POW CSE 36: Dt Struturs Top #10 T Dynm (Equvln) Duo: Unon-y-Sz & Pt Comprsson Wk!! Luk MDowll Summr Qurtr 003 M! ZING Wt s Goo Mz? Mz Construton lortm Gvn: ollton o rooms V Conntons twn t rooms (ntlly ll los) E W wnt to ul ollton o onntons to knok own, E? E, su tt on unqu pt onnts vry two rooms T Prolm, Formlly I n v not yt n onnt r two lmnts n t sm st? Wl s rmn n E { (, ) = RmovRnomWll() ( n v not n onnt ) { (, ) to E Mrk n s onnt Mrk n s onnt Form t unon o two sts Dsjont Sts DT Fn(x) Rturns st ntr Fn(x) = Fn(y) x n y r n t sm st Unon(, ) rumnts r st ntrs How o w unon t sts ontnn x n y? MkNwSt(tm) Crt nw st ontnn only tm Dsjont Sts Forml Proprts Equvln proprty Evry lmnt o DS lons to xtly on st Dynm quvln proprty T st o n lmnt n n tr xuton o unon n(4) unon(3,6) {1,4, {6 {5,9,10 {,3 {7 {,3,6 1
Our Mo Mz Construton lortm Exmpl Wl s rmn n E (, ) = RmovRnomWll () ( Fn()!= Fn() ) E? = E? U (, ) Unon( Fn(), Fn() ) Construt t mz on t rt Intlly (t nm o st s unrln): {{{{{{{{{ 1 6 11 9 1 5 Orr o s n lu Exmpl, ontnu {{{{{{{{{ n()? n()? n()? n() so: 1 to E? unon(, ) Rsult: 11 9 1 5 Orr o s n lu 6 DS DT Tr Rprsntton Mntn orst o up-trs E st s tr Wt s t st ntr? Fn Implmntton Fn(x) Wlk prnts o x to t root Unon Implmntton + Unon(, ) Jon t two trs Sn n r lry t roots o tr, ts s sy! Runtm: Runtm:
Mor o t Exmpl unon(,) 1 6 11 9 1 5 T Fnl Mz Oo sry! Su r mz! Mn-Exrs ssum unon lwys kps rst rumnt s t root 1. Strtn wt stnt sts,,,,,, Unon(,) Unon(,) Unon(,) Fn() Unon(,) Unon(,) Unon(,) Fn(). Must Fn() lwys rturn t sm vlu? 3. Coul Unon v on ttr jo? Nty stor trk orst o up-trs n sly stor n n rry. Us stl to mp no nms to rry ns 0 () 1 () () 3 () 4 () 5 () 6 () 7 () () up-nx: -1 0-1 0 1-1 -1 7 3
nt Fn(Ojt x) { nt xid = Tl[x]; wl(up[xid]!= -1) { xid = up[xid]; Implmntton vo Unon(nt x, nt y) { up[y] = x; Improvn Unon Coul w o ttr jo on ts unon? rturn xid; Unon-y-sz Co nt Unon(nt x, nt y) { // I up[x] n up[y] rn t ot // -1, ts lortm s n troul (sz[x] > sz[y]) { up[y] = x; sz[x] += sz[y]; ls { up[x] = y; sz[y] += sz[x]; nw runtm or Unon(): nw runtm or Fn(): Unon-y-Sz Fn nlyss Fns r O(mx no pt) ll nos strt t pt 0 Dpt nrss Only wn prt o smllr tr n unon Only y on (1) lvl t tm How mny tms n ts ppn??, unon runtm = Improvn Fn n() Pt Comprsson! Wt - wt s tr to mprov? Wl w r nn, oul w o nytn ls? 4
Exrs Us unon-y-sz. Kp t rst rumnt s root tr s t. How mny nos os Fn ss? 1. Strtn wt stnt sts,,,,,, Unon(,) Unon(,) Unon(,) Unon(,) Fn() Unon(,) Unon(,) Unon(,) Unon(,) Fn() Fn() Fn(). Moy t ov to lso us Pt Comprsson. Dos t lp? 3. Usn unon-y-sz, wt s t worst s pt o ny no? Construt squn o unon oprtons tt prous ts or pt o 5. Pt Comprsson Co Intrlu: Rlly Slow Funton nt Fn(Ojt x) { // x ttr n // t st! nt xid = Tl[x]; nt = xid; // Gt t root or // ts st wl(up[xid]!= -1) { xid = up[xid]; // Cn t prnt or // ll nos lon t pt wl(up[]!= -1) { tmp = up[]; up[] = xid; = tmp; rturn xid; (Nw?) runtm or Fn(): krmnn rt rlly unton (x, y) wt t nvrs? (x, y) w s rlly smll How st os? (x, y) row??(x, y) = 4 or x r lrr tn t numr o toms n t unvrs ( 300 )? sows up n: Computton Gomtry (sur omplxty) Comntors o squns Complx Complxty o Unon-y-Sz + Pt Comprsson Trjn prov tt, wt ts optmztons, m unon n n oprtons on st o n lmnts v worst s omplxty o O(m?? (m, n)) For ll prtl purposs ts s mortz onstnt tm: O(m?4) or m oprtons! In som prtl ss, on or ot optmztons s unnssry, us trs o not nturlly t vry p. Dsjont Sts DT Summry lso known s Unon-Fn or Dsjont St Unon/Fn Smpl, nt mplmntton Wt unon-y-sz n pt omprsson Grt symptot ouns Kn o wr t rst ln, ut lots o ppltons 5