Approximate Sorting. Institute for Theoretical Computer Science, ETH Zürich, CH-8092 Zürich

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1 Approximate Sortig Joachim Giese, Eva Schuberth, ad Miloš Stojaković Istitute for Theoretical Computer Sciece, ETH Zürich, CH-809 Zürich Abstract. We show that ay radomized algorithm to approximate ay give rakig of items withi expected Spearma s footrule distace /ν() eeds at least (mi{log ν(), log } 6) comparisos. This boud is tight up to a costat factor sice there exists a determiistic algorithm that shows that 6(log ν() + 1) comparisos are always sufficiet. Keywords. Sortig, Rakig, Spearma s footrule metric, Kedall s tau metric 1 Itroductio Our motivatio to study approximate sortig comes from the followig market research applicatio. We wat to fid out how a respodet raks a set of products. I order to simulate real buyig situatios the respodet is preseted pairs of products out of which he has to choose oe that he prefers, i.e., he has to perform paired comparisos. The respodet s rakig is the recostructed from the sequece of his choices. That is, a procedure that presets a sequece of product pairs to the respodet i order to obtai the product rakig is othig else tha a compariso based sortig algorithm. We ca measure the efficiecy of such a algorithm i terms of the umber of comparisos eeded i order to obtai the rakig. The iformatio theoretic lower boud o sortig [7] states that there is o procedure that ca determie a rakig by posig less tha log e paired compariso questios to the respodet, i.e., i geeral Ω( log ) comparisos are eeded. Eve for oly moderately large that easily is too much sice respodets ofte get wor out after a certai umber of questios ad do ot aswer further questios faithfully aymore. O the other had, it might be eough to kow the respodet s rakig approximately. I this paper we pursue the questio of how may comparisos are ecessary ad sufficiet i order to approximately rak products. I order to give sese to the term approximately we eed some metric to compare rakigs. Assume that we are dealig with products. Sice a rakig is a permutatio of the products, this meas that we eed a metric o the permutatio group S. Not all of the metrics, e.g., the Hammig distace Partly supported by the Swiss Natioal Sciece Foudatio uder the grat Robust Algorithms for Cojoit Aalysis ad by the joit Berli/Zurich graduate program Combiatorics, Geometry ad Computatio, fiaced by ETH Zurich ad the Germa Sciece Foudatio (DFG).

2 that couts how may products are raked differetly, are meaigful for our applicatio. For example, if i the respodet s rakig oe exchages every secod product with its predecessor, the the resultig rakig has maximal Hammig distace to the origial oe. Nevertheless, this rakig still tells a lot about the respodet s prefereces. I marketig applicatios Kedall s tau metric [4] is frequetly used sice it seems to capture the ituitive otio of closeess of two rakigs ad also arises aturally i the statistics of certai radom rakigs [8]. Our results. Istead of workig with Kedall s metric we use Spearma s footrule metric [4] which essetially is equivalet to Kedall s metric, sice the two metrics are withi a costat factor of each other [4]. The maximal distace betwee ay two rakigs of products i Spearma s footrule metric is less tha. We show that i order to obtai a rakig at distace /ν() to the respodet s rakig with ay strategy, a respodet has i geeral to perform at least (mi{log ν(), log } 6) comparisos. Moreover, if we allow the strategy to be radomized such that the obtaied rakig is at expected distace /ν() to the respodet s rakig, we ca show that the same boud o the miimum umber of comparisos holds. O the other had, there is a determiistic strategy (algorithm), suggested i [], that shows that 6(log ν() + 1) comparisos are always sufficiet. Related work. At first glace our work seems related to work doe o pre-sortig. I pre-sortig the goal is to pre-process the data such that fewer comparisos are eeded afterwards to sort them. For example i [5] it is show that with O(1) pre-processig oe ca save Θ() comparisos for Quicksort o average. Pre-processig ca be see as computig a partial order o the data that helps for a give sortig algorithm to reduce the umber of ecessary comparisos. The structural quatity that determies how may comparisos are eeded i geeral to fid the rakig give a partial order is the umber of liear extesios of the partial order, i.e., the umber of rakigs cosistet with the partial order. Actually, the logarithm of this umber is a lower boud o the umber of comparisos eeded i geeral [6]. Here we study aother structural measure, amely, the maximum diameter i the Spearma s metric of the set of rakigs cosistet with a partial order. Our results shows that with o( log ) comparisos oe ca make this diameter asymptotically smaller tha the diameter of the set of all rakigs. That is ot the case for the umber of liear extesios which stays i Θ( log ). Notatio. The logarithm log i this paper is assumed to be biary, ad by id we deote the idetity (icreasig) permutatio of []. Algorithm The idea of the ASort algorithm is to partitio the products ito a sorted sequece of equal-sized bis such that the elemets i each bi have smaller

3 rak tha ay elemet i subsequet bis. This approach was suggested by Chazelle [] for ear-sortig. The output of the algorithm is the sequece of bis. Note that we do ot specify the orderig of elemets iside each bi, but cosider ay rakig cosistet with the orderig of the bis. We will show that ay such rakig approximates the actual rakig of the elemets i terms of Spearma s footrule metric D(π, id) = D(π) = i π(i), where π(i) is the rak of the elemet of rak i i a approximate rakig, i.e., i π(i) measures deviatio of the approximated rak from the actual rak. Note that for ay rakig the distace i the Spearma s footrule metric to id is at most. Sice for every i the value i π(i) is bouded by divided by the umber of bis, we see that the approximatio quality depeds o the umber of bis. The algorithm ASort iteratively performs a umber of media searches, each time placig the media ito the right positio i the rakig. Here the media of elemets is defied to be the elemet of rak +1. i=1 ASort (B : set, m : it) 1 B 01 := B // B ij is the j th bi i the i th roud for i := 1 to m do 3 for j := 1 to i 1 do 4 compute the media of B (i 1)j 5 B i(j 1) := {x B (i 1)j x media} 6 B i(j) := {x B (i 1)j x > media} 7 ed for 8 ed for 9 retur B m1,..., B m( m ) To compute the media i lie 4 ad to partitio the elemets i lie 5 ad 6 we use the determiistic algorithm by Blum et al. [1] that performs at most 5.73 comparisos i order to compute the media of elemets ad to partitio them accordig to the media. We ote that i puttig the algorithm ASort to practice oe may wat to use a differet media algorithm, like, e.g., RadomizedSelect [3]. I the followig we determie the umber of comparisos the algorithm ASort eeds o iput B with B = i order to guaratee a prescribed approximatio error of the actual rakig for ay rakig cosistet with the orderig of the bis B m1,..., B m( m ) computed by the algorithm. Lemma 1. For every x B ij, where 0 i m ad 1 j i, it holds j 1 B ik + 1 rak(x) j B ik.

4 Proof. The lemma ca be prove by iductio o the umber of rouds. By costructio, the elemets i B 11 have rak at least 1 ad at most +1 = B 11 ad the elemets i B 1 have rak at least = B ad at most = B 11 + B 1. Now assume that the statemet holds after the (i 1) th roud. The algorithm partitios every bi B (i 1)j ito two bis B i(j 1) ad B i(j). Agai by costructio the elemets i bi B i(j 1) have rak at least j 1 j 1 B (i 1)k + 1 = ( B i(k 1) + B i(k) ) + 1 = ad at most (j 1) 1 B ik + B i(j 1) = j 1 B ik. (j 1) 1 B ik + 1, Similarly, the elemets i bi B i(j) have rak at least j 1 B ik + 1 ad at most j B (i 1)k = j B ik. Lemma. i B ij i for 0 i m ad 1 j i Proof. We prove by iductio that i ay roud i the sizes of ay two bis differ by at most 1, i.e., Bij B ik 1 for 0 i m ad 1 j, k i. The statemet of the lemma the follows sice by a averagig argumet ad the itegrality of the bi sizes, the size of each bi must by of size either i or i. For i = 1 the elemets of B are partitioed either ito two equal sized bis if is eve, or ito two bis whose sizes differ by 1 if is odd. Now assume that the statemet holds for i 1. Take two bis B (i 1)j ad B (i 1)k. We distiguish two cases. Case 1. B (i 1)j ad B (i 1)k have the same size c. If c is eve, the both bis get split up ito two bis each ad the resultig four bis all have the same size. If c is odd, the each of the bis gets split up ito two bis of sizes c ad c, respectively, which differ by 1. Case. Without loss of geerality, B (i 1)j = c ad B (i 1)k = c + 1. If c is eve, the B (i 1)j gets split up ito two bis both of size c ad B (i 1)k gets split up ito two bis of size c ad c + 1, respectively. If c is odd, the B (i 1)j gets split up ito two subsets of size c+1 ad c+1 1, respectively, ad B (i 1)k gets split up ito two bis of size c+1. I ay case the bis differ i size by at most 1. Lemma 3. I m rouds the algorithm ASort performs less tha 6m comparisos.

5 Proof. The algorithm by Blum et al. [1] eeds at most 5.73 comparisos to fid the media of elemets ad to partitio the elemets with respect to the media. I the i th roud ASort partitios the elemets i every bi B ij, 1 j i with respect to their media. Thus the i th roud eeds at most 5.73 B ij = 5.73 B ij = i j=1 i j=1 comparisos. As the algorithm rus for m rouds the overall umber of comparisos is less tha 6m. Theorem 1. Let r = ν(). Ay rakig cosistet with the orderig of the bis computed by ASort i log ν() + 1 rouds, i.e., with less tha 6(log ν() + 1) comparisos, has a Spearma s footrule distace of at most r to the actual rakig of the elemets from B. Proof. Usig the defiitio of Spearma s footrule metric ad Lemmas 1 ad we ca coclude that the distace of the rakig of the elemets i B to ay rakig cosistet with the orderig of the bis computed by ASort i m rouds ca be bouded by m j=1 B mj m ( m ) m 1 ( m + 1 ) m 1 ( m = m 1. ), sice m Pluggig i log ν() + 1 for m gives a distace less tha r as claimed i the statemet of the theorem. The claim for the umber of comparisos follows from Lemma 3. 3 Lower Boud For r > 0, by B D (id, r) we deote the ball cetered at id of radius r with respect to the Spearma s footrule metric, so B D (id, r) := {π S : D(π, id) r}. Next we estimate the umber of permutatios i a ball of radius r.

6 Lemma 4. ( r ) BD (id, r) e ( ) e(r + ). Proof. Every permutatio π S is uiquely determied by the sequece {π(i) i} i. Hece, for ay sequece of o-egative itegers d i, i = 1,...,, there are at most permutatios π S satisfyig π(i) i = d i. If d D (π, id) r, the i π(i) i r. Sice the umber of sequeces of o-egative itegers whose sum is at most r is ( ) r+, we have ( ) ( ) r + e(r + ) B D (id, r). Next, we give a lower boud o the size of B D (id, r). Let s := r, ad let us first assume that is divisible by s. We divide the idex set [] ito s blocks of size /s, such that for every i {1,,..., s} the ith block cosists of elemets (i 1) s + 1, (i 1) s +,..., i s. For every s permutatios π 1, π,..., π s S /s we defie the permutatio ρ S to be the cocateatio of the permutatios applied to correspodig blocks, so ρ := π 1 (b 1 )π (b )... π s (b s ). Note that the distace of ρ to id with respect to Spearma s footrule metric is at most /s r, sice ρ(i) i /s, for every i []. Obviously, for every choice of π 1, π,..., π s we get a differet permutatio ρ, which meas that we have at least (( ) ) s ( r )! s e differet permutatios i B D (id, r). If is ot divisible by s, we divide [] ito s blocks of size either /s or /s, agai apply a arbitrary permutatio o each of them ad we ca obtai the same boud i a aalogous fashio. Theorem. Let A be a radomized approximate sortig algorithm, let ν = ν() be a fuctio, ad let r = r() = ν(). If for every iput permutatio π S the expected Spearma s footrule distace of the output to id is at most r, the the algorithm performs at least (mi{log ν, log } 6) comparisos. Proof. Let k be the smallest iteger such that A performs at most k comparisos for every iput. For a cotradictio, let us assume that k < (mi{log ν, log } 6). First, we are goig to prove 1! > k Sice log ν 6 > k/, we have ν 6 ( ) e(r + ). (1) > k/ ad sice ν = r we get e r > k/ e. ()

7 O the other had, from log 6 > k/ we get > k/ implyig 6 Puttig () ad (3) together, we obtai Hece e e > k/ e. (3) 1 ( )! > k e e(r + ) > k/. ( ) e(r + ), provig (1). By R we deote the source of radom bits for A. Oe ca see R as the set of all ifiite 0-1 sequeces, ad the the algorithm is give a radom elemet of R alog with the iput. For a permutatio π S ad α R, by A(π, α) we deote the output of the algorithm with iput π ad radom bits α. We fix α R ad ru the algorithm for every permutatio π S. Note that with the radom bits fixed the algorithm is determiistic. For every compariso made by the algorithm there are two possible outcomes. We partitio the set of all permutatios S ito classes such that all permutatios i a class have the same outcomes of all the comparisos the algorithm makes. Sice there is o radomess ivolved, we have that for every class C there exists a σ S such that for every π C we have A(π, α) = σ π. I particular, this implies that the set {A(π, α) : π C} is of size C. O the other had, sice the algorithm i this settig is determiistic ad the umber of comparisos of the algorithm is at most k, there ca be at most k classes. Hece, each permutatio i S is the output for at most k differet iput permutatios. From Lemma 4 we have B D (id, r) ( e(4r+) ), ad this together with (1) implies that at least ( ) e(r + )! k > 1! iput permutatios have output at distace to id more tha r. Now, if both the radom bits α R ad the iput permutatio π S are chose at radom, the expected distace of the output A(π, α) to id is more tha r. Therefore, there exists a permutatio π 0 such that for a radomly chose α R the expected distace d D (A(π 0, α), id) is more tha r. Cotradictio. 4 Coclusio Motivated by a applicatio i market research we studied the problem to approximate a rakig of items. The metric we use to compare rakigs is Spearma s footrule metric, which is withi a costat factor to Kedall s tau metric that is frequetly used i marketig research. We showed that ay radomized algorithm eeds at least (mi{log ν(), log } 6) comparisos to approximate

8 a give rakig of items withi expected distace /ν(). This result is complemeted by a algorithm that shows that 6(log ν() + 1) comparisos are always sufficiet. I particular, this meas that i some cases substatially less comparisos have to be performed tha for sortig exactly, provided that a sufficietly large error is allowed. That is, as log as the desired expected error is of order α for costat α oe eeds Ω( log ) comparisos, which asymptotically is ot better tha sortig exactly. But to achieve expected error of order o(1) oly o( log ) comparisos are eeded. Ackowldegmets. We are idebted to Jiří Matoušek for commets ad isights that made this paper possible. Refereces 1. M. Blum, R. W. Floyd, V. Pratt, R. L. Rivest, ad R. E. Tarja. Liear time bouds for media computatios. I STOC 7: Proceedigs of the fourth aual ACM symposium o Theory of computig, pages ACM Press, B. Chazelle. The soft heap: A approximate priority queue with optimal error rate. Joural of the ACM, 47(6): , T. H. Corme, C. E. Leiserso, ad R. L. Rivest. Itroductio to Algorithms. The MIT Press/McGraw-Hill, P. Diacois ad R. L. Graham. Spearma s footrule as a measure of disarray. Joural of the Royal Statistical Society, 39():6 68, H. K. Hwag, B. Y. Yag, ad Y. N. Yeh. Presortig algorithms: a average-case poit of view. Theoretical Computer Sciece, 4(1-):9 40, J. Kah ad J. H. Kim. Etropy ad sortig. I STOC 9: Proceedigs of the twety-fourth aual ACM symposium o Theory of computig, pages ACM Press, D. E. Kuth. The Art of Computer Programmig, volume 3. Addiso Wesley, C. L. Mallows. No-ull rakig models. Biometrica, 44: , 1957.

Fundamenta Informaticae XXI (2001) IOS Press

Fundamenta Informaticae XXI (2001) IOS Press Fudameta Iformaticae XXI (2001) 1001 1006 1001 IOS Press Approximate Sortig Joachim Giese Max Plak Istitute for Computer Sciece Saarbrücke, Germay Eva Schuberth Istitute for Theoretical Computer Sciece