ANALYSIS OF THE UNION-FIND ALGORITHM

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1 Lecture Note 06 EECS 4101/5101 Istructor: Ady Mirzi ANALYSIS OF THE UNION-FIND ALGORITHM I this hdout we re goig to lyze the worst cse time complexity of the UNION-FIND lgorithm tht uses forest of up trees (i.e. trees where ech ode hs oly poiter to its pret) with weight (or size) lcig for UNION d pth compressio for FIND. (Note: The ook uses UNION y rk, which is other vlid pproch.) The lgorithms re: procedure Mke-Set (x) 1. size[x] 1 2. pret[x] x ed. procedure UNION(, ) {with weight lcig } { d re roots of two distict trees i the forest.} {Mkes the root of the smller tree child of the root of the lrger tree.} 1. if size[] <size[] the 2. pret[] 3. size[] size[] + size[] ed. fuctio FIND(x) {with pth compressio } {Returs the root of the tree tht cotis ode x.} 1. if pret[x] x the 2. pret[x] FIND( pret[x]) 3. retur pret[x] ed. size() size() UNION (,) with weight lcig x = x 1 x 2 x 3 x k-1 x = root k FIND(x) with pth compressio x k = root OR x = x 1 x 2 x 3... x k-1 if size() < size() if size() size() Throughout the hdout, is the umer of differet elemets i ll the sets (i.e. the totl umer of odes i ll the trees i the forest tht represets the sets. I other words, the totl umer

2 -2- of Mke-Set opertios), d m deotes the totl umer of Mke-Set, UNION d FIND opertios i sequece of such opertios. Note tht there c e t most 1 UNION opertios, sice ech such opertio reduces the umer of trees i the forest y oe. Lemm 1: Assume, strtig with the iitil forest, we perform umer of UNION opertios. If we use weight lcig whe mergig trees, y ode i the forest with height h will hve 2 h descedets. Proof: Iductio o the umer of UNION opertios. Bsis: (No UNION opertios re performed.) The ech tree hs 1=2 0 odes, s wted. Iductio Step: Assume the iductio hypothesis holds so fr, d the ext UNION opertio mkes the root of tree T 1 child of the root of other tree T 2.Let us cll the resultig tree T. The height d umer of descedets of ll odes i T remis the sme s efore, except for the root of T. Let us ssume T i,for i = 1, 2,hs size (i.e., umer of odes) s i,d height h i.by the iductio hypothesis we must hve (i) s i 2 h i,for i = 1, 2. Ad ecuse of the weight lcig we must hve (ii) s 2 s 1. The root of T hs s = s 1 + s 2 descedets d hs height h =mx ( h 1 + 1,h 2 ). The from (i) d (ii) we coclude: s = s 1 + s 2 2s h 1 & s = s 1 + s 2 s 2 2 h 2 Therefore, s mx ( 2 1+h 1,2 h 2 )=2 h.this completes the iductive proof. Corollry 0: Assume, strtig with the iitil forest, we perform ritrry umer of UNION d FIND opertios. If we use weight lcig whe mergig trees, y tree i the forest with height h will hve 2 h odes. Proof: The clim follows from Lemm 1 y oservig tht FIND opertio does ot chge the umer of odes i tree d it c ot icrese the height of tree (it my decrese it). Corollry 1: I forest creted y usig the weight lcig rule, y tree with odes hs height lg. Corollry 2: The UNION-FIND lgorithm usig oly weight lcig (ut o pth compressio) tkes O( + mlg)time i the worst cse for ritrry sequece of m UNION-FIND opertios. Proof: Ech UNION opertio tkes O(1 ) time. Ech FIND opertio c tke t most O(lg ) time. I the rest of this hdout we lyze the UNION-FIND lgorithm tht uses oth weight lcig (for UNION) d pth compressio (for FIND). To help us i our lysis, let us defie the super-expoetil d super-logrithmic fuctios. The super-expoetil, exp * (), is defied recursively s follows: exp * (0)=1, d for i >0 exp * (i)=2 exp* (i 1) ;thus exp * () isstck of 2 s. (Let us lso defie the oudry cse exp * ( 1) = 1.) The super-logrithm, lg * ()= mi i such tht exp * (i). Remrk: Note tht exp * grows very rpidly, wheres lg * grows very slowly: exp * (5 ) = , while log * ( )=5. We hve >> ,where the ltter qutity is the estimted umer

3 -3- of toms i the uiverse. Thus lg * () 5for ll prcticl. However, evetully lg * () goes to ifiity s does, ut t lmost uimgily slow rte of growth. Fct 1: Assume r 0 d g 0 re itegers. The, lg * (r) =g if d oly if exp * (g 1) < r exp * (g). Let s e sequece of UNION-FIND opertios. Defiitio: The rk of ode x (i the sequece s), rk(x), is defied s follows:. Let s e the sequece of opertios resultig whe we remove ll FIND opertios from s.. Execute s, usig weight lcig (sice there re o FIND s there will e o pth compressio). c. The rk of x (i s) isthe height of ode x i the forest resultig from the executio of s. Put other wy. Perform sequece s i two differet wys. Oe with pth compressio whe doig FIND opertios, oe without pth compressio. (The UNION opertios i oth re doe with weight lcig.) Cll the resultig UNION-FIND forests, the compressed forest d the ucompressed forest, respectively. The rk(x) is the height of ode x i the fil ucompressed forest. Lemm 2: For y sequece s, there re t most /2 r odes of rk r. Proof: Let s ethe sequece tht results from s if we delete ll the FIND opertios. Cosider the forest produced whe we execute s. By Lemm 1, ech ode of rk r hs 2 r descedets. I forest two odes of the sme height hve distict descedets. I prticulr, distict odes of rk r must hve disjoit sets of descedets. Sice there re odes i totl, there re t most /2 r disjoit sets ech of size 2 r.hece there re t most /2 r odes of rk r. Lemm 3: If durig the executio of sequece s, ode x is ever proper descedet of ode y, the rk(x)<rk(y) i s. Proof: Simply oserve tht if pth compressio i the executio of s cuses x to ecome proper descedet of y, surely x will e proper descedet of y t the forest resultig i the ed of executig s (tht does ot ivolve y pth compressio). Thus the height of x i tht forest is less th the height of y d therefore rk(x)<rk(y) i s, swted. We wt to clculte upper oud o the worst cse time complexity to process sequece s of m opertios o forest of size. First of ll, oserve tht ech Mke-Set d UNION tkes oly O(1 ) time d we c hve Mke-Set d t most 1 UNION opertios. So these opertios will cotriute t most O() time. Let s ow cosider the complexity of t most m FIND opertios. It is useful to thik of odes s eig i groups ccordig to their rk. I prticulr we defie the group umer of ode x, group(x)=lg * (rk(x)). The time for FIND(x) opertio, where x is ode, is proportiol to the umer of odes i the pth from x to the root of its tree. Suppose these odes re x 1 = x, x 2,..., x k = root, where x i+1 = pret(x i ), for 1 i < k. Wewill pportio the cost (i.e., time) for FIND(x) tothe opertio itself d to the odes x 1, x 2,..., x k ccordig to the followig rule: For ech 1 i k: (i) If x i = root (i.e. i= k) orif group(x i ) group(x i+1 ), the chrge 1 uit (of time) to the opertio FIND(x) itself.

4 -4- (ii) If group(x i )=group(x i+1 ), the chrge 1 uit (of time) to ode x i. The time complexity of processig the FIND opertios c the e otied y summig the cost uits pportioed to ech opertio d the cost uits pportioed to ech ode. From Lemm 2, the mximum rk of y ode is lg. Therefore the umer of differet groups is t most lg * ( lg ). This is, the, the mximum umer of uits pportioed to y FIND opertio. Therefore, for the totl of t most m such opertios the umer of uits chrged to the FIND opertios is t most O(mlg * (lg )) = O(m lg * ) (1). Next cosider the cost uits pportioed to the odes. Ech time pth compressio cuses ode x to move up, i.e. to cquire ew pret, the ew pret of x hs, y Lemm 3, higher rk th its previous pret (the previous pret ws proper descedet of the ew pret efore the pth compressio). This mes tht x will e chrged y rule (ii) t most s my times s there re distict rks i group(x). After tht, xmust ecome the child of ode i differet group d theceforth y further move ups of x will e ccouted for y rule (i). (Note tht, gi y Lemm 3, oce x hs cquired pret i differet group th it, ll susequet prets of x will lso e i differet group th x, sice they hve progressively higher rks.) Let g = group(x). By Fct 1, The umer of differet rks i group g is exp * (g) exp * (g 1) (this is the umer of (iteger) rks r such tht lg * (r)=g). The, y the ove discussio, the mximum umer of uits chrged to y ode i group g is exp * (g) exp * (g 1). Now let s clculte the umer of odes i group g, N(g). By Lemm 2 we hve: N(g) = exp * (g) Σ r=exp * (g 1)+1 2 exp* (g 1) = 2 r exp * (g) 2 exp*(g 1)+1 [ ] Thus, the totl umer of uits chrged to odes of group g is t most N(g) (exp * (g) exp * (g 1)) exp * (g) (exp* (g) exp * (g 1)) = O(). We hve lredy see tht the umer of differet groups is lg * ( lg ) d therefore the totl umer of cost uits chrged to ll odes y rule (ii) is O( lg * ( lg )) = O(lg * (lg )) = O( lg * ) (2). By summig (1) d (2) we get tht the worst cse time complexity to process t most m FIND opertios is O((m + ) lg * ). The ltter is O(m lg * )sice m. Sice, s we lredy poited out, O() UNIONs tke oly O() time, we coclude: Theorem 1: The totl worst-cse time complexity to process ritrry sequece of mmke- Set, UNION d FIND opertios, of which re Mke-Set s, usig weight lcig d pth compressio is O( mlg * ).

5 -5- Biliogrphy [CLRS] Chpter 21. [Weiss] Chpter 8. [Tr83] [Tr79] R.E. Trj, Dt Structures d Network Algorithms, CBMS-NSF, SIAM Moogrph, 1983, (Chpter 2). R.E. Trj, Applictios of pth compressio o lced trees, Jourl of ACM, Vol. 26, No. 4, Oct. 1979, pp