On Sorting an Intransitive Total Ordered Set Using Semi-Heap

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1 O Sortig a Itrasitive Total Ordered Set Usig Semi-Heap Jie Wu Departmet of Computer Sciece ad Egieerig Florida Atlatic Uiversity Boca Rato, FL jie@cse.fau.edu Abstract The problem of sortig a itrasitive total ordered set, a geeralizatio of regular sortig, is cosidered. This geeralized sortig is based o the fact that there exists a special liear orderig for ay itrasitive total ordered set. A ew data structure called semi-heap is proposed to costruct a optimal ( log ) sortig algorithm. Fially, we propose a cost-optimal parallel algorithm usig semi-heap. The ru time of this algorithm is () with (log ) processors uder the EREW RAM model.. Itroductio Sortig is oe of the fudametal problems i computer sciece ad may differet solutios for sortig have bee proposed [, ]. Basically, give a sequece of umbers ( ; ; :::; ) as a iput, a sortig algorithm geerates a permutatio (reorderig) ( ; ; :::; ) of the iput sequece such that :::. We cosider a geeralizatio of the sortig problem by replacig with, where is a total order without the trasitive property, i.e., it is itrasitive. That is, if i j ad j k, it is ot ecessary that i k. The total order requires that for ay two elemets i ad j, either i j or j i, but ot both (atisymmetric). The set N of elemets exhibitig itrasitive total order ca be represeted by a directed graph, where i j represets a directed edge from vertex i to vertex j.the uderlyig graph is a complete graph. This graph is also called a touramet [], represetig a touramet of players where every possible pair of players plays oe game to decide the wier (ad the loser) betwee them. Sortig o N correspods to fidig a Hamiltoia path i the touramet. This work was support i part by NSF grat CCR 99. Hell ad Rosefeld [] proved that the boud of fidig a Hamiltoia path is ( log ), the same complexity as the regular sortig. They also cosidered bouds o fidig some geeralized Hamiltoia paths. It is easy to prove that may regular sortig algorithms ca be used to fid a Hamiltoia path i a touramet, such as bubble sort, isertio sort, biary isertio sort, ad merge sort. Amog parallel sortig algorithms, eve-odd merge sort ca still be applied. However, heapsort ad quicksort caot be used. Bar-Noy ad Naor [] studied differet parallel solutios based o differet models ad the umber of processors. Theyshowedthatuderthe CRCW RAM model, the geeralized sortig problem ca be solved i (log ) usig () processors. Other fast parallel algorithms ca be foud i []. I this paper, we propose a ew data structure called semi-heap, which is a extesio of a regular heap structure. We itroduce a optimal ( log ) algorithm to determie a Hamiltoia path i a touramet based o the semi-heap structure. The, we propose a cost-optimal parallel algorithm based o the semi-heap structure that takes () i ru time usig (log ) processors i the EREW RAM model. A implemetatio of the cost-optimal parallel algorithm i the etwork model with a liear array of processors is also show.. Semi-Heap Data Structure I this sectio, we first show the existece of a Hamiltoia path i ay give touramet ad the propose the semi-heap data structure. ropositio: Cosider a set N (jnj = ) with ay two elemets i ad j, either i j or j i, but ot both. The elemets i N ca be arraged i a liear order :::, : The propositio states that a Hamiltoia path exists i ay give touramet, but ot ecessary for a Hamiltoia circle. That is, we ca always arrage players i a liear

2 A[l[i)] i i i+ heapsize (c) (d) Figure. A directed graph with a complete uderlyig graph. A semi-heap structure as a set of overlappig triagles. order from left to right such that each player beats the oe to its right. Figure shows a directed graph with five vertices. Oe liear order is.whe is trasitive, the liear order arragemet is reduced to a regular sortig problem. Ulike the regular sortig problem, more tha oe solutio exists for the geeralized sortig problem. For example, is aother liear order for the example of Figure. Cosider three elemets ; ; i N, deote = maxf ; ; g if ad. Note that i a total order without the trasitive property, the maximum elemet may ot exist amog three elemets. For example, if,,ad, maxf ; ; g does ot exist. Next we itroduce a ew cocept of the maximum elemet based o. Defiitio : = maxf ; ; g if both = maxf ; ; g ad = maxf ; ; g are false. Note that whe i = maxf ; ; g are false for all i =; ;,every i is a maximum elemet. A semi-heap is ay array object that ca be viewed as a complete biary tree, like a regular heap. A complete biary tree of height h is a biary tree that is full dow to level h,, with level h filled i from left to right. However, the regular heap property is chaged. Let L( ) ad R( ) represet left ad right child odes of, respectively. Whe a child, say R( ), does ot exist, the relatio R( ) automatically holds. Defiitio : A semi-heap for a give itrasitive total order is a complete biary tree. For every ode i the tree, = maxf ;L( );R( )g. Whe a array A is used to represet a semi-heap, l(i) ad r(i) are used as idices of the left ad right child odes of i; they ca be computed simply by l(i) =iad r(i) = i +. Figure shows a semi-heap with elemets. A semi-heap ca be viewed as a set of overlappig triagles, with each triagle cosistig of,, Figure. Four possible cofiguratios of a triagle i a semi-heap.. Figure shows four possible cofiguratios of a triagle uder relatio. I this figure, if is true, a directed edge is draw from to. Note that = maxf;;g for all cases. I cases ad coditio = maxf;;g also holds. To simplify the presetatio, we fill i a special symbol, with a smaller value tha ay oe i the semi-heap for etries that are outside the semi-heap. That is, A[j] is true for all i iside the semi-heap ad all j outside the semi-heap. Specifically, is a elemet of the semi-heap if i heapsize. A[j] is a elemet outside the semiheap if j > heapsize.. Geeralized Sortig Usig Semi-Heap Although a semi-heap resembles a heap, the traditioal heapsort algorithm caot be directly applied to a semiheap to geerate a geeralized sorted sequece. Recall that with the trasitive property, root A[] of the heap is always the maximum elemet i the heap, i.e., the player at the root beats all the other players i the touramet. Whe we discard the root, it is replaced by the last elemet A[] i the heap, ad the, the heap is recostructed by pushig A[] dow i the heap, if ecessary, so that the ew root is the maximum elemet amog the remaiig elemets. However, i a semi-heap, we may face a situatio i which A[] beats all A[], A[], ada[], which is a impossible situatio i a regular heap. A[], the ew root, caot be selected (ad is removed from the semi-heap) i the ext roud to be placed after A[], the previously selected elemet, because A[] beats A[]. O the other had, because A[] beats A[], its left child, ad A[], its right child, A[] caot be pushed dow i the semi-heap. Therefore, a differet strategy has to be developed for semi-heap. We follow closely the otatio used i Corme, Leiserso, ad Rivest s book []. The sortig usig semiheap cosists of four modules: SEMI-HEAIFY(A; i), BUILD-SEMI-HEA(A), RELACE(A; i), ad SEMI- HEA-SORT(A). SEMI-HEAIFY(A; i) costructs a

3 A[l(l(i)] semi-heap A[r(l(i))] semi-heap semi-heap Figure. The costructio of a semi-heap usig SEMI-HEAIFY. semi-heap rooted at, provided that biary trees rooted at ad are semi-heaps (see Figure ). The cost of SEMI-HEAIFY is the height of ode, measured by the umber of edges o the logest simple dowward path from the ode to a leaf. That is, the cost of SEMI-HEAIFY is (log ), where = heapsize. BUILD-SEMI-HEA uses the procedure SEMI-HEAIFY i a bottom-up maer to covert a arbitrary array A ito a semi-heap. The cost of BUILD-SEMI-HEA is (), which is the same cost of buildig a regular heap. Geeralized sortig is doe through SEMI-HEA-SORT by repeatly pritig ad removig the root of the biary tree (which is iitially a semi-heap). The root is replaced by either its leftchild or rightchild through RELACE. The selected child is replaced by oe of its child odes. The process cotiues util reachig oe of the leaf odes ad the etry for that leaf ode is replaced by, i.e., that leaf ode is removed from the tree. A ew tree derived is o loger a semi-heap; however, each overlappig triagle i the tree still meets the maximum elemet requiremet i Defiitio. The cost of RELACE is the height of the curret tree, which is bouded by the height of the origial semi-heap, (log ). Therefore, the cost of SEMI-HEA- SORT is ( log ). Without loss of geerality, we assume that. SEMI-HEAIFY(A; i) if = maxf;;g the fid wier such that A[wier], maxf;;g exchage! A[wier] SEMI-HEAIFY(A; wier) BUILD-SEMI-HEA(A) for i,b heapsize c dowto do SEMI-HEAIFY(A; i) RELACE(A; i) if ( = ) ^ ( = ) the, else if ( ) ^ ( ) the, RELACE(A; l[i]) else, RELACE(A; r[i]) SEMI-HEA-SORT(A) BUILD-SEMI-HEA(A) while (A[l()] = ) _ (A[r()] = ) do prit(a[]) RELACE(A, ) prit(a[]) Theorem : BUILD-SEMI-HEA costructs a semi-heap for ay give complete biary tree. roof: The procedure BUILD-SEMI-HEA goes through odes that have at least oe child ode ad rus SEMI- HEAIFY o these odes. The order i which these odes are processed guaratees that the subtrees rooted at child odes of are semi-heap before SEMI-HEAIFY rus at. Whe SEMI-HEAIFY is called at, if is the maximum elemet amog,, ad based o, the biary treerooted at is automatically a semiheap. Otherwise ad without loss of geerality, oe of the child odes, say, is the wier amog three, i.e., beats both ad. I this case, is swapped with, which esures that ode ad its child odes satisfy the semi-heap property. However, ode ow has the origial ad thus the subtree rooted at may violate the semi-heap property. Therefore, SEMI-HEAIFY must be called recursively o that subtree. A ew problem (that does ot appear i the origial heap structure) is how to esure that the resultat root,after applyig SEMI-HEAIFY at, will ot violate the semi-heap property amog,, ad. I a regular heap, is the maximum elemet i the tree rooted at, the heap property amog,, ad automatically holds. I a semi-heap, we eed to prove that the ewly selected root (other tha the origial value ), which is either A[l(l(i))] or A[r(l(i))] i the origial tree, caot beat both (the origial )ad. I fact, we prove that (the origial ) always beats the ewly selected (the origial A[l(l(i))] or A[r(l(i))]). We cosider the followig two cases i the origial tree with a semi-heap rooted at (see Figure ): If beats both A[l(l(i))] ad A[r(l(i))]. The problem is solved, because i the resultat tree, ode becomes ad either A[l(l(i))] or A[r(l(i))] becomes. If beats oly oe child ode, the without loss of geerality, we assume that (which is

4 A:... A:... Figure. A example tree: the iitial cofiguratio, the semi-heap cofiguratio, after applyig BUILD-SEMI-HEA (c) (d) ow ) beats A[l(l(i))], A[l(l(i))] beats A[r(l(i))], ad A[r(l(i))] beats. To select a wier amog the origial (ow ), A[l(l(i))], A[r(l(i))], other tha, A[l(l(i))] is the oly choice (sice A[r(l(i))] has lost to A[l(l(i))]). Cosequetly, A[l(l(i))] becomes the ewly selected root of the left subtree of, based o the assumptio, (the origial ) beats (the origial A[l(l(i))]) i the resultat tree. Cosider a complete biary tree with eight vertices, i.e., heapsize =. The iitial cofiguratio of array A is,,,,,,,ad. The touramet is represeted by a matrix M give below, where M [i; j] = if i beats j (i.e., i j )adm[i; j] =if i is beate by j (i.e., j i ). M [i; i] =, represets a impossible situatio. Note that M [i; j] =ifad olyif M [j; i] =. M = C A Figure shows the iitial cofiguratio of this complete biary tree i array A, where the correspodig tree structure is represeted by a set of overlappig triagles. Three edges amog three vertices i each triagle represet touramet results betwee three pairs of players i the triagle. That is, a edge directed from i to j exists if M [i; j] = i matrix M. Relatioships betwee two vertices from differet triagles are ot show i the figure. Figure shows the resultat semi-heap after applyig BUILD-SEMI-HEA. A[j] is filled with for j. Actually, itis sufficietto defiethe size ofa to be heapsize.... (e)... Figure. A step-by-step applicatio of RELACE(A; i) i the example of Figure. A step-by-step applicatio of RELACE(A; ) to the example of Figure is show i Figure, where the selected (prited) elemets are placed beside the root i a left-toright order. I this example, the fial output sequece is. Oce all elemets are prited, all etries i array A are filled with. The correctess of this result ca be easily verified through the give matrix M. Note that although the RELACE process destroys the semi-heap structure (sice the resultat tree is o loger a complete biary tree), each overlappig triagle i the correspodig biary tree still maitais oe of the four possible cofiguratios of a semi-heap as show i Figure. Therefore, it always geerates a geeralized sorted sequece for ay give semi-heap. Theorem : For ay give semi-heap, SEMI-HEA-SORT geerates a geeralized sorted sequece. roof: It suffices to show that RELACE always replaces the curret root by a elemet beate by the root. I additio, each overlappig triagle i the biary tree is still oe of the four possible cofiguratios of a triagle i a semi-heap, i.e., the root of each triagle is the maximum elemet based o i the triagle. Based o the defiitio of RELACE, the curret root is replaced by for cases ad (c) ad by for cases ad (d) of (f)

5 Figure. The replacig elemet, say, is itself replaced by a elemet i the triagle rooted at. This process cotiues iteratively dow the semi-heap. I additio, the ew root beats both of its child odes (if ay). This property esures whe a child ode is missig (i.e., the correspodig triagle cotais oly two odes), ca still be replaced by aother child ode without causig ay problem. Therefore, the root of each triagle is still the maximum elemet based o i the triagle.. arallel Geeralized Sortig Usig Semi- Heap We itroduce i this sectio a cost-optimal parallel sortig algorithm usig semi-heap i the EREW RAM model. A sortig algorithm is cost-optimal if the product of ru time ad the umber of processors is ( log ), the boud for sequetial solutios. Specifically, the pipelie techique is used to reduce the ru time of the sequetial algorithm from ( log ) to () usig (log ) processors with differet processors hadlig activities of differet levels of the heap. Because procedure BUILD-SEMI-HEA(A) takes oly (), o speed up is ecessary for this part. rocedure SEMI-HEA-SORT ca be improved by assigig oe processor to each level of the biary tree, which iitially is a semi-heap. RELACE(A; ) is pipelied level to level ad this procedure is called at every other step, because each ode is shared by two processors at adjacet levels, a passive step is iserted betwee two calls. The ru time of SEMI-HEA-SORT is reduced to () usig (log ) processors. This parallel algorithm rus o the CREW RAM model, sice two adjacet processors may access (read) vertices i two overlappig triagles of the tree. However, simultaeous accesses ca be avoided by creatig a copy of each vertex that appears i two overlappig triagles. The ehaced versio rus o the EREW RAM model. We use the etwork model to illustrate the parallel algorithm. The etwork model [] ca be viewed as a graph where each ode represets a processor, ad each directed edge ( i ; j ) represets a two-way commuicatio lik betwee processors i ad j. It is easy to covert the algorithm back to the EREW RAM model by replacig sed ad receive commads i the etwork model by read ad write commads i the EREW RAM model. Shared elemets are duplicated ad stored i local memory of adjacet processors. rocessors are coected as a liear array, where each processor commuicates with up to two adjacet processors. The level of each ode i the semi-heap is its distace to the root. Clearly, h = dlog( +)e is the maximum level ad is called the depth of the semi-heap. A liear array of h processors are used which are labeled as ; ; :::; h,. rocessor i has a copy of elemets i levels i ad i +of the semi-heap. I geeral, i is assiged with i triagles (i.e., i cosecutive elemets i array A). I the proposed parallel algorithm, each processor alterates betwee a active step ad a passive step. rocessors with eve ID s take active steps i eve steps, while with odd ID s take active steps i odd steps. That is, at a eve step, processors,,,... take the active step ad processors,,... take the passive step. The role of active ad passive amog these processors exchages i the ext step, which is a odd step. Active ad passive steps iclude the followig activities: At a active step, each processor performs local update ad seds relevat messages to two adjacet processors (if they exist). At a passive step, each processor receives messages from two adjacet processors (if they exist) ad saves them. I the implemetatio usig the etwork model, processor iitiates the sortig process ad the rest i s are activated i sequece. rocessor also geerates a termiatio sigal which is passed dow the liear array of processors oce the job is completed. To make our algorithm more geeral, some activities are ot ordered withi a step. at a active step (starts from step ):. rits root A[].. If both child odes are, A[] is replaced by, ad the, seds a termiatio sigal to ad stops. If at least oe child ode is ot, replaces A[] by oe of two child odes, A[] or A[], followig the rule i RELACE. If A[] is selected, seds id =to processor ; otherwise, id =is set. I the ext step (a passive step), receives (id; replacemet) from, ad the, performs the update A[id] :=replacemet. i, i>, i a passive step: If i receives (id; replacemet) from i+, it performs the update A[id] :=replacemet. If i receives sigal id = i from i,, it performs the followig activities i ext active step:. If both child odes are, is replaced by ; otherwise, is replaced by either or A[i +]based the replacemet rule.. Sed (i; ) to i,.. If either or A[i +]is selected to replace, the correspodig id (i or i +) is set to i+, provided i is ot the last processor (i.e., i = h, ); otherwise, the selected elemet is replaced by.

6 If i receives the termiatio sigal from i,, itforwards the termiatio sigal to the ext processor i+ (if it exists) i the ext active step, ad the, i stops. (, ) Note that i the above algorithm, although each processor is assiged a differet umber of triagles, its workload stays the same: each processor operates o at most two triagles i a passive step ad at most oe triagle i a active step. Whe a child ode exceeds the boudary of the semi-heap, it has a default value of ad o replacemet is eeded. The step-by-step illustratio of the above algorithm is show i Figure for the first three steps of Figure, where the semi-heap is represeted as a tree structure without showig the detail orietatio of each triagle. I this example, each step of Figure correspods to two steps i Figure. Replacemet activities are show usig dashed lies. Theorem : The proposed parallel implemetatio is costoptimal with a ru time of () usig (log ) processors. roof: It is clear that (log ) processors are used. Also, oe elemet is selected (prited) i every other step ad all elemets are prited i steps, ad hece, the ru time is (). Becausetheproductofrutime adtheumberof processors used matches the lower boud ( log ) for a sequetial algorithm, the proposed parallel implemetatio is cost-optimal. The proposed implemetatio ca be exteded without havig to idetify the last processor. This extesio ca be doe by addig oe extra processor h which hadles the last level of the semi-heap (this last level is also duplicated). Clearly, each child ode of ay elemet i the last level is a. Therefore, o other processor will be activated by h. Also, each processor ca termiate itself without usig a termiatio sigal origiated from. i termiates itself oce all i elemets (that it cotrols) become ; however, the bookkeepig process is more complicated tha the oe i the origial desig.. Coclusios We have proposed a data structure called semi-heap which is a geeralizatio of the traditioal heap structure. The semi-heap structure is used to solve a geeralized sortig problem. We have show that the geeralized sortig problem ca be solved optimally usig semi-heap. The solutio ca be easily exteded to a cost-optimal EREW RAM algorithm with () i ru time usig (log ) processors. A implemetatio of this parallel algorithm uder the etwork model is show, where processors are coected as a liear array. We are curretly studyig the problem of geeralized mergig, where the relatio betwee elemets does ot have the trasitive property. The result of this study will be reported i a separate paper [9]. (, ) (, ) (c) (e) (, ) (, ) (d) Figure. A step-by-step illustratio of the first three steps of Figure. Refereces [] A. Bar-Noy ad J. Naor. Sortig, miimal feedback sets, ad Hamilto paths i touramets. SIAM Joural of Discrete Mathematics., (), Feb. 99, -. [] J. A. Body ad U.S.R. Murthy. Graph Theory ad Applicatios. The Macmilla ress. 9. [] T. H. Corme, C. E. Leiserso, ad R. L. Rivest. Itroductio to Algorithms. The MIT ress. 99. []. Hell ad M. Rosefeld. The complexity of fidig geeralized paths i touramets. Joural of Algorithms. 9,, -9. [] J. JaJa. A Itroductio to arallel Algorithms. Addiso-Wesley ublishig Compay. 99. [] D. Kuth. The Art of Computer rogrammig, Vol, Sortig ad Searchig. Addiso-Wesley ublishig Compay, secod editio. 99. [] D. Soroker. Fast parallel algorithms for fidig Hamilto paths ad cycles i a touramet. Joural of Algorithms. 9, -. [] J. Wu. Distributed Systems Desig. The CRC ress [9] J. Wu ad S. Olariu. O optimal merge of two itrasitive sequeces. i preparatio. (f)