split split (a) (b) split split (c) (d)

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1 International Journal of Foundations of Comuter Science c World Scientic Publishing Comany ON COST-OPTIMAL MERGE OF TWO INTRANSITIVE SORTED SEQUENCES JIE WU Deartment of Comuter Science and Engineering Florida Atlantic University Boca Raton, FL jie@cse.fau.edu and STEPHEN OLARIU Deartment of Comuter Science Old Dominion University Norfolk, VA olariu@cs.odu.edu Received (received date) Revised (revised date) Communicated by Editor's name ABSTRACT The roblem of merging two intransitive sorted seuences (that is, to generate a sorted total order without the transitive roerty) is considered. A cost-otimal arallel merging algorithm is roosed under the EREW PRAM model. This algorithm has a run time of O(log 2 n)using O(n= log 2 n) rocessors. The cost-otimal merge in the strong sense is still an oen roblem. Keywords: Hamiltonian ath, merging, PRAM, sorting, tournament. 1. Introduction In this aer we consider the roblem of merging two intransitive sorted seuences. Consider a total order! on set N, but without the transitive roerty. That is, if u i! u j and u j! u k, it is not necessary that u i! u k. The total order reuires that for any two elements u i and u j, either u i! u j or u j! u i. A intransitive sorted seuence is a seuence of elements, u 1 u 2 :::u n, in N, such that u 1! u 2! :::! u n : It has been roved that for any subset of N, the elements in the subset can be arranged in a sorted seuence (more than one may exist). The intransitive total This work was suorted in art by NSF grant CCR and grant ANI

2 u 2 u 1 u 3 u 5 u 4 Figure 1: A tournament of ve layers. order is also called a tournament which is a directed grah with its underlying grah comletely connected. Each layer in a tournament is reresented by a vertex. An edge, u i! u j, exists if layer u i beats layer u j. A sorted seuence corresonds to a Hamiltonian ath of the grah. Figure 1 shows a tournament of ve layers. One intransitive total order is u 3! u 4! u 2! u 5! u 1. When! is transitive, the intransitive total order arrangement is reduced to a regular sorting roblem. Unlike the regular sorting roblem, more than one solution exists for the generalized sorting roblem. For examle, u 1! u 3! u 2! u 5! u 4 is another intransitive total order for the examle of Figure 1. A arallel algorithm is cost-otimal for a given roblem if the roduct of the run time and the number of rocessors used matches the seuential comlexity of the roblem, regardless of the run time of the arallel algorithm. A arallel algorithm is cost-otimal in the strong sense, or strongly cost-otimal, if its run time cannot be imroved by any other cost-otimal arallel algorithms. In this aer, we consider the roblem of merging, which deals with merging two given sorted subseuences into one sorted seuence. Sorting and merging are related, for examle, the two-way merge strategy can be used to generate a two-way merge sort. However, sorting and merging are dierent with dierent lower bounds on comutational comlexity. For examle, under the CREW PRAM model, the lower bound on the regular sorting is O(log n), whereas the lower bound on the regular merging is O(log log n). Merging two intransitive sorted subseuences oses new challenges. The traditional merging by ranking [3] and bitonic merging [2] cannot be alied. Because both concets of ranking and bitonic seuences use the transitive roerty. In this aer, we roose a divide-and-conuer aroach called slit-and-merge that reeatedly slits a air of merging seuences into two indeendent airs of merging subseuences. This slit-and-merge algorithm has a run time of O(log 2 n) using 2

3 first =1 slit last =n slit first =1 (a) last =m (b) slit slit cut cut+1 (c) (d) Figure 2: Three ossible slit-and-merge situations. O(n= log 2 n) rocessors under the EREW PRAM model. Clearly, this solution is cost-otimal. This aer is organized as follows: Section 2 rooses the cost-otmal merge rocess. Section 3 discusses related work and some oen roblems. The aer concludes in Section Cost-Otimal Merge We use [1::n] and [1::m] to reresent two given sorted seuences and [i] reresents a secic element. We rst introduce a slit-and-merge rocess that slits a air of seuences into two indeendent airs of subseuences. Two airs are merged once two subseuences in each airhave been merged. This rocess is done recursively by calling the slit-and-merge rocess. Let [slit] be the center of seuence and it is called the slitting oint. first (first )andlast (last ) denote indices for the rst and last elements of seuence (), resectively. The general stes are the following: Select the center element of as the slitting oint and it is denoted as [slit]. If [slit] beats the rst element of (i.e., [slit]! [first ]), then subseuence [first :: slit] (the white subseuence in Figure 2 (a)) is merged with an emty subseuence of, followed by merging [slit +1::last ]with [first ::last ] (the two gray subseuences in Figure 2 (a)). If [slit] is beaten by the last element of (i.e., [last ]! [slit]), then [first ::slit ; 1] is merged with [first ::last ], followed by [slit::last ] 3

4 ar merge([first ::last ], [first ::last ]): 1. case of 2. (first 6 last ): return [first ::last ] 3. (first 6 last ): return [first ::last ] 4. end of case 5. slit := b first+last c 2 6. if [slit]! [first ] then 7. [first ::slit] k ar merge([first ::last ] [slit +1::last ]) 8. else if [last ]! [slit] then 9. ar merge([first ::last ] [first ::slit ; 1]) k [slit::last ] 10. else cut := cut([slit] [first ::last ]) 11. ar merge([first ::cut] [first ::slit])k 12. ar merge([cut +1::last ] [slit +1::last ]) cut(s r[first::last]): 1. cut := b first+last c 2 2. if r[cut]! s ^ s! r[cut +1]then 3. return cut 4. else if r[cut]! s then 5. cut(s r[cut::last]) 6. else cut(s r[first::cut]) (see Figure 2 (b)). If both of the above two cases fail (see Figure 2 (c)), we use a binary search on to nd a cutting oint (denoted as cut as shown in Figure 2 (d)) in, thatis, [cut]! [slit] and [slit]! [cut+1]. The nal seuence is constructed by merging [first ::slit] with [first ::cut] (two white subseuences in Figure 2 (d)), followed by the result of merging [slit+1::last ] with [cut +1::last ] (two gray subseuences in Figure 2 (d)). The role of and is alternated in the subseuent recursive merging rocess to ensure the sizes of both and are reduced by at least half in every two consecutive rounds. We use k as a concatenation oeration to combine two seuences. The merging rocess starts by calling ar merge([1::n] [1::m]). To simlify our discussion, we assume n = m. The slitting rocess is done by locating the slitting oint at and the cutting oint at. The cutting oint is located through a binary search rocess (cut). It is not necessary that merging seuences have to be slit into ieces of unit size. ar merge can be modied based on the following slit-and-merge rocess, where ar merge terminates whenever the size of both seuences is less than or eual to a redened value, and then, an otimal seuential merging algorithm is alied. 4

5 level 1 O( log n )... level 2 level 3 leaf Figure 3: A samle slit-and-merge tree. slit-and-merge: Use ar merge([1::n] [1::n]) however, the recursive call is terminated whenever the sizes of both and are less than or eual to log 2 n. When a recursive call terminates, an otimal seuential merge algorithm is alied to combine every air of subseuences. To suort the above slit-and-merge strategy in the ar merge algorithm. The following statement is inserted to the case statement: [(last ; first ) < log 2 n] ^ [(last ; first ) < log 2 n]: se merge([first ::last ] [first ::last ]), where se merge is a regular otimal seuential merge algorithm. Figure 3 shows a samle slit-and-merge tree, where each node corresonds to a slit-and-merge rocess. Each node (rocess) may generate u to two nodes (two slit-and-merge rocesses), and therefore, this is a binary tree. The deth of the tree is bounded by O(log n), since the size of each seuence ( or ) is reduced by half in every two consecutive rounds. An otimal seuential merging algorithm is alied to each leaf in the tree. Theorem 1: The run time for the slit-and-merge algorithm is O(log 2 n) with O(n= log 2 n) rocessors under the EREW PRAM model. Proof: The slit-and-merge rocess consists of two hases: arallel slit and seuential merge. At the arallel slit hase, consider a new tree by deleting all leaf nodes (gray nodes in Figure 3) of the slit-and-merge tree. Since the number of elements associated with two seuences at each node is bounded by (log 2 n) in the new tree, whereas the total number of elements in two original seuences is O(n), the number of leaf nodes in this new tree is bounded O(n= log 2 n), that 5

6 is, the number of arallel slits is bounded by that number. Therefore, there is a sucient number of rocessors to handle concurrent slitting activities (within the same level). We only need to calculate the run time of the longest ath. Note that each node in a ath corresonds to a cutting rocess (that identies a cut) using a binary search. The size of each seuence is reduced by at least half in two consecutive rounds. Therefore, the overall cost is bounded by dlog ne + dlog n=2e + dlog n=2e + dlog n=4e + dlog n=4e + ::: which iso(log 2 n). Seuential merge is used at each leaf of the slit-and-merge tree. Throughout the slit-and-merge rocess, no concurrent read or write is needed conseuently, only the EREW PRAM model is needed. The cost of a seuential merge at each leaf node is eual to the number of elements in the two subseuences to be merged, which is bounded by 2log 2 n = O(log 2 n). However, the number of leaf nodes could be more than O(n= log 2 n). In fact, the size of both and is reduced by at least half by two consecutive rounds generating u to four new branches in the slit-and-merge tree. Consider a comlete binary tree with a deth of 2log(n= log 2 n), the total number of leaf nodes is 2 2log(n= log2 n) = (n= log 2 n) 2, which is clearly more than n= log 2 n (the number of rocessors). We divide leaf nodes into two grous: a leaf node with a size of log 2 n or more is assigned to grou one and a node with a size less than log 2 n is assigned to grou two. Each leaf node in grou one is assigned to a distinct rocessor. Leaf nodes in grou two are assigned in seuence to a rocessor until its load is no less than log 2 n (but less than 2log 2 n) and, then, a new rocessor is used for the assignment. Clearly, these seuential merges can be done within 2 log 2 n = O(log 2 n) using no more than 2n= log 2 n = O(n= log 2 n) rocessors. Therefore, the overall run time is O(log 2 n) using O(n= log 2 n) rocessors. Because the roduct of the run time and the number of rocessors is O(n) which matches the lower bound for seuential comutation, the roosed algorithm is cost-otimal. 3. Related Work and Oen Problems Cole [3] shows that the cost-otimal merge of regular sorted seuences in the strong sense under the EREW PRAM model is O(log n) using O(n= log n) rocessors. It is not clear that our solution is cost-otimal in the strong sense for merging two intransitive sorted seuences. Therefore, the cost-otimal solution in the strong sense still remains oen. Several arallel algorithms have been roosed [1, 4, 6] to determine a Hamiltonian ath in a tournament. Wu [7] rooses a ielined solution under the EREW PRAM model using a new data structure called semi-hea. Basically, in semi-hea, the notion of max is relaced by max! dened on the intrasitive order!. Secically, a semi-hea for a given intransitive total order! is a comlete binary tree. For every node u in the tree, u =max!fu L(u) R(u)g, where L(u) andr(u) are left and right child of u, resectively. max! is dened as u =max!fu L(u) R(u)g 6

7 if both L(u) = maxfu L(u) R(u)g and R(u) = maxfu L(u) R(u)g are false, where max is the regular maximum function. Wu and Sheng [8] roose the notion of the sorted seuence of kings. Akingu in a tournament isalayer who beats (!) any other layer v directly or indirectly that is, either u! v or there exists a third layer w such thatu! w and w! v. A sorted seuence of kings in a tournament ofk layers is a seuence of layers, u 1 u 2 :::u n,suchthatu i! u i+1 and u i in a king in sub-tournament fu i u i+1 ::: u n g for i =1 2 ::: n; 1. Clearly, the sorted seuence of kings adds extra constraints on the intransitive sorted seuence. It has been roved that the sorted seuence of kings exists in any tournament andano(n 2 ) solution based in a modied insertion sort is given in [8]. On the other hand, otimal merge of two sorted seuence of kings is still an uncharted territory. Wu [7] also shows the intransitive sorted seuence as an aroximation for ranking layers in a tournament. In general, the tournament ranking roblem [5] is a dicult one without exhibiting \fairness". Suose 1 2 ::: nis a ranking of layers with 1 reresenting the chamion and i reresenting the ith lace winner. Without loss of generality, we assume that layer u i is ranked in the ith lace. For any air of layers u i u j with i<j,ahainess means that u i beats u j while an uset means that u j beats u i. Clearly, a good ranking should have the minimum numberoftotal usets. A median order is dened as a ranking of layers with a minimum number of total usets. However, the roblem of nding a median order in a tournament is NP-comlete. It is shown in [7] that any median order must be an intransitive sorted seuence. 4. Conclusion In this aer, we have rovided a cost-otimal merge of two intransitive sorted seuences, which is a secial total order without the transitive roerty. The roosed solution is based on the EREW PRAM model with a run time of O(log 2 n) using (n= log 2 n) rocessors. We have also discussed other related roblems including sorted seuence of kings and tournament ranking roblem. Finally, we have ointed out that the cost-otimal merge in the strong sense is still an oen roblem. References 1. A. Bar-Noy and J. Naor. Sorting, minimal feedback sets, and Hamilton aths in tournaments. SIAM Journal on Discrete Mathematics. 3, (1), Feb. 1990, K. Batcher. Sorting networks and their alications. Proc. of the AFIPS Sring Joint Comuting Conference. 1968, R. Cole. Parallel merge sort. SIAM Journal on Comuting. 17, 4, 1988, P. Hell and M. Rosenfeld. The comlexity of nding generalized aths in tournaments. Journal of Algorithms. 1983, 4, K. B. Reid and L. W. Beineke. Tournaments. Chater 7 in: L. W. Beineke and R. Wilson, eds., Selected Toics in Grah Theory, Academic Press, New York, D. Soroker. Fast arallel algorithms for nding Hamilton aths and cycles in a 7

8 tournament. Journal of Algorithms. 1988, J. Wu. On sorting an intransitive total ordered set using semi-hea. Proc. of IEEE International Parallel and Distributed Processing Symosium. May 2000, J. Wu and L. Sheng. An ecient sorting algorithm for a seuence of kings in a tournament. Information Processing Letters. 79, 6, 2001,