Structure in solution spaces: Three lessons from Jean-Claude

Transcription

1 Struture in solution spes: Three lessons from Jen-Clue Dvi Eppstein Computer Siene Deprtment, Univ. of Cliforni, Irvine Conferene on Meningfulness n Lerning Spes: A Triute to the Work of Jen-Clue Flmgne Ferury 27 28, 2014

2 My speilty: Algorithm esign The lgorithm esign proess: Fin omputtionl tsk in nee of solution Astrt wy unimportnt etils Often, nive lgorithm exists ut is too slow Design lgorithms tht re fster (sle etter with prolem size) without srifiing solution qulity CC-BY imge 2012 Itlin GP - Ferrri pit.jpg y Frneso Cripp from Wikimei ommons

3 ALEKS ir AC ABCDEFG A ABCDEFGH ABCDEF ABCDE ABCDF ABDFH ABCE ABCD ABDF ACE ABC ABD AB Ø ACBDEFH B ABCDFH Limite to qusi-orinl lerning spes: Wht stuent knows is represente s finite set, the set of onepts the stuent hs mstere Lerning spe: the fmily of sets tht oul possily e the stte of knowlege of some stuent Qusi-orinl: the intersetion or union of ny two sets in the fmily is nother set in the fmily

4 Wht s wrong with qusi-orinl spes? Closure uner unions mkes sense psyhologilly, ut losure uner intersetions oes not This uses the spes to hve more sets thn they shoul (intersetions tht n t relly hppen) The extr sets inrese the numer of test questions neee to ssess stuent, slow own the ssessment lultions, n le to inuries in the ssessments Beuse of these prolems, JCF ws esperte to eliminte this restrition.

5 If qusi-orinl spes re, why use them? Mthemtilly, qusi-orinl spes form istriutive ltties Birkhoff s representtion theorem: the sets in these spes n e represente s ownwr-lose susets of prtil orer ABCDEFGH ABCDEFG ACBDEFH ABCDEF ABCDFH G H ABCDE ABCDF ABDFH E F ABCE ABCD ABDF C D ACE ABC ABD A B AC AB A B Ø

6 But wht oes it men? A prtil orer on set of onepts to e lerne esries prerequisite reltion A stuent will only eome rey to mster onept fter he or she hs mstere ll its prerequisites CC-BY-NC imge from

7 Avntges of using the unerlying prtil orer It s onise Only the prerequisite reltion nees to e ommunite to lient softwre It s fst Key omputtionl ottlenek: listing ll sttes in the lerning spe Time per stte #onepts / mhine wor size [s implemente in erly versions of ALEKS] Cn theoretilly e improve to O(log #onepts) per stte [Squire 1995]

8 Lesson I When your stte spe forms istriutive lttie, fin out wht the unerlying prtil orer mens, n tke vntge of it for fst n spe-effiient lgorithms

9 Applition: Retngulr rtogrms Stylize geogrphi regions s retngles Retngle res represent numeril t [Risz, The retngulr sttistil rtogrm, Geog. Rev. 1934] Styliztion emphsizes the ft tht it s not mp The simpliity of the shpes mkes res esy to ompre

10 Formliztion of rtogrm onstrution Fin prtition of retngle into smller retngles, stisfying: Ajeny: Geogrphilly jent regions shoul sty jent Orienttion: Avoi gross geogrphi misplement (e.g. Cliforni shoul not e north of Oregon) Are universlity: Cn just to ny esire set of res while preserving jeny Are-universl Not re-universl

11 Comintoril lnguge for esriing lyouts Augment jeny grph with four extr verties, one per sie of outer retngle Color sie-y-sie jenies lue, orient left to right Color ove jenies re, orient top to ottom Lyouts for given set of jenies orrespon 1-for-1 with lelings in whih the four olors n orienttions hve the orret lokwise orer t ll verties

12 Lol hnges from one lyout to nother Chnge olors/orienttions within qurilterl Correspons to twisting either the ounry etween two regions (s shown) or retngle surroune y four others

13 The istriutive lttie of lyouts n lelings (,0) (,0) (,0) (,1) (,0) (,0) (,0) (,0) (,0) (,1) (,0) (,0) (,0) (,0) (,1) (,0) (,0) (,0) (,0) (,1) (,0) (,0) (,0) (,0) (,1) (,0) (,0) (,0) (,0) (,1) (,0) (,0) (,0) (,0) (,1) (,0) (,0) (,0) (,0) (,1)

14 Wht oes it men? Elements = lyouts Neighors = lyouts tht iffer y single twist (,0) (,0) (,1) (,0) (,0) Upwr in lttie orer = twist ounterlokwise Elements of prtil orer = numer of times eh ounry hs een twiste Are-universl if n only if no ege twists re possile

15 Results of pplying Lesson I Although there my e exponentilly mny lyouts, the unerlying prtil orer hs polynomil size n n e onstrute in polynomil time By working t the prtil orer level of strtion, n effiiently fin n re-universl lyout (if it exists) with ritrry onstrints on ounry orienttions E., Mumfor, Spekmnn, & Vereek, Are-universl n onstrine retngulr lyouts, SIAM J. Comput. 2012

16 Beyon qusi-orinl spes My ontriution to ALEKS: Inste of prerequisites, esrie lerning spe y its lerning sequenes: orerings in whih ll onepts in the spe oul e lerne Sttes = unions of prefixes of the lerning sequenes ABC AB AC BC This exmple hs four lerning sequenes: A B C, A C B, C A B, n C B A A C Only two, A B C n C B A, suffie to efine the whole spe Ø

17 Avntges of the lerning sequene formultion Cple of representing every lerning spe tht is essile (n e lerne one onept t time) n lose uner unions Mthemtilly, suh spe forms n ntimtroi As with qusi-orinl spes, n list ll sttes quikly (the key step in stuent ssessment) Still quite onise Cn onstrut esription using the smllest possile set of lerning sequenes in time polynomil in the numer of sttes

18 Reltion to qusi-orinl spes n prtil orers A lerning sequene of qusi-orinl spe is liner extension of its unerlying prtil orer, or equivlently topologil orering of its prerequisite reltion. (A sequene of the verties of irete yli grph suh tht eh ege is oriente from erlier to lter in the sequene.) Thus, lerning sequenes provie nturl metho of generlizing liner extensions n topologil orerings to more generl spes

19 Lesson II Antimtrois re goo wy of esriing sets of orerings. When prolem involves liner extensions of prtil orers or topologil orers of irete yli grphs, generlize to ntimtrois n lerning sequenes.

20 Exmple: Burr puzzle isssemly sequenes

21 Applition of Lesson II: The 1/3 2/3 onjeture Conjeture: every prtil orer tht is not totl orer hs two elements x n y suh tht the numer of liner extensions with x erlier thn y is etween 1/3 n 2/3 of the totl numer Formulte inepenently y Kislitsyn (1968), Fremn (ir 1976), n Linil (1984) Equivlently, in omprison sorting, it is lwys possile to reue the numer of potentil output sorte orerings y 2/3 ftor, y mking single well-hosen omprison (As onsequene, every prtil orer n e sorte in numer of omprisons logrithmi in its numer of liner extensions.)

22 The 1/3 2/3 onjeture for ntimtrois Conjeture: every ntimtroi tht is not totl orer hs two elements x n y suh tht the numer of lerning sequenes with x erlier thn y is etween 1/3 n 2/3 of the totl numer [E., Antimtrois n lne pirs, Orer 2014] {,,,,e} {,,,,e} {,,,} {,,,} {,,,e} {,,,e} {,,,} {,,,e} {,,,e} {,,} {,,} {,,} {,,} {,,} {,,} {,,} {,,} {,,} {,} {,} {,} {,} {,} {,} {} {} {} {} {} {} Ø Ø Ø Three ntimtrois for whih the onjeture is tight

23 True for: Prtil results on the onjeture Antimtrois efine y two lerning sequenes (generlizing with-two prtil orers) Antimtrois of height two (generlizing height-two prtil orers) Antimtrois with t most six elements (y omputer serh) Severl lsses of ntimtrois efine from grph serhing Exmple: Elimintion orerings of mximl plnr grphs The re verties hve 2 neighors n n sfely e remove. The yellow verties hve to wit until some neighors hve een remove.

24 Beyon lerning spes Mei: systems of sttes n trnsitions tht n e emee in istne-preserving wy into Hmming ue {0, 1} n Every lerning spe is meium, ut not onversely < < < < <,, <, < < <,, < < <, <,, < < < < < Meium of voter preferenes with frozen sttes {0, 1} 19 From Flmgne, Regenwetter, n Grofmn, 1997

25 Fst shortest pths in mei The meium struture mkes fining shortest pths etween ll pirs of sttes esier thn in ritrry stte-trnsition systems E. & Flmgne, Dis. Appl. Mth By omining it-prllel reth-first-serh se leling phse with the fst shortest pth lgorithm, n reognize whether stte-trnsition system forms meium in qurti time E., SODA 2008 & J. Grph. Alg. Appl. 2011

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27 Lesson III Lrge numers of ifferent stte-trnsition systems hve the struture of meium When they o, the unerlying hyperue emeing llows fst onstrution of shortest pths

28 Unexpline ihotomy in omputtionl omplexity Three importnt lsses of omputtionl prolems P: prolems tht n e solve in polynomil time NP: prolems tht n e solve in exponentil time y simple rute-fore serh NP-hr: t lest s iffiult s ll prolems in NP Most omputtionl prolems tht hve een stuie re either known to e in P or known to e NP-hr One of the rre exeptions: rottion istne in inry trees / flip istne in polygon tringultions

29 Rottion in inry trees Sttes: inry trees hving given orere sequene of keys Trnsitions: swp prent-hil reltion etween two noes (rerrnging their three other hilren to preserve key sequene) CC-BY-SA-imge Tree rottion.png y Rmsmy from Wikimei ommons

30 Binry trees n polygon tringultions Binry trees with n leves orrespon y plnr grph ulity to tringultions of n (n + 1)-sie polygon The one extr sie mrks the root of the tree Tree rottions flips in the tringultion Flip: retringulte the qurilterl forme y two jent tringles

31 Flip istne: Distne in the flip grph

32 Generlize flip istne to non-onvex point sets E.g. in this se the point set is 3 3 gri The generlize prolem is NP-hr Luiw & Pthk, CCCG 2012 Pilz, Comp. Geom But mye some other speil ses re esier?

33 Applition of Lesson III to flip istne Tringultions n flips form meium if n only if the point set oes not inlue the verties of n empty onvex pentgon Inlues ll onvex susets of n integer gri, n some other sets: When this is true, we n ompute flip istne in polynomil time E., SoCG 2007 & J. Comp. Geom. 2010

34 Conlusions To ompute effiiently with lrge stte spes, one must unerstn their mthemtil struture The strutures ientifie y JCF inluing qusi-orinl spes, lerning spes, n mei pper uiquitously oth in soil siene pplitions n eyon Ientifying one of these strutures in n pplition is the first step to fining effiient lgorithms for tht pplition Thnk you, Jen-Clue!