BRICS. Fast Meldable Priority Queues. Basic Research in Computer Science. BRICS RS G. S. Brodal: Fast Meldable Priority Queues

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1 BRIS RS G. S. Brodal: Fast Meldable Priority Queues BRIS Basic Researc in omputer Science Fast Meldable Priority Queues Gert Stølting Brodal BRIS Report Series RS ISSN February 1995

2 opyrigt c 1995, BRIS, Department of omputer Science University of Aarus. All rigts reserved. Reproduction of all or part of tis work is permitted for educational or researc use on condition tat tis copyrigt notice is included in any copy. See back inner page for a list of recent publications in te BRIS Report Series. opies may be obtained by contacting: BRIS Department of omputer Science University of Aarus Ny Munkegade, building 540 DK Aarus Denmark Telepone: Telefax: Internet: BRIS@daimi.aau.dk BRIS publications are in general accessible troug WWW and anonymous FTP: ttp:// ftp ftp.brics.aau.dk (cd pub/bris)

3 Fast Meldable Priority Queues Gert Stlting Brodal BRIS y Department of omputer Science, University of Aarus Ny Munkegade, DK-8000 Arus, Denmark 15t February 1995 Abstract We present priority queues tat support te operations MakeQueue, FindMin, Insert and Meld in worst case time O(1) and Delete and DeleteMin in worst case time O(log n). Tey can be implemented on te pointer macine and require linear space. Te time bounds are optimal for all implementations were Meld takes worst case time o(n). To our knowledge tis is te rst priority queue implementation tat supports Meld in worst case constant time and DeleteMin in logaritmic time. Tis work was partially supported by te ESPRIT II Basic Researc Actions Program of te E under contract no (project ALOM II) and by te Danis Natural Science Researc ouncil (Grant No ). y Basic Researc in omputer Science, entre of te Danis National Researc Foundation. 1

4 Introduction We consider te problem of implementing meldable priority queues. Te operations tat sould be supported are: MakeQueue reates a new empty priority queue. FindMin(Q) Returns te minimum element contained in priority queue Q. Insert(Q; e) Inserts element e into priority queue Q. Meld(Q 1 ; Q 2 ) Melds te priority queues Q 1 and Q 2 to one priority queue and returns te new priority queue. DeleteMin(Q) Deletes te minimum element of Q and returns te element. Delete(Q; e) Deletes element e from priority queue Q provided tat it is known were e is stored in Q (priority queues do not support te searcing for an element). Te implementation of priority queues is a classical problem in data structures. A few references are [13, 12, 8, 7, 5, 6, 10]. In te amortised sense, [11], te best performance is acieved by binomial eaps [12]. Tey support Delete and DeleteMin in amortised time O(log n) and all oter operations in amortised constant time. If we want to perform Insert in worst case constant time two ecient data structures exist. Te implicit priority queues of arlsson and Munro [2] and te relaxed eaps of Driscoll et al. [5], but neiter of tese support Meld eciently. However tey do support MakeQueue, FindMin and Insert in worst case constant time and Delete and DeleteMin in worst case time O(log n). Our implementation beats te above by supporting MakeQueue, Find- Min, Insert and Meld in worst case time O(1) and Delete and Delete- Min in worst case time O(log n). Te computational model is te pointer macine and te space requirement is linear in te number of elements contained in te priority queues. We assume tat te priority queues contain elements from a totally ordered universe. Te only allowed operation on te elements is te comparisons of two elements. We assume tat comparisons can be performed 2

5 in worst case constant time. For simplicity we assume tat all priority queues are nonempty. For a given operation we let n denote te size of te priority queue of maximum size involved in te operation. In Sect. 1 we describe te data structure and in Sect. 2 we sow ow to implement te operations. In Sect. 3 we sow tat our construction is optimal. Section 4 contains some nal remarks. 1 Te Data Structure Our basic representation of a priority queue is a eap ordered tree were eac node contains one element. Tis is sligtly dierent from binomial eaps [12] and Fibonacci eaps [8] were te representation is a forest of eap ordered trees. Wit eac node we associate a rank and we partition te sons of a node into two types, type i and type ii. Te eap ordered tree must satisfy te following structural constraints. a) A node as at most one son of type i. Tis son may be of arbitrary rank. b) Te sons of type ii of a node of rank r ave all rank less tan r. c) For a xed node or rank r, let n i denote te number of sons of type ii tat ave rank i. We maintain te regularity constraint tat i) 8i : (0 i < r ) 1 n i 3); ii) 8i; j : (i < j ^ n i = n j = 3 ) 9k : i < k < j ^ n k = 1); iii) 8i : (n i = 3 ) 9k : k < i ^ n k = 1): d) Te root as rank zero. Te eap order implies tat te minimum element is at te root. Properties a), b) and c) bound te degree of a node by tree times te rank of te node plus one. Te size of te subtree rooted at a node is controlled by property c). Lemma 1 sows tat te size is at least exponential in te rank. Te last two properties are essential to acieve Meld in worst case constant time. Te regularity constraint c) is a variation of te regularity constraint tat Guibas et al. [9] used in teir construction of nger searc trees. Te idea is tat between two ranks were tree sons ave 3

6 equal rank tere is a rank of wic tere only is one son. Figure 1 sows a eap ordered tree tat satises te requirements a) to d) (te elements contained in te tree are omitted). 0 3 (( (((( H XX H X H 2 H Figure 1: A eap ordered tree satisfying te properties a) to d). A box denotes a son of type i, a circle denotes a son of type ii, and te numbers are te ranks of te nodes. 0 Lemma 1 Any subtree rooted at a node of rank r as size 2 r. Proof: Te proof is a simple induction in te structure of te tree. By c.i) leaves ave rank zero and te lemma is true. For a node of rank r property c.i) implies tat te node as at least one son of eac rank less tan r. By induction we get tat te size is at least 1 + P r?1 i=0 2 i = 2 r. 2 orollary 1 Te only son of te root of a tree containing n elements as rank at most blog(n? 1)c. We now describe te details of ow to represent a eap ordered tree. A son of type i is always te rigtmost son. Te sons of type ii appear in increasing rank order from rigt to left. See Fig. 1 and Fig. 2 for examples. A node consists of te following seven elds: 1) te element associated wit te node, 2) te rank of te node, 3) te type of te node, 4) a pointer to te fater node, 5) a pointer to te leftmost son and 6) a pointer to te next sibling to te left. Te next sibling pointer of te leftmost son points to te rigtmost son in te list. Tis enables te access to te rigtmost son of a node in constant time too. Field 7) is used to maintain a single linked list of triples of sons of type ii tat ave 4

7 6 leftmost son fater next next triple Figure 2: Te arrangement of te sons of a node. equal rank (see Fig. 2). Te nodes appear in increasing rank order. We only maintain tese pointers for te rigtmost son and for te rigtmost son in a triple of sons of equal rank. Figure 2 sows an example of ow te sons of a node are arranged. In te next section we describe ow to implement te operations. Tere are two essential transformations. Te rst transformation is to add a son of rank r to a node of rank r. Because we ave a pointer to te leftmost son of a node (tat as rank r? 1 wen r > 0) tis can be done in constant time. Notice tat tis transformation cannot create tree sons of equal rank. Te second transformation is to nd te smallest rank i were tree sons ave equal rank. Two of te sons are replaced by a son of rank i + 1. Because we maintain a single linked list of triples of nodes of equal rank we can also do tis in constant time. 2 Operations In tis section we describe ow to implement te dierent operations. Te basic operation we use is to link two nodes of equal rank r. Tis is done by comparing te elements associated wit te two nodes and making te node wit te largest element a son of te oter node. By increasing te rank of te node wit te smallest element to r + 1 te properties a) to d) are satised. Te operation is illustrated in Fig. 3. Tis is similar to te linking of trees in binomial eaps and Fibonacci eaps [12, 8]. We now describe ow to implement te operations. MakeQueue is trivial. We just return te null pointer. FindMin(Q) returns te element located at te root of te tree 5

8 r r + 1 r r " " > Figure 3: Te linking of two nodes of equal rank. representing Q. Insert(Q; e) is equal to Meld Q wit a priority queue only consisting of a rank zero node containing e. Meld(Q 1 ; Q 2 ) can be implemented in two steps. In te rst we insert one of te eap ordered trees into te oter eap ordered tree. Tis can violate property c) at one node because te node gets one additional son of rank zero. In te second step we reestablis property c) at te node. Figure 4 sows an example of te rst step. e 1 e 2 e 1 e 0 1 e 0 2 > b e 2 b e 0 T 1 1 T 2 T 1 e 0 2 T 2 Figure 4: Te rst step of a Meld operation (te case e 1 e 2 < e 0 1 e0 2). Let e 1 and e 2 denote te roots of te trees representing Q 1 and Q 2 and let e 0 1 and e 0 2 denote te only sons of e 1 and e 2. Assume w.l.o.g. tat e 1 is te smallest element. If e 2 e 0 1 we let e 2 become a rank zero son of e 0 1, oterwise e 2 < e 0 1. If e 0 2 < e0 1 we can intercange te subtrees rooted at e 0 2 and e 0 1, so w.l.o.g. we assume e 1 e 2 < e 0 1 e 0 2. In tis case we make e 2 a rank zero son of e 0 1 and swap te elements e 0 1 and e 2 (see Fig. 4). We ave assumed tat te sizes of Q 1 and Q 2 are at least two, but te oter cases are just simplied cases of te general case. 6

9 Te only invariants tat can be violated now are te invariants b) and c) at te son of te root because it as got one additional rank zero son. Let v denote te son of te root. If v ad rank zero we can satisfy te invariants by setting te rank of v to one. Oterwise only c) can be violated at v. Let n i denote te number of sons of v tat ave rank i. By linking two nodes of rank i were i is te smallest rank were n i = 3 it is easy to verify tat c) can be reestablised. Te linking reduces n i by two and increments n i+1 by one. If we let (n r?1 ; : : : ; n 0 ) be a string in f1; 2; 3g te following table sows tat c) is reestablised after te above described transformations. We let x denote a string in f1; 2; 3g and y i strings in f1; 2g. Te table sows all te possible cases. Recall tat c) states tat between every two n i = 3 tere is at least one n i = 1. Te dierent cases are also considered in [9]. y 1 1 > y 1 2 y 2 13y 1 1 > y 2 21y 1 2 y 2 23y 1 1 > y 2 31y 1 2 x3y 2 13y 1 1 > x3y 2 21y 1 2 x3y 3 1y 2 23y 1 1 > x3y 3 1y 2 31y 1 2 y 1 12 > y 1 21 y 1 22 > y 1 31 x3y 1 12 > x3y 1 21 x3y 2 1y 1 22 > x3y 2 1y 1 31 After te linking only b) can be violated at v because a son of rank r as been created. Tis problem can be solved by increasing te rank of v by one. Because of te given representation Meld can be performed in worst case time O(1). DeleteMin(Q) removes te root e 1 of te tree representing Q. Te problem is tat now property d) can be violated because te new root e 2 can ave arbitrary rank. Tis problem is solved by te following transformations. First we remove te root e 2. Tis element later on becomes te new root of rank zero. At most O(log n) trees can be created by 7

10 removing te root. Among tese trees te root tat contains te minimum element e 3 is found and removed. Tis again creates at most O(log n) trees. We now nd te root (e 4 ) of maximum rank among all te trees and replaces it by te element e 3. A rank zero node containing e 4 is created. Te tree of maximum rank and wit root e 3 is made te only son of e 2. All oter trees are made sons of te node containing e 3. Notice tat all te new sons of e 3 ave rank less tan te rank of e 3. By iterated linking of sons of equal rank were tere are tree sons wit equal rank, we can guarantee tat n i 2 f1; 2g for all i less tan te rank of e 3. Possibly, we ave to increase te rank of e 3. Finally, we return te element e 1. e 1 e 2, lt, e, l 4. T. l B e... B. 3 T B T. T..... e 2 e 2 e 2 e 3... PP e 3 e B B A > > e B > A A e 4 B B B Figure 5: Te implementation of DeleteMin. Because te number of trees is at most O(log n) DeleteMin can be performed in worst case time O(log n). Figure 5 illustrates ow DeleteMin is performed. Delete(Q; e) can be implemented similar to DeleteMin. If e is te root we just perform DeleteMin. Oterwise we start by bubbling e upwards in te tree. We replace e wit its fater until te fater of e as rank less tan or equal to te rank of e. Now, e is te arbitrarily ranked son of its fater. Tis allows us to replace e wit an arbitrary ranked node, provided tat te eap order is still satised. Because te rank of e increases for eac bubble step, and te rank of a node is bounded by blog(n? 1)c, tis can be performed in time O(log n). We can now replace e wit te meld of te sons of e as described in te implementation of DeleteMin. Tis again can be performed in worst case time O(log n). 8

11 To summarise, we ave te teorem: Teorem 1 Tere exists an implementation of priority queues tat supports MakeQueue, FindMin, Insert and Meld in worst case time O(1) and DeleteMin and Delete in worst case time O(log n). Te implementation requires linear space and can be implemented on te pointer macine. 3 Optimality Te following teorem sows tat if Meld is required to be nontrivial, i.e. to take worst case sublinear time, ten DeleteMin must take worst case logaritmic time. Tis sows tat te construction described in te previous sections is optimal among all implementations were Meld takes sublinear time. If Meld is allowed to take linear time it is possible to support Delete- Min in worst case constant time by using te nger searc trees of Dietz and Raman [3]. By using teir data structure MakeQueue, FindMin, DeleteMin, Delete can be supported in worst case time O(1), Insert in worst case time O(log n) and Meld in worst case time O(n). Teorem 2 If Meld can be performed in worst case time o(n) ten DeleteMin cannot be performed in worst case time o(log n). Proof: Te proof is by contradiction. Assume Meld takes worst case time o(n) and DeleteMin takes worst cast time o(log n). We sow tat tis implies a contradiction wit te (n log n) lower bound on comparison based sorting. Assume we ave n elements tat we want to sort. Assume w.l.o.g. tat n is a power of 2, n = 2 k. We can sort te elements by te following list of priority queue operations. First, create n priority queues eac containing one of te n elements (eac creation takes worst case time O(1)). Ten join te n priority queues to one priority queue by n?1 Meld operations. Te Meld operations are done bottom-up by always melding two priority queues of smallest size. Finally, perform n DeleteMin operations. Te elements are now sorted. 9

12 Te total time for tis sequence of operations is: X nt MakeQueue + k?1 i=0 X 2 k?1?i T Meld (2 i ) + n T DeleteMin (i) = o(n log n): Tis contradicts te lower bound on comparison based sorting. 2 i=1 4 onclusion We ave presented an implementation of meldable priority queues were Meld takes worst case time O(1) and DeleteMin worst case time O(log n). Anoter interesting operation to consider is DecreaseKey. Our data structure supports DecreaseKey in worst case time O(log n), because DecreaseKey can be implemented in terms of a Delete operation followed by an Insert operation. Relaxed eaps [5] support DecreaseKey in worst case time O(1) but do not support Meld. But it is easy to see tat relaxed eaps can be extended to support Meld in worst case time O(log n). Te problem to consider is if it is possible to support bot DecreaseKey and Meld simultaneously in worst case constant time. As a simple consequence of our construction we get a new implementation of meldable double ended priority queues, wic is a data type tat allows bot FindMin/FindMax and DeleteMin/DeleteMax [1, 4]. For eac queue we just ave to maintain two eap ordered trees as described in section 1. One tree ordered wit respect to minimum and te oter wit respect to maximum. If we let bot trees contain all elements and te elements know teir positions in bot trees we get te following corollary. orollary 2 An implementation of meldable double ended priority queues exists tat supports MakeQueue, FindMin, FindMax, Insert and Meld in worst case time O(1) and DeleteMin, DeleteMax, Delete, DecreaseKey and IncreaseKey in worst case time O(log n). 10

13 References [1] M. D. Atkinson, J.-R. Sack, N. Santoro, and T. Strototte. Minmax eaps and generalized priority queues. ommunications of te AM, 29(10):996{1000, [2] Svante arlsson and J. Ian Munro. An implicit binomial queue wit constant insertion time. In Proc. 1st Scandinavian Worksop on Algoritm Teory (SWAT), volume 318 of Lecture Notes in omputer Science, pages 1{13. Springer Verlag, Berlin, [3] Paul F. Dietz and Rajeev Raman. A constant update time nger searc tree. In Advances in omputing and Information - II '90, volume 468 of Lecture Notes in omputer Science, pages 100{109. Springer Verlag, Berlin, [4] Yuzeng Ding and Mark Allen Weiss. Te relaxed min-max eap. ATA Informatica, 30:215{231, [5] James R. Driscoll, Harold N. Gabow, Rut Srairman, and Robert E. Tarjan. Relaxed eaps: An alternative to bonacci eaps wit applications to parallel computation. ommunications of te AM, 31(11):1343{1354, [6] Micael J. Fiscer and Micael S. Paterson. Fisspear: A priority queue algoritm. Journal of te AM, 41(1):3{30, [7] Micael L. Fredman, Robert Sedgewick, Daniel D. Sleator, and Robert E. Tarjan. Te pairing eap: A new form of self{adjusting eap. Algoritmica, 1:111{129, [8] Micael L. Fredman and Robert Endre Tarjan. Fibonacci eaps and teir uses in improved network optimization algoritms. In Proc. 25rd Ann. Symp. on Foundations of omputer Science (FOS), pages 338{346, [9] Leo J. Guibas, Edward M. Mcreigt, Micael F. Plass, and Janet R. Roberts. A new representation for linear lists. In Proc. 9tAnn. AM Symp. on Teory of omputing (STO), pages 49{60,

14 [10] Peter Hyer. A general tecnique for implementation of ecient priority queues. Tecnical Report IMADA-94-33, Odense University, [11] Robert Endre Tarjan. Amortized computational complexity. SIAM Journal on Algebraic and Discrete Metods, 6:306{318, [12] Jean Vuillemin. A data structure for manipulating priority queues. ommunications of te AM, 21(4):309{315, [13] J. W. J. Williams. Algoritm 232: Heapsort. ommunications of te AM, 7(6):347{348,

15 Recent Publications in te BRIS Report Series RS Gert Stølting Brodal. Fast Meldable Priority Queues. February pp. RS Alberto Apostolico and Dany Breslauer. An Optimal O(log log n) Time Parallel Algoritm for Detecting all Squares in a String. February pp. To appear in SIAM Journal on omputing. RS Dany Breslauer and Devdatt P. Dubasi. Transforming omparison Model Lower Bounds to te Parallel-Random- Access-Macine. February pp. RS-95-9 RS-95-8 RS-95-7 RS-95-6 RS-95-5 RS-95-4 RS-95-3 RS-95-2 RS-95-1 Lars R. Knudsen. Partial and Higer Order Differentials and Applications to te DES. February pp. Ole I. Hougaard, Micael I. Scwartzbac, and Hosein Askari. Type Inference of Turbo Pascal. February pp. David A. Basin and Nils Klarlund. Hardware Verification using Monadic Second-Order Logic. January pp. Igor Walukiewicz. A omplete Deductive System for te -alculus. January pp. Luca Aceto and Anna Ingólfsdóttir. A omplete Equational Axiomatization for Prefix Iteration wit Silent Steps. January pp. Mogens Nielsen and Glynn Winskel. Petri Nets and Bisimulations. January pp. To appear in TS. Anna Ingólfsdóttir. A Semantic Teory for Value Passing Processes, Late Approac, Part I: A Denotational Model and Its omplete Axiomatization. January pp. François Laroussinie, Kim G. Larsen, and arsten Weise. From Timed Automata to Logic - and Back. January pp. Gudmund Skovbjerg Frandsen, Tore Husfeldt, Peter Bro Miltersen, Teis Raue, and Søren Skyum. Dynamic Algoritms for te Dyck Languages. January pp.