Comparator Networks for Binary Heap. Construction? Abstract. Comparator networks for constructing binary heaps of size n

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1 Compaato Netwoks fo Binay Heap Constuction? Geth Stlting Bodal 1;?? and M. Cistina Pinotti 2;??? 1 Max-Planck-Institut fu Infomatik, Im Stadtwald, D Saabucken, Gemany 2 Istituto di Elaboazione della Infomazione, CNR, Pisa, Italy Abstact. Compaato netwoks fo constucting binay heaps of size n ae pesented which have size O(n log log n) and depth O(log n). A lowe bound of n log log n?o(n) fo the size of any heap constuction netwok is also poven, implying that the netwoks pesented ae within a constant facto of optimal. We give a tight elation between the leading constants in the size of selection netwoks and in the size of heap constuction netwoks. Intoduction The heap data stuctue, intoduced in 1964 by Williams [17], has been extensively investigated in the liteatue due to its many applications and intiguing patial ode. Algoithms fo heap management insetion, minimum deletion, and constuction have been discussed in seveal models of computation. Fo the heap constuction algoithm, Floyd has given a sequential algoithm building the tee in a bottom-up fashion in linea time, which is clealy optimal. On the weak shaed memoy machine model, EREW-PRAM, Olaiu and Wen can build a heap of size n in time O(log n) and optimal wok [14]. On the poweful CRCW- PRAM model, the best-known heap constuction algoithm was given by Raman and Dietz and takes O(log log n) time [6]. The same time pefomance holds fo the paallel compaison tee model [5]. Finally Dietz showed that O((n)), whee (n) is the invese of Ackeman's function, is the expected time equied to build a heap in the andomized paallel compaison tee model [5]. All the above paallel algoithms achieve optimal wok O(n), and the time optimality of the deteministic algoithms can be agued by eduction fom the selection of the minimum element in a set. In this pape we addess the heap constuction poblem fo the simplest paallel model of computation, namely compaato netwoks. Compaato netwoks pefom only compaison opeations, which may occu simultaneously.? This eseach was done while the st autho was visiting the Istituto di Elaboazione della Infomazione, CNR, Pisa.?? Suppoted by the Calsbeg foundation (Gant No /20). Patially suppoted by the ESPRIT Long Tem Reseach Pogam of the EU unde contact No (ALCOM-IT). bodal@mpi-sb.mpg.de.??? pinotti@iei.pi.cn.it.

2 The most studied compaato netwoks ae soting and meging netwoks. In the ealy 1960's, Batche poposed the odd-even mege algoithm to mege two sequences of n and m elements, n m, which can be implemented by a meging netwok of size O((m + n) log m). In the ealy 1970's Floyd [12] and Yao [18] poved the asymptotic optimality of Batche's netwoks. The lowe bound has ecently been impoved by Miltesen, Pateson and Taui [13], closing the longstanding facto-of-two gap between uppe and lowe bounds. It is notewothy to ecall, that mege can be solved in the compaison tee model with a tee of depth m + n? 1. Batche also showed how his mege algoithm could be used to implement soting netwoks with size O(n log 2 n) and depth O(log 2 n) to sot n inputs [12]. Fo a long time, the question emained open as to whethe soting netwoks with size O(n log n) and depth O(log n) existed. In 1983, Ajtai, Komlos and Szemeedi [1] pesented soting netwoks with size O(n log n) and depth O(log n) to sot n elements. This esult, although patially unsatisfying due to big constants hidden by the O-notation, eveals that the soting poblem equies the same amount of wok in both compaison tee and compaato netwok models. Selection, soting and meging ae stictly elated poblems. Seveal sequential algoithms with linea wok have been discussed fo selection. The st is due to Blum et al. [4] and equies 5:43n compaisons. This esult was late impoved by Schonhage et al. to 3n [16] and by Do and Zwick to 2:95n [7, 8]. Bent and John poved a lowe bound of 2n fo this poblem [3]. Do and Zwick [9] impoved it to (2 + )n [9]. Fo a suvey of pevious wok on lowe bounds in the compaison tee model, see the pape by Do and Zwick [9]. An (n; t)-selection netwok is a compaato netwok that selects the t smallest elements in a set of n elements. Alekseev [2] poved that an (n; t)-selection netwok has at least size (n? t)dlog(t + 1)e. 1 Fo t = (n ) and 0 < < 1, the existence of a wok optimal selection netwok immediately follows by the soting netwoks of Ajtai et al. Howeve, since selection netwoks do not need to do as much as soting netwoks, and due to the big constant hidden by the soting netwoks in [1], selection netwoks with impoved constant factos in both depth and size have been developed. In paticula, Pippenge poposes a (n; bn=2c)-selection netwok with size 2n log n and depth O(log 2 n) [15]. Moe ecently, Jimbo and Maouka have constucted a (n; bn=2c)-selection netwok of depth O(log n) and of size at most Cn log n + O(n), fo any abitay C > 3= log 3 1:89, which impoves Pippenge's constuction by a constant facto in size and at the same time by an ode in depth [11]. The peceding summay shows that wok optimal compaato netwoks have been studied fo meging, soting, and selection. Although the heap data stuctue has histoically been stictly elated to these poblems, we ae not awae of any compaato netwok fo the heap constuction poblem. In this scenaio, we show that heap constuction can be done by compaato netwoks of size O(n log log n) and depth O(log n), and that ou netwoks each optimal size by educing the poblem of selecting the smallest log n elements to heap constuc- 1 All logaithms thoughout this pape have base 2

3 tion. Finally, since nding the minimum equies at least a netwok of size n? 1 and depth dlog ne, ou heap constuction netwoks also have optimal depth. 1 Peliminaies Let us eview some denitions, and agee on some notations used thoughout the pape. A binay tee of size n is a tee with n nodes, each of degee at most two. A node x of a binay tee belongs to level k if the longest simple path fom the oot to x has k edges. The height of the tee is the numbe of edges in the longest simple path stating at the oot of the tee. The subtee T x ooted at node x at level k is the tee induced by the descendants of x. A complete binay tee is a binay tee in which all the leaves ae at the same level and all the intenal nodes have degee two. Clealy, it has height blog nc. A heap shaped binay tee of height h is a binay tee whose h? 1 uppemost levels ae completed lled and the h-th level is lled fom the left to the ight. In a heap odeed binay tee, each node contains one element which is geate o equal to the element at its paent. Finally, a binay heap is dened as a heap-shaped and heap-odeed binay tee [17], which can be stoed in an aay H as an implicit tee of size n, as depicted in Fig. 1. The element of the oot of the tee is at index 1 of the aay, (i.e., oot is stoed in H[1]), and given an index i of a node x, the indices of its left and ight childen ae 2i and 2i + 1, espectively. A compaato netwok with n inputs and size s is a collection of n hoizontal lines, one fo each input, and s compaatos. A compaato between line i and j, biey i : j, compaes the cuent values on lines i and j and is dawn as a vetical line connecting lines i and j. Afte the compaison i : j, the minimum value is put on line i, while the maximum ends up on line j. Finally, a compaato netwok has depth d, if d is the lagest numbe of compaatos that any input element can pass though. Assuming that each compaato poduces its output in constant time, the depth of a compaato netwok is the unning time of such a netwok. Fom now on, let us efe to compaato netwoks simply as netwoks. Fo a compehensive account of compaato netwoks, see [12, pp ]. 2 Sequential Heap Constuction It is well known that an implicit epesentation of a binay heap H of size n can be built in linea sequential time by the heap constuction algoithm of Floyd [10]. Because we base ou heap constuction netwoks on Floyd's algoithm, we ephase it as follows: Assuming that the two binay tees ooted at the childen of a node i ae heaps, the heap-ode popety in the subheap ooted at i can be eestablished simply by bubbling down the element H[i]. We let the bubbling down pocedue be denoted Siftdown. At each step, Siftdown detemines the smallest of the

4 elements H[i]; H[2i], and H[2i+1]. If H[i] is the smallest, then the subtee ooted at node i is a heap and the Siftdown pocedue teminates. Othewise, the child with the smallest element and H[i] ae exchanged. The node exchanged with H[i], howeve, may violate the heap ode at this point. Theefoe, the Siftdown pocedue is ecusively invoked on that subtee. We can now apply Siftdown in a bottom-up manne to convet an aay H stoing n elements into a binay heap. Since the elements in the subaay H[(bn=2c + 1) ::n] ae all leaves, each is a 1-element heap to begin with. Then, the emaining nodes of the tee ae visited to un the Siftdown pocedue on each one. Since the nodes ae pocessed level by level in a bottom up fashion, it is guaanteed that the subtees ooted at the childen of the node i ae heaps befoe Siftdown uns at that node. In conclusion, obseve that the Siftdown outine invoked on a subheap of height i pefoms 2i compaisons in the wost case, and that the wost case unning time of the heap constuction algoithm of Floyd descibed above is P blog nc i=0 n 2 i 2i = O(n), which is optimal. 2m 1 H H H H m5 13m 2 3? @ 14 m m AA A 10 m 26m A 32m 15 m A m 9 9m m 15 Fig. 1. A binay heap of size 15 and its implicit epesentation. 3 Heap Constuction Netwoks of Size n log n In this section we pesent heap constuction netwoks which have size at most nblog nc and depth 4blog nc? 2. Notice that any soting netwok could also be used as a heap constuction netwok. The netwoks pesented in this section ae used in Sect. 4 to constuct impoved heap constuction netwoks of size O(n log log n), and in Sect. 5 to give a eduction fom selection to heap constuction. Lemma 1 gives a netwok implementation of the sifting down algoithm used in the heap constuction algoithm by Floyd [10].

5 Lemma 1. Let T be a binay tee of size n and height h. If the subtees ooted at the childen of the oot satisfy heap ode, then the elements of T can be eaanged to satisfy heap ode with a netwok of size n? 1 and depth 2h. At depth 2i + 1 and 2i + 2 of the netwok the compaatos ae only between nodes at level i and i + 1 in T. All compaatos coespond to edges of T, and fo each edge thee is exactly one compaato. Poof. If the tee has height zeo, no compaato is equied. Othewise let be the oot and u and v the childen of. If u o v is not pesent, the steps below which would involve v o u ae skipped. Fist we apply the compaatos : u and : v. Because T u and T v wee assumed to be heap odeed subtees, now has the minimum. Afte the two compaatos the heap ode can be violated at the oots of both T u and T v. We theefoe ecusively apply the above to the subtees T u and T v. Notice that the two ecusively constucted netwoks involve disjoint nodes and theefoe can be pefomed in paallel. If only has one child we still chage the netwok depth two to compae with its childen to guaantee that all compaisons done in paallel by the netwok coespond to edges between nodes at the same levels in T. The depth of the netwok is two plus the depth of the deepest ecusively constucted netwok. By induction it follows that the depth of the netwok is 2h, and that the netwok at depth 2i + 1 and 2i + 2 only pefoms compaisons between nodes at level i and i + 1 in T. Futhemoe, the netwok contains exactly one compaato fo each edge of T. ut Notice that the netwok has n? 1 compaatos while the coesponding algoithm of Floyd only needs h compaisons. By eplacing the sifting down algoithm in Floyd's heap constuction algoithm by the sifting down netwoks of Lemma 1, we get the following lemma. Lemma 2. Let T be a binay tee of size n and height h which does not satisfy heap ode, and let n i be the numbe of nodes at level i in T. Then a netwok exists of size Ph i=0 i n i and depth 4h? 2 which eaanges the elements of T to satisfy heap ode. All compaatos coespond to edges of T. Poof. Initially all nodes at level h of T by denition ae heap odeed binay tees of height zeo. Iteatively fo each level i = h?1; : : : ; 0 we apply the sifting down netwoks of Lemma 1 in paallel to the 2 i subtees ooted at level i of T, to make these subtees satisfy heap ode. The esulting tee then satises heap ode. By Lemma 1 all compaatos coespond to edges of T. The edge between a node v at level i and its paent coesponds to a set of compaatos in the esulting netwok. These compaatos ae pefomed exactly when we apply the sifting down netwoks of Lemma 1 to an ancesto of v, i.e., thee ae exactly i compaatos coesponding to this edge. The total numbe of compaatos is Ph i=0 i n i. By Lemma 1 the depth of the netwok is Ph i=0 2i = h2 + h. But because the netwoks constucted by Lemma 1 poceeds top-down on T, having exactly

6 depth two fo each level of T, the applications of Lemma 1 can be pipelined. Afte the st two compaatos of the applications of Lemma 1 to subtees ooted at level i, the applications of Lemma 1 to subtees ooted at level i? 1 can be initiated. The application of Lemma 1 to the oot of the tee can theefoe be initiated at depth 2(h? 1) + 1 of the netwok, i.e., the netwok has depth 2(h? 1) + 2h = 4h? 2. ut Theoem 1. Thee exists a heap constuction netwok of size at most nblog nc and depth 4blog nc? 2. All compaatos coespond to edges of T. Poof. Let the n input lines epesent a heap shaped binay tee of height blog nc. The theoem then follows fom Lemma 2. ut In Fig. 2 we show the netwok of Theoem 1 fo n = 15. The netwok has size 34 and depth 10. Notice that the st two compaatos of the application of Lemma 1 to the oot of the tee (1 : 2 and 1 : 3) ae done in paallel with the thid and fouth compaato of the applications of Lemma 1 to the subtees ooted at nodes 2 and 3. x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x Fig. 2. A heap constuction netwok fo n = 15. All compaatos ae of the fom i : j, whee i < j. 4 Heap Constuction Netwoks of Size O(n log log n) In the following we give impoved heap constuction netwoks of depth O(log n) and size O(n log log n). The impoved netwoks ae obtained by combining the netwoks of Theoem 1 with ecient selection netwoks. An abitay soting netwok is obviously also an (n; t)-selection netwok, e.g., the soting netwok of size O(n log n) by Ajtai et al. [1]. Due to the lage constants involved in the soting netwok of Ajtai et al., Pippenge [15] and

7 Jimbo and Mauoka [11] have developed specialized (n; bn=2c)-selection netwoks of size O(n log n) whee the involved constants ae of easonable size. The following lemma was developed by Jimbo and Mauoka [11]. Lemma 3 (Jimbo and Mauoka). Fo an abitay constant C > 3= log3 1:89, thee exist (n; bn=2c)-selection netwoks of size at most Cn log n + O(n) and depth O(log n). Unfotunately, neithe Pippenge [15] o Jimbo and Mauoka [11] give bounds fo geneal (n; t)-selection netwoks. The following lemma is a consequence of Lemma 3, and is sucient fo ou puposes. Lemma 4. Fo an abitay constant C > 6= log 3 3:79, thee exist (n; t)- selection netwoks of size Cn log t + O(n) and depth O(log n log t). Poof. The n input lines ae patitioned into dn=te blocks B 1 ; : : : ; B dn=te of size t each. By applying the selection netwoks of Lemma 3 to B 1 [ B 2 we nd the t least elements of B 1 [ B 2. By combining the dn=te blocks in a teewise fashion with dn=te?1 applications of Lemma 3 to 2t elements, we nd the t least elements of the n inputs. The esulting netwok has size (dn=te? 1)(C 2t log 2t+O(2t)) = 2Cn log t + O(n) and depth O(log n log t), fo C > 3= log 3. ut We need the following denition. Let P be an abitay connected subset of nodes of a binay tee T which contains the oot of T. Let x 1 x 2 x jpj be the set of elements in P, and let x 0 1 x0 2 x0 be the set of elements in P jpj afte applying a netwok N to T. We dene a netwok N to be heap-convegent, if N fo all possible inputs, all connected subsets P of nodes of T containing the oot of T, and i = 1; : : : ; jpj satises x 0 x i i. Notice that soting netwoks ae not heap-convegent. If P is the path to the ightmost node in the lowest level of a tee, then P always contains the maximum element afte applying a soting netwok, but the maximum element could initially be anywhee in the tee. Lemma 5. A compaato coesponding to an edge in a binay tee T is a heapconvegent netwok. Poof. Let the compaato be u : v, whee v is a child of u in T. If P does not contain u it does not contain v eithe, implying that the elements in P ae unchanged. If P contains both u and v, the set of elements is also unchanged. If P contains u but not v, the compaato u : v can only eplace the element at u with a smalle element fom v in which case x 0 x i i fo all i = 1; : : : ; jpj. ut Because the netwoks constucted by Theoem 1 only contain compaatos coesponding to tee edges and heap convegence is a tansitive popety we immediately have the following coollay: Coollay 1. The netwoks constucted by Theoem 1 ae heap-convegent. Theoem 2. If fo some constants C and d, thee exist (n; t)-selection netwoks of size Cn log t + O(n) and depth O(log d n), then thee exist heap constuction netwoks of size Cn log log n+o(n log log log n) and depth 4 log n+o(log d log n).

8 Poof. Assume without loss of geneality that n 4. Let the n input lines epesent a heap shaped binay tee T of height h = blog nc, and let k = dlog he 1. The heap constuction netwok poceeds in thee phases. 1. To each subtee T v ooted at level h? 2k + 1, apply in paallel (jt v j; 2 k? 1)- selection netwoks, such that all elements at the uppe k levels of T v become less than o equal to all elements at the emaining levels of T v. 2. Apply the heap constuction netwoks of Theoem 1 to the uppemost h? k levels of T. 3. In paallel apply Theoem 1 to each subtee T v ooted at level h? 2k + 1. It follows immediately fom Step 2 that the uppemost h? 2k levels of the tee satisfy heap ode and fom Step 3 that each subtee ooted at level h? 2k + 1 satises heap ode. What emains to be poven fo the coectness of the algoithm is that fo all nodes v at level h? 2k + 1, the subtee T v only contains elements which ae geate o equal to the elements on the path fom the oot to v. Afte Step 1, the 2 k? 1 least elements e 0 e 2 k?2 of T v ae at the uppemost k levels of T v, which ae exactly the levels of T v which ovelap with Step 2. Let p 0 p h?2k denote the elements on the path fom the oot to v (excluding v) afte Step 2. Because the netwok applied in Step 2 is heapconvegent and 2 k? 2 h? 2k, we have p i e i fo i = 0; : : :; h? 2k by letting P consist of the path fom the oot to v togethe with the uppe k levels of T v. We conclude that afte Step 2 all elements on the path fom the oot to v ae smalle than o equal to all the elements in T v, and that afte Step 3, T satises heap ode. Fom Theoem 1 we get the following uppe bound on the size and depth of the esulting netwok. The size is bounded by? Cn log 2k + O(n) + O n 2 k log n 2 k +? n log 2 2k + O(n) ; which is (C + 2)n log log n + O(n), and the depth is bounded by O log d 2 2k + (4(h? k)? 2) + (4(2k? 1)? 2) ; which is 4 log n + O(log d log n). The \+2" in the size bound comes fom the application of the heap constuction netwoks of Theoem 1 in Step 3. If we instead apply the above constuction ecusively in Step 3, we get heap constuction netwoks of size Cn log log n + (C + 2)n log log log n + O(n) and depth 4 log n + O(log d log n). ut Notice that in Steps 1 and 3 we could have used abitay soting netwoks, but in Step 2 it is essential that the heap constuction netwok used is heapconvegent. By applying the constuction ecusively O(log n) times the asymptotic size could be slightly impoved, but the constant in font of n log log n would still be C. Fom Lemma 4 we get the following coollay:

9 Coollay 2. Fo an abitay constant C > 6= log 3 3:79, thee exist heap constuction netwoks of size Cn log log n+o(n log log log n) and depth 4 log n+ O(log 2 log n). 5 A Lowe Bound fo the Size of Heap Constuction Netwoks We now pove that the constuction of the pevious section is optimal. Let S(n; t) denote the minimal size of (n; t)-selection netwoks, and let H(n) denote the minimal size of heap constuction netwoks on n inputs. The following lowe bound on S(n; t) is due to Alekseev [2]. Lemma 6 (Alekseev). S(n; t) (n? t)dlog(t + 1)e. Theoem 3. H(n) S(n; blog nc)? O(n). Poof. The theoem is poven by giving a eduction fom (n; t)-selection to heap constuction. We pove that (n; t)-selection can be done by netwoks with size H(n) + 2 t+1? 2t? 2. Fist we constuct a heap ove the n inputs with a netwok of size H(n), and make the obsevation that the t least elements can only be at levels 0; : : : ; t? 1 of the heap. The minimum is at the oot, i.e., at output line one. To nd the second least element we conside the implicit heap given by the lines n; 2; 3; : : :; 2 t?1. Notice that the oot is now line n. By applying the sifting down netwok of Lemma 1 to the levels 0; : : : ; t? 1 of this tee the emaining t? 1 least inputs ae at levels 0; : : :; t? 2 of this tee. The second least element is now at output line n. By iteatively letting the oot be lines n? 1; n? 2; : : :; n? t? 2, and by applying Lemma 1 to tees of deceasing height, the t least elements will appea in soted ode at output lines 1; n; n? 1; n? 2; : : :; n? t + 2. If the t smallest inputs ae equied to appea at the st t output lines, the netwok lines ae pemuted accodingly. The total numbe of compaatos fo the t? 1 applications of Lemma 1 is Xt?1 i=0 (2 i+1? 2) = 2 t+1? 2t? 2 : We conclude that the esulting (n; t)-selection netwok has size H(n) + 2 t+1? 2t?2, implying H(n) S(n; t)?2 t+1 +2t+2. By letting t = blog nc the theoem follows. ut By combining Lemma 6 and Theoem 3, we get the following coollay. Coollay 3. H(n) n log log n? O(n).

10 6 Conclusion The paallel constuction of heaps has been addessed fo seveal paallel models of computation: EREW-PRAM [14], CRCW-PRAM [6], the paallel compaison tee model and the andomized paallel compaison tee model [5]. These algoithms all achieve optimal O(n) wok. In this pape we have addessed the poblem fo the most simple paallel model of computation, namely compaato netwoks. As opposed to meging and selection, which both can be solved in sequential linea time but equie netwoks of size (n log n), we have shown that heap constuction can be done by netwoks of size O(n log log n) and depth O(log n), and that this is optimal. By combining the esults of Theoem 2 and Theoem 3, we get the following chaacteization of the leading constant in the size of heap constuction netwoks compaed to the leading constant in the size of (n; t)- selection netwoks. Theoem 4. If fo constants C 1 and C 2, then C 1 n log t? O(n) S(n; t) C 2 n log t + O(n) ; C 1 n log log n? O(n) H(n) C 2 n log log n + O(n log log log n) : Acknowledgment Thanks to Pete Sandes fo his comments on an ealie daft of this pape. Refeences 1. Miklos Ajtai, Janos Komlos, and Ende Szemeedi. Soting in c log n paallel steps. Combinatoica, 3:1{19, Vladimi Evgen'evich Alekseev. Soting algoithms with minimum memoy. Kibenetika, 5(5):99{103, Samuel W. Bent and John W. John. Finding the median equies 2n compaisons. In Poc. 17th Ann. ACM Symp. on Theoy of Computing (STOC), pages 213{216, Manuel Blum, Robet W. Floyd, Vaughan Patt, Ronald L. Rivest, and Robet Ende Tajan. Time bounds fo selection. Jounal of Compute and System Sciences, 7:448{461, Paul F. Dietz. Heap constuction in the paallel compaison tee model. In Poc. 3d Scandinavian Wokshop on Algoithm Theoy (SWAT), volume 621 of Lectue Notes in Compute Science, pages 140{150. Spinge Velag, Belin, Paul F. Dietz and Rajeev Raman. Vey fast optimal paallel algoithms fo heap constuction. In Poc. 6th Symposium on Paallel and Distibuted Pocessing, pages 514{521, Doit Do and Ui Zwick. Selecting the median. In Poc. 6th ACM-SIAM Symposium on Discete Algoithms (SODA), pages 28{37, 1995.

11 8. Doit Do and Ui Zwick. Finding the alpha n-th lagest element. Combinatoica, 16:41{58, Doit Do and Ui Zwick. Median selection equies (2 + )n compaisons. In Poc. 37th Ann. Symp. on Foundations of Compute Science (FOCS), pages 125{134, Robet W. Floyd. Algoithm 245: Teesot3. Communications of the ACM, 7(12):701, Shuji Jimbo and Akia Mauoka. A method of constucting selection netwoks with O(log n) depth. SIAM Jounal of Computing, 25(4):709{739, Donald E. Knuth. The At of Compute Pogamming, Volume III: Soting and Seaching. Addison-Wesley, Reading, MA, Pete Bo Miltesen, Mike Pateson, and Jun Taui. The asymptotic complexity of meging netwoks. Jounal of the ACM, 43(1):147{165, Stephan Olaiu and Zhaofang Wen. Optimal paallel initialization algoithms fo a class of pioity queues. IEEE Tansactions on Paallel and Distibuted Systems, 2:423{429, Nicholas Pippenge. Selection netwoks. SIAM Jounal of Computing, 20(5):878{ 887, Anold Schonhage, Michael S. Pateson, and Nicholas Pippenge. Finding the median. Jounal of Compute and System Sciences, 13:184{199, John William Joseph Williams. Algoithm 232: Heapsot. Communications of the ACM, 7(6):347{348, Andew C. Yao and Fances F. Yao. Lowe bounds on meging netwoks. Jounal of the ACM, 23:566{571, 1976.