ntworks : Expliit ormul or n lvs, yls, m gs ross yls + symptoti vlution or n lvs: 0.134 2.943 n n n-1 Smpl & Stl, TCBB, 2006 Unroot lvl-2 ntworks : Expliit ormul or n lvs : n+i-1 4i+j-1 i k p q -4 s 9 i 23 k -21 q (n-1)! Σ i j k p q s 21 2 9 (-1) p 46 0 s q p k i n-1 j=n-1-i-k-p-q-s 0 i 0 numr o lvs 2 3 4 5 6 7 unroot lvl-1-2 15 192 3 450 79 740 root lvl-1 3 36 723 20 280 730 755 32 171 580 unroot lvl-2-9 282 14 697 1 071 720 100 461 195 Bouvl, Gmtt, Brry & Pul, mnusript, 2011

Equivln twn root n unroot lvl g i h rooting Rooting: - hoosing root - hoosing n orinttion or th gs g h i Gmtt, Brry & Pul, mnusript, 2011

Equivln twn root n unroot lvl g i h rooting Rooting: - hoosing root - hoosing n orinttion or th gs g h i - mny possil rootings (possily xponntil in th lvl) - sm lvl (invrint) Gmtt, Brry & Pul, mnusript, 2011

Outlin Astrt n xpliit phylognti ntworks Lvl-k ntworks Unroot lvl-1 ntworks n irulr split systms Ronstrution rom triplts n qurtts

Splits in unroot lvl-k ntworks Split: Split o tr ontin in th ntwork? Lvs sprt y miniml ut in th ntwork? Woolly, Pos & Crnll, PLoS On, 2008 Brns & Cornlsn, DAM, 2009 i h g

Splits in unroot lvl-k ntworks Biprtition : Split o tr ontin in th ntwork Lvs sprt y miniml ut in th ntwork i h g quivlnt! i h g Gmtt, Brry & Pul, mnusript, 2011

Rprsnting splits y unroot ntworks rl ntwork g h i trs g h i g h i

Rprsnting splits y unroot ntworks rl ntwork g h i trs g h i g h i splits SplitsTr i h g split ntwork

Rprsnting splits y unroot ntworks rl ntwork unroot lvl-1 ntwork i g h i h trs splits g h i splits g h i SplitsTr ghi i g ghi h g split ntwork

Splits in unroot lvl-1 ntworks Splits Σ(N) o lvl-1 ntwork i h g N i h g Cirulr split systm Σ

Splits in unroot lvl-1 ntworks Splits Σ(N) o lvl-1 ntwork i h g Σ(N) irulr N i h g Cirulr split systm Σ Gmtt, Brry & Pul, mnusript, 2011

Splits in unroot lvl-1 ntworks Splits Σ(N) o lvl-1 ntwork i h g Σ(N) irulr N i h g Cirulr split systm Σ Thr xists lvl-1 ntwork N suh tht Σ Σ(N) Gmtt, Brry & Pul, mnusript, 2011

Outlin Astrt n xpliit phylognti ntworks Lvl-k ntworks Unroot lvl-1 ntworks n irulr split systms Ronstrution rom triplts n qurtts

Comintoril phylognti ntwork ronstrution spis 1 : AATTGCAG TAGCCCAAAAT spis 2 : ACCTGCAG TAGACCAAT spis 3 : GCTTGCCG TAGACAAGAAT spis 4 : ATTTGCAG AAGACCAAAT spis 5 : TAGACAAGAAT spis 6 : ACTTGCAG TAGCACAAAAT spis 7 : ACCTGGTG TAAAAT G1 G2 {gn squns} {trs} T1 T2 HOGENOM ts Duyr, Durt, Pnl, Gouy, Rhnmnn & Prrièr, BioIn, 2005 ntwork ontins th trs + optiml

Ronstrution rom triplts / qurtts {gn squns} {trs} {triplts} {qurtts} ntwork ontins th qurtts/triplts + optiml

Ronstrution rom triplts / qurtts {gn squns} {trs} {triplts} {qurtts} triplt ntwork ontins th qurtts/triplts + optiml

Ronstrution rom triplts / qurtts {gn squns} {trs} triplt {triplts} {qurtts} qurtt ntwork ontins th qurtts/triplts + optiml

Ronstrution rom triplts / qurtts {trs} {triplts} {qurtts} ntwork N ntwork N' N'=N?

Ronstrution rom triplts / qurtts A ntwork ontining ll qurtts o tr T os not lwys ontin T. T N N ontins ll qurtts o T ut not T

Ronstrution rom triplts / qurtts {trs} {triplts} {qurtts} ntwork N ntwork N' N'=N? Not lwys, ut {N} {N'}

Ronstrution rom triplts / qurtts Fining ll triplts o root ntwork: O(n 3 ) Byrk, Gwryhowski, Hur & Klk, JDA, 2010

Ronstrution rom triplts / qurtts Fining ll triplts o root ntwork: O(n 3 ) Byrk, Gwryhowski, Hur & Klk, JDA, 2010 Fining ll qurtts o n unroot ntwork? i h g qurtt

Ronstrution rom triplts / qurtts Fining ll triplts o root ntwork: O(n 3 ) Byrk, Gwryhowski, Hur & Klk, JDA, 2010 Fining ll qurtts o n unroot ntwork? i h g qurtt 2-isjoint pths -,-

Ronstrution rom triplts / qurtts Fining ll triplts o root ntwork: O(n 3 ) Byrk, Gwryhowski, Hur & Klk, JDA, 2010 Fining ll qurtts o n unroot ntwork: O(n 6 ) 2-Disjoint Pths in grph o gr 3: O(n(1+α(n,n))) h g i Tholy, SOFSEM'09, 2009 qurtt 2-isjoint pths -,-

Tr ronstrution unroot rom qurtts root rom triplts gnrl NP-omplt Stl, JOC, 1992 polynomil Aho, Sgiv, Szymnski & Ullmn, SJOC, 1981 Hnzingr, King & Wrnow, ALG, 1999 Jnsson, Ng, Skn & Sung, ALG, 2005 ns t lst on qurtt or h st o 4 lvs O(n 4 ) Brry & Gsul, TCS, 2000 O(n 3 ) Aho t l., SJOC, 1981

Ronstrution o lvl-k ntworks unroot rom qurtts root rom triplts lvl 1 lvl k>1 lvl 1 lvl k>1 gnrl NP-omplt Grünwl, Moulton & Spillnr, DAM, 2009? NP-omplt Jnsson, Nguyn & Sung, SJOC, 2006 NP-omplt Vn Irsl, Klk & Mnih, JBCB, 2009 ns t lst on qurtt or h st o 4 lvs? (omposition in polynomil tim)? O(n 3 ) Jnsson, Nguyn & Sung, SJOC, 2006 O(n 5k+4 ) To & Hi, CPM'09 omplt ll qurtts o th ntwork O(n 4 )? (omposition in polynomil tim) O(n 3 ) Jnsson, Nguyn & Sung, SJOC, 2006 O(n 3k+3 ) Vn Irsl & Klk, ALG, 2010 Gmtt, Brry & Pul, mnusript, 2011

Qurtt st omposition A A SN-split o th qurtt st Q: For ll lvs x,y A, z,t A, Th only qurtt o Q on {x,y,z,t} is xy zt Root ontxt: SN-st Jnsson & Sung, TCS, 2006 To & Hi, CPM'09 Q is ns qurtt st SN-splits o Q r omptil (n rprsnt y n unroot tr) Computing th SN-splits: O(n 4 ) vrint o th Q* lgorithm Brry & Gsul, TCS, 2001 Root ontxt: Computing th SN-sts: O(n 3 ) Jnsson, Nguyn & Sung, TCS, 2006 Gmtt, Brry & Pul, mnusript, 2011

Qurtt st omposition Q(N), th st o ll qurtts ontin in n unroot lvl-k ntwork N SN-splits o Q(N) r ijtivly ssoit with ut-gs o N. N h g gh gh gh gh h g gh gh gh g h gh gh Gmtt, Brry & Pul, mnusript, 2011

Blo ronstrution rom qurtts W n sprt th los o N rom Q(N). How to ronstrut h lo o N? Lvl-1 ntwork ronstrution rom th omplt qurtt st: N, lvl-1 ntwork Fining n orring o th lvs roun th yl: O(n 2 ) Qurtt irulr puzzling Fix our lvs,,,:

Blo ronstrution rom qurtts W n sprt th los o N rom Q(N). How to ronstrut h lo o N? Lvl-1 ntwork ronstrution rom th omplt qurtt st: N, lvl-1 ntwork Fining n orring o th lvs roun th yl: O(n 2 ) x Qurtt irulr puzzling Fix our lvs,,, (, ) For h othr l: For h possil position on th yl: Tst in O(1) i it is th orrt on. x + x?

Link with non-btwnnss Btwnnss: Input: - st X o lmnts - st C o twnnss onstrints: twn n Output: - orring σ whih rspts th onstrints o C

Link with non-btwnnss Non-Btwnnss: Input: - st X o lmnts - st C o non-twnnss onstrints: not twn n Output: - orring σ whih rspts th onstrints o C

Link with non-btwnnss Cirulr Non-Btwnnss: Input: - st X o lmnts - st C o irulr non-twnnss onstrints: sn rom, not twn n Output: - irulr orring σ whih rspts th onstrints o C

Link with non-btwnnss Cirulr Non-Btwnnss: Input: - st X o lmnts - st C o irulr non-twnnss onstrints: sn rom, not twn n Output: - irulr orring σ whih rspts th onstrints o C quivlnt to unroot lvl-1 lo ronstrution rom qurtts

Link with non-btwnnss Gnrl s Dns s Btwnnss Non-Btwnnss Cirulr Non-Btwnnss NP-omplt Optrny, JOC, 1979 NP-omplt Guttmn & Muhr, IFIP-TCS'06 NP-omplt Grünwl, Moulton & Spillnr, DAM, 2009 polynomil??

Thnk you! Couthors o ths rsults: Vinnt Brry & Christoph Pul (Montpllir), Mthil Bouvl (Borux) Unroot lvl-2 ntwork uilt rom th mnusript with TrClou SplitsTr T-Rx http://www.trlou.org