Recall from Last Time: Disjoint Set ADT

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Allen Leonard
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1 Ltur 21: Unon n Fn twn Up-Trs Toy s An: Plntn n rown orst o Up-Trs Unon-n n Fn-n Extn xmpl Implmntn Unon/Fn Smrt Unon n Fn Unon-y-sz/t n Pt Comprsson Run Tm Anlyss s tou s t ts! Covr n Cptr 8 o t txtook Rll rom Lst Tm: Dsjont St ADT Stors N unqu lmnts. Two oprtons: Fn: Gvn n lmnt, rturn t nm o ts quvln lss (ts st) Unon: Gvn t nms o two quvln lsss, mr tm nto on lss Exmpl: Fn(4) Intl Clsss = {1,4,8, {2,3, 8 {6, {7, {10,9,5 Nm o quv. lss unrln Unon(3,6) {1,4,8 {10,5,9 {6 {7 {2,3,6 {2,3 Som o t mtrl on ts sls r ourtsy o: S. Wolmn, CSE 326, Up-Tr Dt Strutur or Dsjont Sts Nt mplmntton trk or Up-Trs E quvln lss (or st) s n up-tr wt ts root s ts rprsnttv mmr (= lss nm) All mmrs o vn st r nos n tt st s uptr Hs tl mps nput t to no.. nput strn! ntr nx NULL NULL NULL {,,,, {, { Up-trs r usully not nry! 3 Forst o up-trs n sly stor n n rry (ll t up ) I no nms r ntrs or rtrs, n us vry smpl, prt s unton: Hs(X) = X up[x]= prnt o X; = 0 root Arry up: 0 1 () 2 () 3 () 4 () 5 () 6 () 7 () 8 () - NULL NULL NULL

Two oprtons: Fn: Gvn n lmnt, rturn t nm o ts quvln lss (ts st) Unon: Gvn t nms o two quvln lsss, mr tm nto on lss Exmpl: Fn(4) Intl Clsss = {1,4,8, {2,3, 8 {6, {7, {10,9,5 Nm o quv.

2 Exmpl o Fn Fn: Just trvrs to t root! Exmpl o Unon Unon: Just n on root rom t otr! Runtm =? Fn() = Fn() = Runtm =? Now: Fn() = Fn() = Unon(,) 0 1 () 2 () 3 () 4 () 5 () 6 () 7 () () 2 () 3 () 4 () 5 () 6 () 7 () 8 Arry up: Arry up: Cn (rom 0) to pont to (= 3) 6 A mor tl xmpl A mor tl xmpl Intl Sts: Unon(,) Unon(,) 7 8

Now: Fn() = Fn() = Unon(,) 0 1 () 2 () 3 () 4 () 5 () 6 () 7 () 8 0 1 () 2 () 3 () 4 () 5

3 A mor tl xmpl A mor tl xmpl Unon(,) But (you sy) n r not roots! My llow n som mplmnttons o Fn rst to t roots Sn Fn() = Fn(), unon lry on! Unon(,) But: wl w r nn, oul w o somtn to sp up Fn() nxt tm? (ol tt tout!) 9 10 A mor tl xmpl (ontnu) A mor tl xmpl Unon(,) Unon(,) 11 12

lry on! Unon(,) But: wl w r nn, oul w o somtn to sp up Fn() nxt tm?

4 A mor tl xmpl Implmntton o Fn n Unon Unon(,) nt Fn(nt X, DsjSt up) { // Assums X = Hs(X_Elmnt) // X_Elmnt oul str/r t. (up[x] <= 0) // Root rturn X; //Rturn root = st nm ls //Fn prnt rturn Fn(up[X], up); vo Unon(DsjSt up, nt X, nt Y) { //Mk sur X, Y r //roots ssrt(up[x] == 0); ssrt(up[y] == 0); up[y] = X; Runtm o Fn: O(mx t) Runtm o Unon: O(1) Ht pns on prvous Unons!Bst s: 1-2, 1-3, 1-4, O(1)!Worst s: 2-1, 3-2, 4-3, O(N) Cn w o ttr? Lt s look k t our xmpl Spn Up Unon/Fn: Unon-y-Sz Unon(,) For M Fns n N-1 Unons, worst s tm s O(MN+N) Cn w sp tns up y n lvr out rown our up-trs? I: In Unon, lwys mk root o lrr tr t nw root Wy? Mnmzs t o t nw up-tr Coulwottrjoon ts Unon? Wt ppn to? Unon(,) Unon-y-Sz! 15 16

up[y] = X; Runtm o Fn: O(mx t) Runtm o Unon: O(1) Ht pns on prvous Unons!Bst s: 1-2, 1-3, 1-4, O(1)!Worst s: 2-1, 3-2, 4-3, O(N) Cn w o ttr?

5 Trk or Storn Sz Inormton Inst o storn 0 n root, stor up-tr sz s ntv vlu n root no Wy not postv vlu? Woul not know rry ntry s sz or prnt pontr Arry up: NULL NULL NULL 0 1 () 2 () 3 () 4 () 5 () 6 () 7 () 8 () Unon-y-Sz Co vo Unon(DsjSt up, nt X, nt Y) { //X, Y r roots //ontnn (-sz) o up-trs ssrt(up[x] < 0); ssrt(up[y] < 0); (-up[x] > -up[y]) { //upt sz o X n root o Y up[x] += up[y]; up[y] = X; ls { //sz o X < sz o Y up[y] += up[x]; up[x] = Y; Nw run tm o Unon =? Nw run tm o Fn =? 18 Unon-y-Sz: Anlyss Unon-y-Ht Fns r O(mx up-tr t) or orst o up-trs ontnn N nos Numr o nos n n up-tr o t usn unon-y-sz s 2 Bs s: =0,trs2 0 = 1 no Inuton ypotss: Assumtruor< Pk up-tr wt mx t Tn, 2 mx t N mx t lo N Fn tks O(lo N) Inuton Stp: Nw tr o t ws orm v unon o two trs o t -1 E tr tn s 2-1 nos y t nuton ypotss So, totl nos =2! Tru or ll Txtook srs ltrntv strty o Unon-y-t Kp trk o t o up-tr n t root nos Unon mks root o up-tr wt rtr t t nw root Sm rsults n smlr mplmntton s Unon-y-Sz Fn s O(lo N) n Unon s O(1) 19 20

(-sz) o up-trs ssrt(up[x] < 0); ssrt(up[y] < 0); (-up[x] > -up[y]) { //upt sz o X n root o Y up[x] += up[y]; up[y] = X; ls { //sz o X < sz o Y up[y] += up[x]; up[x] = Y; Nw run tm o Unon =?

6 Spn Up Fn: Pt Comprsson I w o M Fns on sm lmnt! O(M lo N) tm Cn w moy Fn to v s-ts so tt nxt Fn wll str? A P.C. xmpl wt mor mt Fn() Pt Comprsson: Pont vrytn lon pt o Fn to root Rus t o ntr ss pt to 1: Fns t str! Déjà vu? I smlr to t on n your ol rn sply tr Fn() Pt omprsson! How to P.C. Pt Comprsson Co How to P.C. Pt Comprsson Co nt Fn(nt X, DsjSt up) { // Assums X = Hs(X_Elmnt) // X_Elmnt oul str/r t. nt Fn(nt X, DsjSt up) { // Assums X = Hs(X_Elmnt) // X_Elmnt oul str/r t. (up[x] <= 0) // Root rturn X; //Rturn root = st nm ls //Fn prnt rturn up[x] = Fn(up[X], up); Mk ll nos lon ss pt pont to root (up[x] <= 0) // Root rturn X; //Rturn root = st nm ls //Fn prnt rturn up[x] = Fn(up[X], up); Collpsnttry pontn to root Trvl moton o ornl Fn Nw runnn tm o Fn =? Fn stll tks O(mx up-tr t) worst s But wt ppns to t tr ts ovr tm? Wt s t mortz runtmofnwomfns? 23 24

$nt Fn(nt X, DsjSt up) { // Assums X = Hs(X_Elmnt) // X_Elmnt oul str/r t.$

7 Anlyss o P.C. wt Unon-y-Sz R. E. Trjn (o t up-trs m) sow tt: Wn ot P.C. n Unon-y-Sz r us, t worst s run tm or squn o M oprtons (Unons or Fns) s Θ(M α(m,n)) Wt s α(m,n)? α(m,n) s t nvrs o Akrmnn s unton Wt s Akrmnn s unton? Drsson: Tm slow-rown untons How st os lo N row? lo N = 4 or N = 16 = 2 4 Grows qut slowly Lt lo (k) N = lo (lo (lo (lo N))) (k los) Lt lo* N = mnmum k su tt lo (k) N 1 How st os lo * Nrow? lo * N = 4 or N = = Grows vry slowly Akrmnn rt rlly xplosv unton A(, j) wos nvrs α(m, N) rows vry, vry slowly (slowr tn lo * N) Howslowosα(M, N) row? α(m, N) = 4 or M ( N) r lrr tn t numr o toms n t unvrs (2 300 )!! Anlyss o P.C. wt Unon-y-Sz Summry o Dsjont St n Unon/Fn R. E. Trjn (o t up-trs m) sow tt: Wn ot P.C. n Unon-y-Sz r us, t worst s run tm or squn o M oprtons (Unons or Fns) s Θ(M α(m,n)) α(m, N) 4 or ll prtl os o M n N Txtook provs wkr rsult o O(M lo* N) tm 7 ps n 8 Lmms! (Ck t out ut no n to know t proo) Amortz run tm pr oprton = totl tm/(# oprtons) = Θ(M α(m,n))/m = Θ(α(M,N)) Θ(1) or ll prtl purposs (onstnt tm!) 27 Dsjont St t strutur rss n mny ppltons wr ojts o ntrst ll nto rnt quvln lsss or sts Cts on mp, ltrl omponnts on p, omputrs n ntwork, popl rlt to otr y loo, t. Two mn oprtons: Unon o two lsss n Fn lss nm or vn lmnt Up-Tr t strutur llows nt rry mplmntton Unons tk O(1) worst s tm, Fns n tk O(N) Unon-y-Sz rus worst s tm or Fn to O(lo N) Unon-y-Sz plus Pt Comprsson llows urtr spup Any squn o M Unon/Fn oprtons rsults n O(1) mortz tm pr oprton (or ll prtl purposs) 28

lo N = 4 or N = 16 = 2 4 Grows qut slowly Lt lo (k) N = lo (lo (lo (lo N))) (k los) Lt lo* N = mnmum k su tt lo (k) N 1 How st os lo * Nrow?

8 Nxt Clss: CSE 373 ts rp (Alo-rytms on Grps) To Do: Fns Homwork #4 (u nxt lss) Fns rn ptr 8 Strt rn ptr 9 29

Reading. Minimum Spanning Trees. Outline. A File Sharing Problem. A Kevin Bacon Problem. Spanning Trees. Section 9.6

Reading. Minimum Spanning Trees. Outline. A File Sharing Problem. A Kevin Bacon Problem. Spanning Trees. Section 9.6 Rin Stion 9.6 Minimum Spnnin Trs Outlin Minimum Spnnin Trs Prim s Alorithm Kruskl s Alorithm Extr:Distriut Shortst-Pth Alorithms A Fil Shrin Prolm Sy unh o usrs wnt to istriut il monst thmslvs. Btwn h