G.S. Brodal. Fast Meldable Priority Queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), vol. 955 of Lecture Notes in Computer Science, pp.282-290, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....introduced Fibonacci heaps Driscoll et al. 8] described a meldable heap data structure they call run relaxed heaps. Runrelaxed heaps support all operations within the same time bounds as Fibonacci heaps but in the worst case except meld, which takes O(log n) worst case time. More recently, Brodal [4] described a di erent data structure that supports meld in O(1) worst case time but the time bound for decrease key is O(log n) in the worst case. The time bound for all other operations are as of Fibonacci heaps but worst case. Ultimately, Brodal [5] gave a data structure matching the time bounds ....

Gerth Stolting Brodal. Fast meldable priority queues. In Workshop on Algorithms and Data Structures, pages 282-290, 1995.

....1 as desired 2. Fix rightmost 2 or 1 by changing x2 to (x 1)0 or x( 1) to (x 1)1. This counter is similar to the spine list and the way it is maintained in Section 3, where 1,2,3,4 nodes correspond to the digits 1,0,1,2 respectively. Recently, in independent and distantly related work Brodal [1, 3] and Brodal and Okasaki [2] have designed heaps that can be melded in constant time. Interestingly, an essential ingredient in all the structures they describe is a redundant counter similar to the ones we use here. It is an intriguing open problem whether one can design a sorted list ....

G. S. Brodal. Fast meldable priority queues. In Proceedings of the 4th International Workshop on Algorithms and data structures (WADS'95), pages 282--290. Springer, 1995. LNCS 955.

....Thus, we bridge this gap by providing the first examples of priority queues which support melding in sub logarithmic worst case time, without allowing linear time for deletion. Added in print: It should be noted that subsequent to the submission of this paper, a priority queue has been given [1] which achieves the optimal worst case complexities of O(1) for Insert and Meld, and O(logn) for Delete min. 2 Binomial Queues We briefly review binomial queues. The family of binomial trees B k , where k = 0; 1; 2; is defined inductively as follows: B 0 consists of just one node, and any ....

Gerth S. Brodal. Fast meldable priority queues. In Algorithms and Data Structures, volume 955 of LNCS, pages 282--290. Springer-Verlag, 1995.

....1 as desired 2. Fix rightmost 2 or 1 by changing x2 to (x 1)0 or x( 1) to (x 1)1. This counter is similar to the spine list and the way it is maintained in Section 3, where 1,2,3,4 nodes correspond to the digits 1,0,1,2 respectively. Recently, in independent and distantly related work Brodal [1, 2] and Brodal and Okasaki [3] have designed heaps which can be melded in constant time. Interestingly, an essential ingredient in all the structures they describe is a redundant counter similar to the ones we use here. It is an intriguing open problem whether one can design a sorted list ....

G. S. Brodal. Fast Meldable Priority Queues. WADS'95 proceedings, LNCS 955, pp.282-290.

....only amortized. Some earlier proposals [4, 5, 8] achieve such bounds in the worst case sense for various subsets of the operations supported by Fibonacci heaps. None of these subsets includes the meld operation. This has been remedied recently by Brodal, who has given worst case solutions, first [2] for the set Insert, Delete min, Find min, Meld, and Delete, and later [3] for the full set of Fibonacci heap operations. The latter result has been achieved independently by Boyapati and Rangan [1] By the lower bound on sorting, no strictly better set of complexities can be found in the ....

....has been achieved independently by Boyapati and Rangan [1] By the lower bound on sorting, no strictly better set of complexities can be found in the comparison model. The recent solutions for worst case efficient meldable priority queues are quite involved, in particular [1, 3] but even in [2], each element needs four pointers and two information fields, and the code for the operations contains many cases. Our purpose here is to present an alternative way of implementing a priority queue achieving the bounds in [2] Our structure uses only E mail address: rolf imada.ou.dk. Supported ....

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Gerth S. Brodal. Fast meldable priority queues. In Proc. WADS 95, volume 955 of LNCS, pages 282--290. Springer-Verlag, 1995.

....k operations in O(log k) time and MultiExtractMin k in O(log log k) time. Key words: Parallel priority queues, constant time operations, binomial trees, pipelined operations. 1 Introduction The construction of priority queues is a classical topic in data structures. Some references are [1,3,5,12 16,19,29,31 33]. A historical overview of implementations has been given by Mehlhorn and Tsakalidis [22] Recently several papers have also considered how to implement priority queues on parallel machines [6,8 11,18,24 28] In this paper we focus on how to achieve optimal 1 A preliminary version of the paper ....

....partition the elements into b(n 1) 6c blocks of size six. In parallel we now construct 17 a rank one tree from each block. The remaining 1 6 elements are stored in Q:L[0] The same block partitioning and linking is now done for the rank one trees. The remaining rank one trees are stored in Q:L[1]. This process continues until no tree remains. Because the resulting forest has no holes, we have r(Q) blog 6 nc and there are at most blog 6 nc 1 iterations because. The resulting forest satis es B 1 and B 3 . By standard techniques it follows that the above construction can be done in O(log ....

Gerth Stlting Brodal. Fast meldable priority queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), volume 955 of Lecture Notes in Computer Science, pages 282-290. Springer Verlag, Berlin, 1995.

....EU ESPRIT LTR project No. 20244 (ALCOM IT) Email: shiva mpi sb.mpg.de. z Tata Institute of Fundamental Research, Mumbai, India. Email: jaikumar tcs.tifr.res.in. Implementation Insert Delete FindMin Doubly linked list 1 1 n Heap [8] log n log n 1 Search tree [5, 7] log n 1 1 Priority queue [2, 3, 4] 1 log n 1 Figure 1: Worst case asymptotic time bounds for different set implementations. the worst case times of the two update operations Insert, Delete and the query operation FindMin. We prove the following lower bound on this tradeoff: any randomized algorithm with expected amortized update ....

Gerth Stølting Brodal. Fast meldable priority queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), volume 955 of Lecture Notes in Computer Science, pages 282--290. Springer Verlag, Berlin, 1995.

....volume 3(3) 1996. Chapter 4 [20] The Randomized Complexity of Maintaining the Minimum, with Shiva Chaudhuri and Jaikumar Radhakrishnan. In Nordic Journal of Computing, Selected Papers of the 5th Scandinavian Workshop on Algorithm Theory (SWAT 96) volume 3(4) pages 337 351, 1996. Chapter 5 [15]: Fast Meldable Priority Queues. In Proc. 4th Workshop on Algorithms and Data Structures, LNCS volume 955, pages 282 290, 1995. Chapter 6 [18] Worst Case Efficient Priority Queues. In Proc. 7th ACM SIAM Symposium on Discrete Algorithms, pages 52 58, 1996. Chapter 7 [17] Priority Queues on ....

....work was partially supported by the EU ESPRIT LTR project No. 20244 (ALCOM IT) Table 4.1: Worst case asymptotic time bounds for different set implementations. Implementation Insert Delete FindMin Doubly linked list 1 1 n Heap [109] log n log n 1 Search tree [48, 73] log n 1 1 Priority queue [15, 27, 43] 1 log n 1 number of comparisons it makes. The input is a sequence of operations, given to the algorithm in an online manner, that is, the algorithm must process the current operation before it receives the next operation in the sequence. The worst case time for an operation is the maximum, over ....

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Gerth Stølting Brodal. Fast meldable priority queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), volume 955 of Lecture Notes in Computer Science, pages 282-- 290. Springer Verlag, Berlin, 1995.

....e and it is known where e is stored in Q. DeleteMin(Q) deletes and returns the minimum element from priority queue Q. Delete(Q; e) deletes element e from priority queue Q provided it is known where e is stored in Q. The construction of priority queues is a classical topic in data structures [1, 2, 3, 4, 5, 6, 7, 10, 12, 15, 16, 17]. A historical overview of implementations can be found in [11] There are many applications of priority queues. Two of the most prominent examples are sorting problems and network optimization problems [13] This work was partially supported by the ESPRIT II Basic Research Actions Program of ....

....amortized constant time for all operations except for the two delete operations which require amortized time O(log n) The data structure we present achieves matching worst case time bounds for all operations. Previously, this was only achieved for various strict subsets of the listed operations [1, 2, 3, 15]. For example the relaxed heaps of Driscoll et al. 3] and the priority queues in [1] achieve the above time bounds in the worst case sense except that in [3] Meld requires worst case time Theta(log n) and in [1] DecreaseKey requires worst case time Theta(log n) Refer to Table 1. If we ignore ....

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Gerth Stølting Brodal. Fast meldable priority queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), volume 955 of Lecture Notes in Computer Science, pages 282--290. Springer Verlag, Berlin, 1995.

....essentially equally hard. We THE RANDOMIZED COMPLEXITY OF MAINTAINING THE MINIMUM 3 Table I: Worst case asymptotic time bounds for different set implementations. Implementation Insert Delete FindMin Doubly linked list 1 1 n Heap [10] log n log n 1 Search tree [6, 8] log n 1 1 Priority queue [3, 4, 5] 1 log n 1 show that any deterministic algorithm with amortized update time at most t requires n=2 4t 3 Gamma 1 comparisons for some FindAny operation. This lower bound is proved using an explicit adversary argument, similar to the one used by Borodin, Guibas, Lynch and Yao [2] The adversary ....

Gerth Stølting Brodal. Fast meldable priority queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), volume 955 of Lecture Notes in Computer Science, pages 282--290. Springer Verlag, Berlin, 1995. THE RANDOMIZED COMPLEXITY OF MAINTAINING THE MINIMUM 15

....(q) Discard the minimum element of queue q. In addition, priority queues supply a value empty representing the empty queue and a predicate isEmpty. For simplicity, we will ignore empty queues except when presenting actual code. Figure 1 displays a Standard ML signature for these priority queues. Brodal (1995) recently introduced the first imperative data structure to support all these operations in O(1) worst case time except deleteMin, which requires O(logn) worst case time. Several previous implementations, most notably Fibonacci heaps (Fredman Tarjan, 1987) had achieved these bounds, but in an ....

.... tradeoffs between the running times of the various operations are also possible, but no comparison based priority queue can support insert in better than O(log n) worst case time or meld in better than O(n) worst case time unless one of findMin or deleteMin takes at least O(log n) worst case time (Brodal, 1995). The bootstrapping process can be elegantly expressed in Standard ML extended with higher order functors and recursive structures, as shown in Figure 9. The higher order nature of Bootstrap is analogous to the higher order nature of AddRootToFun, while the recursion between RootedQ and Q captures ....

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Brodal, G. S. (1995) Fast meldable priority queues. Workshop on Algorithms and Data Structures. LNCS 955, pp. 282--290. Springer-Verlag.

....most 3 comparisons. Our proofs use various averaging arguments which are used to derive general combinatorial properties of trees. These are presented in Sect. 2.2. Implementation Insert Delete FindMin Doubly linked list 1 1 n Heap [8] log n log n 1 Search tree [5, 7] log n 1 1 Priority queue [2, 3, 4] 1 log n 1 Fig. 1. Worst case asymptotic time bounds for different set implementations. 2 Preliminaries 2.1 Definitions and notation For a rooted tree T , let leaves(T ) be the set of leaves of T . For a vertex, v in T , define deg(v) to be the number of children of v. Define, for l 2 leaves(T ....

Gerth Stølting Brodal. Fast meldable priority queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), volume 955 of Lecture Notes in Computer Science, pages 282--290. Springer Verlag, Berlin, 1995.

....of the priority queues adopt to a processor array of size O(log n) supporting the operations MakeQueue, Insert, Meld, FindMin, ExtractMin, Delete and DecreaseKey in constant time. 1 Introduction The construction of priority queues is a classical topic in data structures. Some references are [1, 2, 6, 7, 8, 9, 19, 20]. A historical overview of implementations can be found in [13] Recently several papers have also considered how to implement priority queues on parallel machines [3, 4, 5, 11, 15, 16, 17, 18] In this paper we focus on how to achieve optimal speedup for the individual priority queue operations ....

....ExtractMin(Q) First the minimumelement min(Q:L[0] is removed. Performing ParUnlink once guarantees that A 2 is satisfied, especially that the new minimum element is contained in Q:L[0] because the new minimum element was either already contained in Q:L[0] or it was the minimum element in Q:L[1]. Finally ParLink performed once reestablishes A 1 . A pseudo code implementation for a CREW PRAM based on the previous discussion is shown in Fig. 2. Notice that the only part of the code requiring concurrent read is to broadcast the values of Q; Q 1 and Q 2 to all the processors. Otherwise the ....

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Gerth Stølting Brodal. Fast meldable priority queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), volume 955 of Lecture Notes in Computer Science, pages 282--290. Springer Verlag, Berlin, 1995.

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G.S. Brodal. Fast Meldable Priority Queues. In Proc. 4th Workshop on Algorithms and Data Structures (WADS), vol. 955 of Lecture Notes in Computer Science, pp.282-290, 1995.

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Gerth Stlting Brodal, Fast Meldable Priority Queues, In Proc. 4th International Workshop on Algorithms and Data Structures, Vol. 955 of Lecture Notes in Computer Science, pp. 282-290. Springer Verlag, Berlin, 1995.

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