The Rand and block distances of pairs of set partitions

Transcription

1 The Rad ad block distaces of pairs of set partitios Frak Ruskey 1 ad Jeifer Woodcock 1 Dept. of Computer Sciece, Uiversity of Victoria, CANADA Abstract. The Rad distaces of two set partitios is the umber of pairs {x, y} such that there is a block i oe partitio cotaiig both x ad y, but x ad y are i differet blocks i the other partitio. Let R(, k deote the umber of distict (uordered pairs of partitios of that have Rad distace k. For fixed k we prove that R(, k ca be expressed as c k,( B where c k, is a o-egative iteger ad B is a Bell umber. For fixed k we prove that there is a costat K such that R(, ( k ca be expressed as a polyomial of degree k i for all K. This polyomial is explicitly determied for 0 k 11. The block distace of two set partitios is the umber of elemets that are ot i commo blocks. We give formulae ad asymptotics based o N(, the umber of pairs of partitios with o blocks i commo. We develop a O( algorithm for computig the block distace. 1 Itroductio ad Motivatio I statistics, particularly as it is applied to cluster aalysis, it is sometimes useful to have a measure of the differece betwee two set partitios []. The Rad distace is oe such measure, ad was itroduced i Rad [8]. I this paper we will iitiate a combiatorial study of the properties of the Rad distace, take over all uordered pairs of partitios of a -set. We will also itroduce aother measure, which we call the block distace, ad determie some of its properties. For example, we will determie a exact expressio for the umber of pairs of partitios that have o blocks i commo. Furthermore, we will show how to compute the block distace efficietly. The Rad distace of two set partitios is the umber of uordered pairs {x, y} such that there is a block i oe partitio cotaiig both x ad y, but x ad y are i differet blocks i the other partitio. We use R(P, Q to deote the Rad distace betwee two set partitios P ad Q. For example, R({{1, }, {}}, {{1}, {, }} = (the pairs are {1, } ad {, } ad R({{1,, }}, {{1}, {}, {}} = (the pairs are {1, }, {1, }, ad {, }. I geeral, if P ad Q are partitios of a -set, the 0 R(P, Q (. Let R(, k be the umber of distict (uordered pairs of partitios of a -set that have Rad distace k. See Table i Sectio.1. This table was computed from exhaustive computer listigs of all partitios of {1,,..., } up Research supported i part by NSERC

2 Frak Ruskey ad Jeifer Woodcock to = 11. The colum sums are ( B. Note that the umbers for fixed are ot uimodal i geeral. We defie the block distace B(P, Q betwee two partitios of as the umber of elemets i the blocks that are ot commo to both P ad Q. For example, B({{1, }, {}}, {{1}, {}, {}} = sice the oly block that is commo to both partitios is {} ad there are elemets i the remaiig blocks. By B(, k we deote the umber of pairs of partitios of that have block distace k. See Table 1 i Sectio. The Rad distace ca be cleverly computed usig a liear umber of arithmetic operatios; see Filkov ad Skiea [] ad we will show that the block distace is also efficietly computable. Orgaizatioally, we will fiish this sectio by givig some backgroud o set partitios. I the succeedig two sectios, we discuss first the block distace ad the the Rad distace. The focus is maily o the elucidatio of some eumerative results alog with a clever O( algorithm for computig the block distace. 1.1 Backgroud o set partitios A partitio of a set S is collectio of disoit subsets of S, say {S 1, S,, S k } whose uio is S. Each S i is referred to as a block. The umber of partitios of a -set ito k blocks is the Stirlig umber (of the secod kid, which is deoted as { k}. We use [] to deote {1,,..., }. I the computer, partitios are usually represeted by restricted growth strigs. We assume that the blocks of a partitio X are umbered S 1, S,..., S k accordig to the size of the smallest elemet i each block. That is, S 1 cotais 1, S cotais the smallest elemet ot i S 1, ad so o. The the restricted growth strig r[1..] of X is defied by takig r[i] to be the distace of the block cotaiig i. The Gray code algorithms for geeratig restricted growth strigs developed i [9] ad discussed i [5] were used to geerate the umbers i Tables 1 ad ; as each strig was geerated the O( algorithms for computig the Rad distace ad the block distace were applied. The -th Bell umber, B, is the total umber of partitios of a -set, irrespective of block size. Thus B = { k k}. The expoetial geeratig fuctio (egf of the Bell umbers is well-kow (e.g., Staley [10], pg. to be B(z = 1 z B! = z ee 1. (1 The umber of pairs of partitios is ( B. For = 1,,,..., 10 these umbers are 0, 1, 10, 105, 16, 050, 816, , 5871, They give the row sums i Tables 1 ad.

3 Rad ad block distaces We use several times a geeralizatio of the fact that if f(z = 0 f z /! is the egf of a sequece f, the zf(z is the egf of the sequece f 1. See Kuth, Graham, Patashik [], page 50. Furthermore, for k 0, z k f(z = k = k! 0 ( 1 ( k + 1f k z Thus k! ( k f k is the -th coefficiet of z k f(z. The Block Distace ( z f k k!. ( Recall that the block distace B(P, Q of two partitios of is the umber of elemets i the blocks that are ot commo to both P ad Q, ad that B(, k is the umber of pairs of partitios of that have block distace k. See Table 1.! \k Table 1. The values of B(, k for 1 k 9. Let N( = B(, ; this is the umber of uordered pairs of partitios that have o blocks i commo. The umerical values of N(, for 0 10, are 0, 0, 1, 7, 65, 811, 176, 588, , 1865, Determiig N( for i = 1,..., is sufficiet to determie B(, k sice, by direct combiatorial cosideratios, ( B(, k = N(k B k. ( k We also ote that ( B = B(, k = k=0 ( N(k B k. ( k k=0

4 Frak Ruskey ad Jeifer Woodcock Lettig N(z be the egf of the N( umbers, from ( we obtai the equatio P (z := ( B z! = z N(zee 1. 0 Ad thus N(z = P (ze 1 ez. (5 The egf e 1 ez is kow; it is the egf of the complemetary Bell umbers (OEIS A The complemetary Bell umbers, C, for = 0, 1,,..., 1 are 1, 1, 0, 1, 1,, 9, 9, 50, 67, 1, 180, 1771, 505, It is kow that C = { } ( 1 k. k k=0 Thus, from (5 we get a closed-form formula for N(, amely N( = =0 ( ( B C = =0 ( ( B { } ( 1 k. k k=0.1 Liear Time Algorithm to Compute the Block Distace I this subsectio we preset a liear time algorithm to compute the block distace of two partitios. Closely related to the restricted growth strig, we defie the block strig, b[1..], of P as follows: b[i] is the smallest elemet i the block cotaiig i. Every block strig has the characterizig property that b[1] = 1, ad for i > 1, b[i] {i, b[1], b[],..., b[i 1]}. It is relatively simple to covert a restricted growth strig ito the correspodig block strig i O( time. The followig code takes as iput a restricted growth fuctio r[1..] ad returs the correspodig block strig b[1..]. It uses a temporary array m[1..] that maitais the ivariat b[i] = m[r[i]]. for i {1,,..., } do m[i] := 0; for i := 1,,..., do if m[r[i]] = 0 the m[r[i]] := i; b[i] := m[r[i]]; Before describig the algorithm for computig the block distace, we ecourage the reader to cosider the followig small example. Suppose P = {1}{}{, }{5, 7}{6}, Q = {1, }{,, 6}{5, 7}.

5 Rad ad block distaces 5 The the restricted growth strigs for P ad Q are ad the block strigs are rp = 1,,,,, 5,, rq = 1, 1,,,,, p = 1,,,, 5, 6, 5, q = 1, 1,,, 5,, 5. Comparig the elemets, we fid that the blocks labelled 1,,, ad 6 are ot commo to P ad Q ad that the block labelled 5 is commo to P ad Q. Sice there are 5 elemets i blocks 1,,, ad 6, the block distace of P ad Q is 5. The algorithm maitais a boolea array C[1..] with the property that, upo termiatio, C[i] is true if i is i a block commo to P ad Q, ad is false otherwise. The block distace is thus equal to the umber of etries i this array that are false. The algorithm makes two passes over p, oe pass over q, ad oe pass over C. Cosider p[i] ad q[i]; there are three mutually exclusive cases: (a p[i] q[i] ad i is ot i a commo block, (b p[i] = q[i] ad i is i a commo block, ad (c p[i] = q[i] ad i is ot i a commo block. (Because we are usig the block strig ad ot the restricted growth strig, it is ot possible that p[i] q[i] ad i is i a commo block. I the first pass we test oly for case (a. I the secod pass we (idirectly distiguish cases (b ad (c. The key observatio is this: If i is ot i a commo block ad p[i] = q[i], the there is some value i such that is i the same block as i i P but is i a differet block tha i i Q, or vice-versa. I other words, p[i] = p[] q[] or p[] q[] = q[i]. Thus, i the first pass C[p[]] ad C[q[]] were set to false. So o the secod pass, we test whether C[p[i]] is false to determie whether i is i a commo block or ot. O the fial pass, we fid the block distace by coutig the umber of false values i C. Below is the code i detail. for i {1,,..., } do C[i] := true for i := 1,,..., do if p[i] q[i] the C[p[i]] := C[q[i]] := false; for i := 1,,..., do if C[p[i]] the C[i] := false; c := 0; for i {1,,..., } do if C[i] the c := c + 1; retur(c; Results o the Rad distace Now recall that R(P, Q is the umber of uordered pairs {x, y} such that there is a block i oe partitio cotaiig both x ad y, but x ad y are i differet blocks i the other partitio, ad that R(, k is the umber of distict (uordered pairs of partitios of a -set that have Rad distace k. See Table. Let R( be the sum of the Rad distace over all uordered pairs of partitios.

6 6 Frak Ruskey ad Jeifer Woodcock k\ Table. The values of R(, k for 11 ad 1 k 55. This table is iverted i the sese that k icreases dow colums ad varies alog the colums.

7 Rad ad block distaces 7 Theorem 1. ( R( = k R(, k = k=0 ( B 1 (B B 1. Proof. Choose a pair {x, y}. The umber of partitios i which this pair appears i the same block is B 1. The umber of partitios i which this pair appears i differet blocks is the differece B B 1. Thus i total, each pair cotributes B 1 (B B 1 to the sum. Sice there are ( ways to choose a pair, the proof is fiished. The average value of the Rad distace is thus R( ( B = ( 1B 1(B B 1. B (B 1 Sice the Bell umbers grow expoetially, R( ( B B 1, B which experimetally appears to be Θ( log..1 Determiig R(, k for small values of k We ow cosider R(, k for small values of k. Clearly R(, 0 = 0. Theorem. For all 1, R(, 1 = ( B. Proof. The oly way that the Rad distace ca be 1 is if there is a block {x, y} i oe partitio ad two blocks {x}, {y} i the other, ad all other blocks i oe partitio are preset i the other. There are ( ways to choose the pair ad B ways to determie the other blocks. Corollary 1. The egf of the R(, 1 umbers is 1 R(, 1 z! = z z B(z = z ee 1. Proof. Apply ( with k =. The previous two results were warm-ups for the more techical results that follow.

8 8 Frak Ruskey ad Jeifer Woodcock Theorem. For fixed k, there are o-egative iteger costats c k, such that, for all 1, k ( R(, k = c k, B. = (1+ 1+8k/ Proof. Ay two partitios P ad Q will have a largest subpartitio X that is commo to both P ad Q. Thus R(P, Q = R(P \ X, Q \ X. I the sum above represets X, give that R(P, Q = k. The lower boud i the summatio follows from the fact that the maximum Rad distace betwee two partitios of is ( ad thus k (. Solvig the implied quadratic yields ( k/, which gives us the lower boud. We hereafter use α = ( k/ for ease of readig. For the upper boud, cosider two partitios P ad Q of a -set that have o block i commo, ad have Rad distace k. We claim that /. Cosider some arbitrary iteger x {1,,..., }. Sice P ad Q have o commo blocks, there is some iteger y that is i the same block as x i oe partitio, ad i aother block i the other partitio. Thus we have distict ordered pairs (x, y, oe for each differet value of x. At least / of them have to be distict as uordered pairs, ad each such uordered pair cotributes 1 to the Rad distace. Thus k / as claimed. From this it follows that k, which is the upper boud i the sum above. Theorem. For all 1, ( ( R(, = 6 B + 6 B. Proof. Theorem tells us that ( ( R(, = c, B + c, B. From the k = row of Table we the have the followig two equatios. ( ( R(, = 6 = c, B 0 + c, B 1 = c, ad ( ( R(, = 0 = c, B 1 + c, B 0 = c, + c,. This system of equatios ca be solved to obtai c, = c, = 6. Corollary. The egf of the R(, umbers is 1 R(, z! = (z + z B(z = (z + z e ez 1.

9 Rad ad block distaces 9 I a similar fashio we ca solve systems of liear equatios to obtai the followig theorems ad corollaries. Theorem 5. For all 1, ( ( ( ( R(, = B + 8 B + 10 B B Corollary. The egf of the R(, umbers is ( ( R(, z z =! 6 + 7z 6 + z5 + z6 z B(z = z 6 + z5 + z6 1 e ez 1. Theorem 6. For all 1, the value of R(, is ( ( ( ( ( B B B B B Corollary. The egf of the R(, umbers is 1 R(, z! = (z + z5 + 1z6 + z7 6 + z8 B(z. 8 Ituitively, c k, is the umber of pairs of -elemet set partitios with o commo blocks that have Rad Distace k. We summarize the kow values of c k, i Table. Although we do t kow the value of c k, i geeral, we ca determie a few specific ifiite sequeces, which are give i the ext lemma. Lemma 1. For all k 1, c k,k = (k 1! (k 1! ad c k,α = R(α, k. Proof. For a pair of k elemet set partitios P ad Q to have Rad distace k with o commo blocks, the k elemets must be paired, ad each pair of elemets is a block i either P or Q. Further, if {a, b} is a block i set P the set B cotais the sigleto blocks {a} ad {b} ad vice versa. Sice the order of the blocks does t matter, we ca assume the blocks (pairs are sorted by their smallest elemets. So, for i = 1,,..., k, oce we have chose the elemets for blocks 1,,..., i 1, the first elemet i block i must be the smallest remaiig elemet ad there are k ((i = k i + 1 choices for the secod elemet i block i. Thus the umber of ways to pair the elemets is k (k 1! (k i + 1 = k 1 (k 1! i=1 If we assume, without loss of geerality, that a pair, say {a, b}, is i partitio P, the there are k 1 uique ways to distribute the remaiig pairs betwee P ad Q. So we have c,k = (k 1! k 1 (k 1! k 1 = (k 1! (k 1!.

10 10 Frak Ruskey ad Jeifer Woodcock k\ Table. Kow values of c k, for 11. The bold value at the begiig of each row is c k,α = R(α, k.

11 Sice B i = 0 whe i < 0, B 0 = 1, ad ( i i = 1, R(α, k = k =α c k, ( α Rad ad block distaces 11 B α = c k,α. Lemma. For all 1, ( ( c (, = R, = 1. For all : For all 5: c ( 1, = R (, c (, = R (, ( 1 = ( = (. (( 1. For all + x: ( ( c ( x, = R, x. Proof. Omitted i this exteded abstract.. The umbers R(, ( k for small k We ow cosider the umbers at the bottom of the colums i Table. Clearly R(, ( = 1 (the pair is {1,... } ad {1}{}... {}. Theorem 7. For all, ( R(, 1 = (, ad R(, = 6. Proof. For, the two partitios are the full set {1,,..., } ad the partitio cosistig of oe pair ad sigleto sets. Theorem 8. For all 5, R(, ( = ad R(, =, R(, 5 =. (( 1 = 1 ( 1( (, 8

12 1 Frak Ruskey ad Jeifer Woodcock Proof. For 5 the two partitios are the full set {1,,..., } ad the partitio cosistig of two pairs ad sigleto sets. The order of the two pairs does ot matter so we have R(, ( = 1 ( ( which ca be show to be equal to the two values give i the statemet of the theorem. The umbers i Theorems 7 ad 8 are a shifted versios of OEIS A00017 ad OEIS A0505, respectively. Theorem 9. For all 6, ( R(, = 1 ( ( 6 ad R(, =, R(5, 7 = 60. (, + (, Proof. For 5 the two partitios are either the full set {1,,..., } ad the partitio cosistig of three pairs ad 6 sigleto sets, or the full set {1,,..., } ad the partitio cosistig of oe triple ad sigleto sets. Theorem 10. For fixed k there is a costat K k such that R(, ( k is a polyomial of degree k i for all K k. Proof. Omitted i this exteded abstract. Ackowledgemets We wish to thak Rod Cafield for helpful discussios. Refereces 1. E.R. Cafield, Egel s iequality for the Bell umbers, Joural of Combiatorial Theory, Series A, 7 ( V. Filkov ad S. Skiea, Itegratig Microarray Data by Cosesus Clusterig, Iteratioal Joural o Artificial Itelligece Tools, 1(: , 00.. R.L. Graham, D.E. Kuth, ad O. Patashik, Cocrete Mathematics, Addiso- Wesley, L. Hubert, Comparig Partitios, Joural of Classificatio, ( D.E. Kuth, The Art of Computer Programmig, Volume : Combiatorial Algorithms, Part 1, Addiso-Wesley, Readig, MA, L. Moser ad A. Wyma, A asymptotic formula for the Bell umbers, Trasactios of the Royal Society of Caada III, 9 (

13 Rad ad block distaces 1 7. A. O. Muagi, Set Partitios with Successios ad Separatios, It. J. Math ad Math. Sc. 005, o. (005, W. Rad, Obective criteria for the evaluatio of clusterig methods, J. America Statistical Assoc., 66 (6: , F. Ruskey, Simple combiatorial Gray codes costructed by reversig sublists, th ISAAC (Iteratioal Symposium o Algorithms ad Computatio, Lecture Notes i Computer Sciece (LNCS, #76 ( Richard R. Staley, Eumerative Combiatorics, Volume 1 Wadsworth, 1986.