The Maximum Number of Squares in a Tree

Transcription

1 The Mximum Numer of Sures in Tree Mxime Crochemore 1,3, Costs S. Iliooulos 1,4, Tomsz Kociumk 2, Mrcin Kuic 2, Jku Rdoszewski 2, Wojciech Rytter 2,5, Wojciech Tyczy«ski 2, nd Tomsz Wle«2,6 1 Det. of Informtics, King's College London, London WC2R 2LS, UK [mxime.crochemore,csi]@dcs.kcl.c.uk 2 Fculty of Mthemtics, Informtics nd Mechnics, University of Wrsw, Wrsw, Polnd [kociumk,kuic,jrd,rytter,w.tyczynski,wlen]@mimuw.edu.l 3 Université Pris-Est, Frnce 4 Fculty of Engineering, Comuting nd Mthemtics, University of Western Austrli, Perth WA 6009, Austrli 5 Fculty of Mthemtics nd Comuter Science, Coernicus University, Toru«, Polnd 6 Lortory of Bioinformtics nd Protein Engineering, Interntionl Institute of Moleculr nd Cell Biology in Wrsw, Polnd Astrct. We show tht the mximum numer of dierent sure sustrings in unrooted lelled trees ehves much dierently thn in words. A sustring in tree corresonds (s its vlue) to simle th. Let s(n) e the mximum numer of dierent sure sustrings in tree of size n. We show tht symtoticlly s(n) is strictly etween liner nd udrtic orders, for some constnts c 1, c 2 > 0 we otin: c 1n 4/3 s(n) c 2n 4/3. 1 Introduction Reetitions re fundmentl notion in comintorics nd lgorithmics on words. The sic tye of reetition re sures: words of the tye zz, where z ε. (By ε we denote the emty word.) In this er we consider sure sustrings corresonding to simle ths in lelled trees. If tree is single th then it is rolem of clssicl reetitions in strings. Comintorics of sures in clssicl strings hs een investigted in [7,9,10] nd for rtil words in [3]. Sures were lso studied in the context of gmes, e.g. in [8]. Reetitions in trees nd grhs hve lredy een considered, for exmle in [4,1,2]. The numer of sure sustrings in generl grhs drmticlly increses it cn e exonentil, even in cse of inry lhet. Assume we hve tree T whose edges re lelled with symols from n lhet Σ. By T we denote the size of the tree, tht is the numer of nodes. If u nd v re two nodes of T, then y vl(u, v) we denote the seuence of lels of edges on the th from u to v. We cll vl(u, v) sustring of T. Figure 1 illustrtes sure sustring in smle tree. We consider only

2 c c c c Fig. 1: There re 4 sure sustrings in this tree. The longest is cc nd it corresonds to th mrked with solid line in the gure. simle ths: this mens tht vertices of th do not reet (though lels cn reet). For tree T, y s(t ) we denote the numer of dierent sure sustrings in T. Let s(n) e the mximum of s(t ) over ll trees of size n. We show tht s(n) = Θ(n 4/3 ). Thus s(n) hs dierent symtotics thn the mximum numer of dierent sure sustrings in stndrd word ( single th tree) of length n, which is known to e Θ(n) [7]. We introduce fmily of trees which we cll coms. The lower ound for s(n) turns out to e relized y trees from this fmily, nd such trees lso ly n imortnt role in the roof of the uer ound. Before we show the generl uer ound, we rovide some intuition ehind this roof y showing the sme uer ound for coms nd for secil sures of the form ( i j ) 2 in generl trees. 2 Bounds for Coms A com is lelled tree tht consists of th clled the sine, with t most one rnch ttched to ech node of the sine. All sine-edges re lelled with the letter. Ech rnch is th strting with the letter, followed y numer of -lelled edges, see Fig. 2. Fig. 2: A com contining 11 sure sustrings. As we show in the theorem elow, there exists fmily T m of coms for which s(t m ) = Ω( T m 4/3 ). From this one esily otins s(n) = Ω(n 4/3 ) for 2

3 ny n. In this section we lso rove n uer ound of O(n 4/3 ) for the numer of sures in com of size n. This roof is extensively used throughout the roof of the sme uer ound for generl trees, given in the following sections. Hence, our fmily of coms T m meets the symtotic uer ound for s(n) for generl trees. For m = k 2 we dene set Z m = {1,..., k} {i k : 1 i k}. Lemm 1. Assume m is sure of ositive integer. Then for ech 0 < j < m there exist u, v Z m such tht u v = j. Proof. Ech numer 0 < j < m cn e written s m, where 0 <, m. This formul corresonds to distnce etween oints nd m. For m = k 2 we dene com T m s follows: T m consists of sine of length m with vertices numered from 1 to m, nd rnches of the form m ttched to ech vertex j Z m of the sine, see Fig. 3. m m m m m Fig. 3: The structure of com T m. Theorem 1. [Lower Bound Theorem] For ech tree T m we hve s(t m ) = Ω( T m 4/3 ). Proof. From Lemm 1, for every 0 < j < m there re two nodes u, v of degree 3 on the sine with distnce(u, v) = j. Hence, T m contins ll sures of the form ( i j i ) 2 for 0 i j nd 0 < j < m. Altogether this gives Ω(m 2 ) dierent sures. Note tht T m = O(m m). Hence, the numer of sure sustrings in T m is Ω( T m 4/3 ). Lemm 2. The numer of sures in com of size n is O(n 4/3 ). Proof. Let T e com of size n. Note tht T contins only sure sustrings of the form ( i ) 2 or ( i j ) 2. The numer of sures of the former tye is 3

4 O(n). We need to ound the numer of sures of the ltter tye (secil sures). Any occurrence of secil sure strts nd ends within two dierent rnches of T. There re t most n 4/3 dierent secil sures for which i < n 2/3 nd j < n 2/3. Hence, it suces to rove tht there re O(n 4/3 ) secil sure sustrings of T for i n 2/3 or j n 2/3, we cll such secil sures long. A rnch of com is clled long if it contins t lest n 2/3 nodes. Note tht there re O(n 1/3 ) long rnches in T. Any occurrence of long secil sure hs t lest one endoint in long rnch. Consider node u locted in rnch B of T nd long rnch B. There is t most one occurrence of long secil sure tht strts in u nd ends within the rnch B. Indeed, if there re i -lelled edges on the th from u to the sine nd k edges on the th connecting the rnches B nd B then the considered sure ( i k i ) 2 uniuely determines its other endoint. Hence, the totl numer of long secil sures is ounded y the numer of nodes u multilied y the numer of long rnches B, tht is, y O(n 4/3 ). This comletes the roof. 3 Prelude to Uer Bound Proof In this section we show tight uer ound for tye of sures clled here secil sures. Along the wy we introduce some rt of the mchinery for the generl roof. Dene doule tree D = (T 1, T 2, R) s lelled tree consisting of two disjoint (excet one vertex) trees T 1, T 2 with common root R. The size of D is dened s D = T 1 + T 2 1. The sustrings of D re dened s vlues of ths which strt within T 1 nd end in T 2. An exmle of doule tree is shown in Fig. 4, T 1 lies elow R (lower tree) while T 2 ove R (uer tree). A directed rooted lelled tree is deterministic if the edges going down from the sme vertex hve dierent lels. Note tht tree is deterministic if nd only if it is trie (lso clled rex tree) of the vlues of the ths from R to the leves. A doule tree is deterministic if ech of the trees T 1, T 2 treted s directed tree with root R is deterministic. A doule deterministic tree is lso clled here D-tree. The doule tree in Fig. 4 is smle D-tree. Lemm 3. For ech doule (ossily nondeterministic) tree there exists D- tree with t most the sme numer of vertices nd the sme set of sustrings (going from T 1 to T 2 ). Proof. For moment let us direct ech tree T i down from R (treted s root). Assume we hve vertex v with edges (v, u), (v, w) going to its children nd lelled with the sme letter. Then we cn glue the vertices u, w. We cn erform such oertion going to-down from the root in BFS 4

5 trversl. Note tht the resulting trees T i re deterministic, their sizes could only decrese, nd the set of the sustrings of the D-tree remins unchnged. A th in tree T is sid to e nchored in node R T if R lies on this th. A sure is nchored in R if it is vlue of th nchored in R. A th from v to u in D-tree is clled D-sure if v T 1, u T 2, vl(v, u) is sure nd its midoint lies within T 1, nd mongst ll such ths of the sme vlue hs its strting node closest to R. Since the D-tree is deterministic, no two D-sures hve the sme vlue. Below we ind the numer of D-sures in D-trees with the numer of sures in ordinry trees. Recll tht centroid of tree T is node R such tht ech comonent of T \ {R} contins t most n/2 nodes. It is well-known fct tht ech tree hs centroid. Lemm 4. Assume tht the numer of D-sures in ny D-tree of size n is O(n 4/3 ). Then the numer of sures in ny tree is lso O(n 4/3 ). Proof. Let T e tree of size n nd let R e its centroid. Consider D-tree D = (T 1, T 2, R) comosed of two coies T 1 nd T 2 of T, determinised s in Lemm 3. Let xx e sure in T nchored in R. Either this sure or its reverse corresonds to D-sure in D. Oviously D = O(n), therefore, y the hyothesis of the lemm, there re O(n 4/3 ) D-sures in this D-tree. Hence, the numer of sures in T nchored in R is lso O(n 4/3 ). Now we need to count the sures in T tht re not nchored in R. After removing the node R, the tree is rtitioned into comonents T 1,..., T k, such tht i T i = n 1 nd T i n/2. Hence, the numer of sures in T cn e written s: s(t ) = O( T 4/3 ) + i s(t i). A solution to this recurrence yields the uer ound s(n) = O(n 4/3 ). T 2 R R T 1 Fig. 4: Illustrtion of Oservtion 1. 5

6 The roof of the ssumtion of the revious lemm is the core of this er. In full generlity it is rovided in the lst section. Here we limit ourselves to very secil tye of sures. There is useful connection etween D-trees nd coms, s exressed y the following oservtion, see Fig. 4. Oservtion 1 Assume we hve D-tree lelled with letters,. Let us tke only ths from vertex in T 1 to R or from R to vertex in T 2 which contin t most one, with other edges lelled with. Then the resulting lelled tree is com (with t most one dditionl rnch ttched to R). Corollry 1. Assume inry lhet {, }. The mximum numer of secil sures in ny tree is O(n 4/3 ). Proof. By Lemm 4, it suces to consider D-tree D = (T 1, T 2, R) nd only secil D-sures in D which hve one letter in T 1 nd one letter in T 2. The numer of other secil D-sures in D is liner. Now the conclusion follows from Lemm 2 nd Oservtion 1. 4 (, )-Reresenttions of Sustrings In this section w is word of length n. We strt y reclling few sic notions of word eriodicity, see e.g. [11]. A order of w is dened s rex of w which is lso sux of w. We sy tht ositive integer is eriod of w = w 1 w 2... w n if w i = w i+ holds for ll 1 i n. A non-emty word w is clled eriodic if it hs eriod such tht 2 w. The rimitive root of word w, denoted root(w), is the shortest word u such tht u k = w for some ositive integer k. We cll word w rimitive if root(w) = w, otherwise it is clled non-rimitive. Assume tht w is eriodic. There exists uniue reresenttion of w: w = () k such tht k 2, ε nd is rimitive. This reresenttion is clled cnonicl reresenttion of w. Here is the shortest eriod of w. We sy tht w is of eriodic tye (, ). The roof of the following Fct 1 cn e found in the endix. Exmle 1. The word hs cnonicl reresenttion () 2, with = nd =. On the other hnd, hs reresenttion () 3 with = ε nd =. Fct 1 Borders of w tht re eriodic elong to O(log n) eriodic tyes. Additionlly, w my hve O(log n) orders which re not eriodic. A eriodic order v of w is clled glol if its eriod is the eriod of the whole word w. In other words, v is glol if v, w re of the sme eriodic tye. If w is of eriodic tye (, ) nd its cnonicl reresenttion is w = () k, then ll its glol orders re () k for 2 k k. 6

7 Denition 1. Let, e such words tht ε nd is rimitive. The reresenttion w = () l y() r is clled the (, )-reresenttion of w if: () l, r 1; () y hs rex ut not () 2 ; (c) y hs sux ut not () 2, see Fig. 5 nd 6. w 1 : w 2 : Fig. 5: The (, )-reresenttions: w 1 = () 2 () 1 nd w 2 = () 2 () 1. In oth cses =, = nd y is mrked grey. y Fig. 6: A schemtic view of (, )-reresenttion. The -symols corresond to the rst mismtch for the continution of the eriod from the left side nd the eriod from the right side. Lemm 5. Assume w hs non-glol eriodic order of eriodic tye (, ). Then w hs (, )-reresenttion w = () l y() r. Moreover: () this (, )-reresenttion is uniue (i.e., l, r nd y re uniue); () y is not rex of () 2 ; (c) y is not sux of () 2 ; (d) ll orders of w of eriodic tye (, ) re: () k for 2 k min(l, r) + 1. A eriodic order is clled mximl if it is the longest order of its eriodic tye. By Fct 1, w hs O(log n) mximl orders. We cll order regulr if it is eriodic nd is neither glol nor mximl. 5 Generlized Coms nd Generl Uer Bound Due to Lemm 4 in this section we re only deling with D-sures in deterministic doule tree D = (T 1, T 2, R) of size n. For node v T 1 we dene the set SQ(v) of ll D-sures which strt in v. Ech D-sure in SQ(v) of vlue xx induces eriod x of vl(v, R), nd thus corresonds to order u of vl(v, R). This D-sure is clled regulr if u is regulr order of vl(v, R). The eriodic tye of D-sure is dened s the eriodic tye of the underlying order u. The following lemm lets us concentrte only on the regulr D-sures. Lemm 6. At most O(n log n) D-sures in D re not regulr. 7

8 Fig. 7: Corresondence etween sures in trees nd orders. We introduce n imortnt notion of generl com. Before we give forml denition, we rovide few sentences of intuition ehind it. Assume vl(v, u) is regulr D-sure of tye (, ). By Lemm 5 we hve the reresenttion vl(v, R) = () l y() r. All D-sures of tye (, ) strting in v corresond to the sme reresenttion. Those sures induce rticulr structure of ths lelled with, nd y in the uer rt of the D-tree D. A similr structure is lso resent in the lower rt. Denition 2. Let,, y stisfy the conditions of Lemm 5. A D-tree (T 1, T 2, R) is clled (,, y)-com if for ech lef v T 1, vl(v, R) = () m y() k for some integers k, m, for ech lef u T 2, vl(r, u) = () m y() k for some integers k, m. For D-tree D contining regulr D-sure of eriodic tye (, ), y Com(D,,, y) we denote the mximl sutree of D tht is (,, y)-com. From now on, wht we clled com will e referred to s stndrd com, while y com we men generl com. Note tht the conditions of Denition 1 nd Lemm 5 in rticulr imly tht neither y is rex of () 2 nor () 2 rex of y. Similrly neither y is sux of () 2 nor () 2 sux of y. Hence, ll coms hve regulr structure, see Fig. 8. Ech (,, y)-com consists of th lelled with () m (for n integer m) nd contining the root, which we cll the sine, nd the rnches which re ths ttched directly to the sine. Some nodes of the coms re rticulrly imortnt for the D-sures. These re nodes v of vlues vl(v, R) = () k y() m in the lower rt, nd nodes u of vlues vl(r, u) = () k y() m in the uer rt (k, m re ritrry nonnegtive integers in oth cses). Such nodes re clled min. For com C, y Min(C) we denote the set of min nodes in C, nd y C we denote Min(C). D-sures in C with oth endoints in min nodes re sid to e induced y the com. The following lemm conrms strong reltion etween coms nd regulr D-sures. 8

9 y y y R y y y Fig. 8: A smle (,, y)-com with the root R; the min nodes re shown s lrger circles; the ended edges re rtilly glued to the sine due to deterministion. All (,, y)-coms re sutrees of this innite D-tree. For = ε, = nd y = we otin stndrd com. Lemm 7. Ech regulr D-sure of tye (, ) in D is induced y the corresonding com Com(D,,, y). Proof. Let vl(v, u) e regulr D-sure in D of tye (, ). By Lemm 5, vl(v, R) hs following reresenttion vl(v, R) = () l y() r. The underlying order is regulr, tht is () k for some 2 k min(l, r), hence the vlue of the D-sure is (() l y() r k ) 2. Thus vl(r, u) = () l k y() r k. Now it is cler tht oth v nd u re min nodes of Com(D,,, y). The following result is simle extension of the uer ound for stndrd coms. Its full roof cn e found in the endix. Lemm 8. A com C induces O( C 4/3 ) D-sures. Finlly, we cn rove the min lemm. Lemm 9 (Key lemm). A D-tree of size n contins O(n 4/3 ) regulr D- sures. Proof. We show tht coms in D-tree re lmost disjoint with regrd to their min nodes. More recisely, due to comintoril roerties of words, ny two dierent such coms cn hve t most two common min nodes in uer rnches, nd sme for lower rnches. Clim 1 Let C = Com(D,,, y) nd C = Com(D,,, y ). Then either C = C or Min(C) Min(C ) 4. 9

10 Now let us divide coms into smll coms, for which C n 0.6, nd the remining ig coms. Due to the following clim, we cn restrict the further nlysis to ig coms. Clim 2 The numer of regulr D-sures in D induced y smll coms is o(n 4/3 ). Let C 1,..., C k denote ll ig coms of D. As conseuence of Clim 1, the totl size of ll these coms, mesured in the numer of min nodes, turns out to e liner in terms of n. Clim 3 For ny D-tree of n nodes, k C i = O(n). Let D e D-tree of size n. Due to Lemm 7, ech regulr D-sure in D is induced y com in D. By Clim 2, there re o(n 4/3 ) such D-sures induced y smll coms. Finlly, y Lemm 8 nd Clim 3, the numer of regulr D-sures induced y ig coms C 1,..., C k of D is ounded y: ( k ( k 4/3 O C i 4/3) = O C i ) = O(n 4/3 ). This comletes the roof of the key lemm. As corollry, y Lemm 4 nd 6 we otin the desired uer ound. Theorem 2. The numer of sures in tree with n nodes is O(n 4/3 ). References 1. Nog Alon nd Jroslw Grytczuk. Breking the rhythm on grhs. Discrete Mthemtics, 308(8): , Nog Alon, Jroslw Grytczuk, Mriusz Hluszczk, nd Oliver Riordn. Nonreetitive colorings of grhs. Rndom Struct. Algorithms, 21(3-4):336346, Frncine Blnchet-Sdri, Roert Mercs, nd Georey Scott. Counting distinct sures in rtil words. Act Cyern., 19(2):465477, Bostjn Bresr, Jroslw Grytczuk, Sndi Klvzr, Stszek Niwczyk, nd Iztok Peterin. Nonreetitive colorings of trees. Discrete Mthemtics, 307(2):163172, Mxime Crochemore, Christohe Hncrt, nd Thierry Lecro. Algorithms on Strings. Cmridge University Press, Mxime Crochemore nd Wojciech Rytter. Jewels of Stringology. World Scientic, A. S. Frenkel nd J. Simson. How mny sures cn string contin? J. of Comintoril Theory Series A, 82:112120, Jroslw Grytczuk, Jku Przyylo, nd Xuding Zhu. Nonreetitive list colourings of ths. Rndom Struct. Algorithms, 38(1-2):162173, Lucin Ilie. A simle roof tht word of length n hs t most 2n distinct sures. J. Com. Theory, Ser. A, 112(1):163164, Lucin Ilie. A note on the numer of sures in word. Theor. Comut. Sci., 380(3):373376, M. Lothire. Comintorics on Words. Addison-Wesley, Reding, MA., U.S.A.,

11 Aendix Simle Comintoril Tools Let us rst recll few well-known roerties of words relted to their eriodic structure, see lso the clssicl ooks [5,6,11]. Fct 2 w hs order of length if nd only if w hs eriod w. Lemm 10 (Periodicity Lemm). If word of length n hs two eriods nd, such tht + n + gcd(, ), then gcd(, ) is lso eriod of the word. Fct 3 (Synchronizing Proerties) () If uv = vu then oth words u, v re owers of the sme rimitive word. () Let ε e rimitive word. Then hs exctly two occurrences in. (c) Let ε, ε e such tht is rimitive. Then hs exctly one occurrence in. Proof of Fct 1 As for the rst rt of the lemm, let u, v e eriodic orders of w such tht u < v 1.5 u. We show tht u nd v re of the sme eriodic tye. Indeed, let v = () k, where d = is the shortest eriod of v, nd u = ( ) k e the cnonicl reresenttions of v nd u. The order u is lso order of v. Due to Fct 2, oth d nd v u re eriods of v. Moreover d < 1 2 v (since k 2) nd v u 1 3 v (since v 1.5 u ). Hence, y the Periodicity Lemm, v u is multile of d. The word u is rex of v, hence u = () l for some l < k. Now, let us show tht l 2. Assume to the contrry tht l 1. Then: (k + 1) + k = v 1.5 u This is clerly contrdiction, since > 0. Hence l 2. Now y the uniueness of cnonicl reresenttions we otin =, = nd k = l. This concludes tht the orders u nd v re of the sme eriodic tye. As for the second rt, let u, v e non-eriodic orders of w such tht u < v. We show tht v > 2 u. Assume to the contrry tht u < v 2 u. As in the revious rt of the roof, we see tht v u is eriod of v. However, 2( v u ) = v + v 2 u v + 2 u 2 u = v, therefore v is eriodic, contrdiction. 11

12 Proof of Lemm 5. Let u = () k e the longest order of w of tye (, ). Clerly is rex of w nd w > u > 2, so let us write w = z. Now, let () l+1 e the mximl ower of tht is rex of z. We hve for some z : w = () l z. Let () r+1 e the mximl ower of tht is sux of z. Now we cn write w = () l y() r. Let us rove tht this reresenttion stises the reuired conditions. We get the following esily: z hs rex nd sux. z hs rex ut not () 2, nd sux y hs rex ut not () 2, nd sux ut not () 2. Let us now show tht y is not rex of () 2. Assume to the contrry. Recll tht is sux of y. Thus we get n occurrence of in. If ε, from Fct 3, we get tht y =. But then, w would e of tye (, ). Therefore = ε. From Fct 3 we conclude tht y = nd w is ower of. This is, however, imossile since w hs non-glol eriodic order () k = k. This contrdiction imlies tht y is not rex of () 2. A symmetric rgument roves tht y is not sux of () 2. Since none of () 2 nd y is rex of the other, () l+2 is not rex of w. Similrly () r+2 is not sux of w. Thus u = () min(l,r)+1. In rticulr l, r 1. Clerly () k for 2 k min(l, r) + 1 re the only eriodic orders of w of the tye (, ). Now it remins to show the uniueness of the reresenttion. Assume there ws nother reresenttion w = () l y () r. Since y hs rex ut not () 2, l + 1 is the lrgest m such tht () m is rex of m, tht is l = l. Similrly r = r nd nlly y = y. Proof of Lemm 6. We show tht in SQ(v) t most O(log n) D-sures re not regulr. Ech D- sure in SQ(v) corresonds to dierent order of vl(v, R). The orders corresonding to non-regulr D-sures re non-eriodic, glol or mximl; we extend these terms to D-sures s in the cse of regulr D-sures nd orders. We hve the following clim. Clim. In SQ(v) t most one D-sure is glol. Proof. Let xx nd x x e vlues of two glol D-sures strting in v. Assume x < x. Let w = vl(v, R) = () k. The glol D-sures re of the form () k. Since glol order is eriodic of eriodic tye (, ), we hve 1 k k 2. Let x = () l nd x = () l, l < l. Let u e n ncestor of v the dened y vl(u, R) = () k 1. A th strting in u nd going to the uer end of x x hs the vlue () 2l 1, which 12

13 hs rex () 2l = xx. We hve l < k 1, so this occurrence hs centre in the lower rt of the D-tree. Hence, it is cndidte for D-sure of the vlue xx. This concludes tht the originl th of vlue xx strting in v could not e D-sure, which is contrdiction. As we noticed in Section 4, only O(log n) orders of vl(v, R) cn e noneriodic or mximl. Hence only O(log n) D-sures strting in v cn corresond to non-regulr order. Thus there cn e only O(n log n) D-sures which re not regulr. Proof of Lemm 8. Let C e (,, y)-com. We cn construct (ε,, )-com C of the sme structure of rnches nd min nodes s C. Clerly, C = C nd the numer of sures induced y oth coms is the sme. But now C is stndrd com. For the com C we hve n uer ound s(c ) = O( C 4/3 ) from Lemm 2. In order to otin n O( C 4/3 ) ound for the numer of sures induced y C, it suces to restrict the roof of tht lemm to secil sures ( i j ) for i 2 nd j 1. This wy we otin n uer ound of O( C 4/3 ) for the numer of D-sures induced y C, conseuently n O( C 4/3 ) uer ound for n ritrry com C. Proof of Clim 1. We strt y showing fct tht is conseuence of the synchronizing roerties of rimitive words. Fct 4 Let,,, e words such tht: ε nd ε, is rimitive, =, nd =. Then = nd =. Proof. First ssume = ε. Then = = =. From Fct 3, since ε, we get = ε. This nturlly imlies tht =. Now ssume tht ε. We hve = = nd from Fct 3c we know tht there is only one occurrence of in. Thus = nd =. Now we show the following uxiliry clim. Clim. If C = Com(D,,, y) nd C = Com(D,,, y ) re dierent coms stisfying =, then Min(C) Min(C ) =. Proof. Assume u is common min node of the two coms. It cn lie either in the uer rt or in the lower rt of D. First, let us consider the rst cse. Let w = vl(r, u). Since nd is rex of w nd nd is sux of w, we get tht = nd =. By Fct 4, = nd 13

14 =. We now know tht w = () l y() r = () l y () r. Assume l l, without the loss of generlity l < l. Since y hs rex, y() r hs rex () 2.i This is imossile y the denition of com. By similr rgument, r = r. Hence y = y, so C nd C cnnot e dierent coms. Now, consider common min node u in the lower rt. Let w = vl(u, R). As reviously we esily otin tht = nd =. This time we hve w = () l y() r = () l y () r. Exctly in the sme wy s efore, we get l = l, r = r nd conclude tht y = y. Now, it suces to show tht if C nd C hve t lest 5 common min nodes, then =. First we show tht no two common min nodes my lie on single rnch (in the uer or in the lower tree). Assume we hve such two nodes u nd u nd u is the lower mong them. Then vl(u, u ) is ower of oth nd. But nd re rimitive, so = root(vl(u, u )) =, which y clim concludes the roof of this cse. Now we show tht no three common min nodes my lie in the uer tree. Assume tht u, u, u re such nodes. Since no two of them cn lie on the sme rnch, they re ligned s in Fig. 9 (u to ermuttion of u, u, u ). u u u v v Fig. 9: Min nodes on three dierent rnches of D-tree. Note tht nodes v nd v need to e rnching nodes of oth coms. Oviously, = root(vl(v, v )) =. This gin concludes the roof of the current cse. Finlly, it remins to show tht no three common min nodes my lie in the lower tree. The roof is exctly the sme s in the revious cse. R Proof of Clim 2. Clim. Assume SQ(v) contins s > 0 regulr D-sures of tye (, ), nd vl(v, R) = w = () l y() r is corresonding reresenttion. Let C = Com(D,,, y). Then Min(C) = Ω(s 2 ). Proof. Let x 1 x 1,..., x s x s e those s regulr D-sures ordered y incresing lengths. As in the roof of Lemm 7 the vlues of these D-sures re of regulr form. Nmely, we hve x i = () l y() k i for some mx(0, r l) 14

15 k 1 <... < k s < r. Let u 1,..., u s e the other endoints of these D-sures. We hve vl(r, u i ) = () l r+k i y() k i. The nodes in the uer tree of C corresonding to ths of the form () l r+k i y() k for 0 k k i re ll distinct min nodes, hence Min(C) ((k 1 +1)+(k 2 +1)+...+(k s +1)) ( s) = Ω(s 2 ). As conseuence, we get tht O(n 0.3 ) D-sures from SQ(v) cn e induced y single smll com. Moreover, y Fct 1, regulr sures strting in v re induced y O(log n) coms. Conseuently, the numer of elements of SQ(v) tht re induced y smll coms is O(n 0.3 ) O(log n) = o(n 1/3 ). In totl, smll coms induce o(n 4/3 ) sures. Proof of Clim 3. We will show the following ineulity: k C i n + 2(k 1)(k 2). (1) From this ineulity, y C i n 0.6, we get k n 0.6 n + 2(k 1)(k 2). Comring symtotics of oth sides of the ineulity, we conclude tht for lmost ll vlues of n (tht is, ll vlues excluding only nite numer) k < n 0.5. For such vlues of k the right side of the ineulity (1) is O(n), which will conclude the roof of the clim rovided tht we show tht ineulity. As for the roof of (1), using Clim 1 we otin tht: k Min(C i ) = k Min(C i ) \ k = Min(C i) \ Conseuently: Min(C j ) Min(C j ) i 1 j=1 i 1 j=1 k ( C i 4 (i 1)) = k C i 2(k 1)(k 2) which is euivlent to the ineulity (1). k C i 2(k 1)(k 2). k Min(C i ) n 15