J. R. Driscoll, H. N. Gabow, R. Shrairman, and R. E. Tarjan. Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation. Communications of the ACM, 31(11):1343--1354, November 1988.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....h) Delete item i from heap h. In the amortized setting Fibonacci heaps [10] support delete in O(log n) time and all other operations in O(1) time. Achieving these time bounds in the worst case turned out to be harder. Shortly after Fredman and Tarjan 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 ....

....and delete in O(log n) time. Although our use of fat heaps does not require them to support meld one can make them support meld in O(log n) time. Thereby they match the time bounds of the run relaxed heaps of Driscoll et al. and provide an alternative to run relaxed heaps in all their applications [8]. Fat heaps are described in Section 4. Using fat heaps we then build a data structure that almost matches the worst case lower bounds. The only gap is in the worst case time bound for meld, which is O(log k n k) worst case time rather than the desired O(k) At a high level this data structure is ....

James R. Driscoll, Harold N. Gabow, Ruth Shrairman, and Robert E. Tarjan. Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation. Communications of the ACM, 31(11):1343-1354, 1988.

....and returns the element. Delete(Q; e) Deletes element e from priority queue Q provided that it is known where e is stored in Q (priority queues do not support the searching for an element) The implementation of priority queues is a classical problem in data structures. A few references are [13, 12, 8, 7, 5, 6, 10]. In the amortised sense, 11] the best performance is achieved by binomial heaps [12] They support Delete and DeleteMin in amortised time O(log n) and all other operations in amortised constant time. If we want to perform Insert in worst case constant time two efficient data structures exist. ....

....support Delete and DeleteMin in amortised time O(log n) and all other operations in amortised constant time. If we want to perform Insert in worst case constant time two efficient data structures exist. The implicit priority queues of Carlsson and Munro [2] and the relaxed heaps of Driscoll et al. [5], but neither of these support Meld efficiently. However they 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 the above by supporting MakeQueue, FindMin, Insert and Meld in worst case time O(1) and ....

[Article contains additional citation context not shown here]

James R. Driscoll, Harold N. Gabow, Ruth Shrairman, and Robert E. Tarjan. Relaxed heaps: An alternative to fibonacci heaps with applications to parallel computation. Communications of the ACM, 31(11):1343--1354, 1988.

....Sect. 7 summarizes the results and sketches some open problems and future improvements. Previous Work PRAM algorithms: There is no parallel O(n log n m) work PRAM algorithm with sublinear running time for general digraphs with non negative edge weights. The best O(n log n m) work solution [9] has running time O(n log n) All known algorithms with polylogarithmic execution time are work inefficient. O(log time and O(n (log log n= log n) work for the algorithm in [11] An O(n) time algorithm requiring O( n m) log n) work was presented in [3] For special classes of ....

....ffi o (v) where ffi o (v) min fc(v; w) v; w) 2 Eg; ffi o (v) is precomputed once and for all upon initialization. The Q i queues are used to efficiently derive the criterion of the OUT version indicating whether a node can be deleted in a phase. The queues are implemented as relaxed heaps [9] because they provide worst case running times: findMin, insert and decreaseKey are performed in O(1) time and delete deleteMin in O(log q) time where q denotes the local queue size. Let r be the number of delete phases which are needed, e.g. for the OUTvariant r = O( n) whp. For the analysis ....

J. R. Driscoll, H. N. Gabow, R. Shrairman, and R. E. Tarjan. Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation. Communications of the ACM, 31(11):1343--1354, 1988.

....for arbitrary directed graphs with random edge weights have been given [16, 24] So far there is no parallel O(n log n m) work PRAM SSSP algorithm with worst case sublinear running time for arbitrary digraphs with nonnegative edge weights. The O(n log n m) work solution by Driscoll et al. [12] has 1 running time O(n log n) An O(n) time algorithm requiring O(m log n) work was presented by Brodal et al. 7] All faster known algorithms require more work, e.g. the approach by Han et al. 18] needs O(log 2 n) time and O(n 3 (log log n= log n) 1=3 ) work. The algorithm of ....

....= dlog 2 ne relaxed heaps D 1 ; D h such that D i is in charge of tentative distances for nodes having in degree in f2 i 1 ; 2 i 1g. A relaxed heap allows insertions and decrease key operations in worst case constant time, deletions of the minimum in worst case logarithmic time [12]. Let M i be the smallest tentative distance in D i (M i = 1 for empty D i ) and let M = min i M i . Then the algorithm computes max : min i fmaxf2 i 1 ; M i Mgg. Subsequently, for each D i , the SSSP algorithm scans all nodes v 2 D i satisfying tent(v) M max . Hence, nodes with ....

J. R. Driscoll, H. N. Gabow, R. Shrairman, and R. E. Tarjan. Relaxed heaps: An alternative to fibonacci heaps with applications to parallel computation. Communications of the ACM, 31, 1988.

....whether or not arc costs are allowed to be negative. In fact, if all arc costs are nonnegative, then Dijkstra s algorithm 2 [20] solves the single source shortest paths problem in nearlinear time O(m n log n) if implemented with e#cient data structures such as Fibonacci heaps [28] see also [21, 9, 81]. We use log to denote logarithms to base e and log 2 to denote logarithms to base 2. In the general case of possibly negative arc costs, Dijkstra s algorithm has exponential worst case running time [47, 76] but the Bellman Ford algorithm [7, 26] solves the single source shortest paths problem ....

....key (a delete min operation) in time O(log Q ) where Q is the number of pairs in the priority queue, and an operation decrease key in constant time. A decrease key operation takes a pointer to a pair (x, k) in the priority queue and allows the replacement of k by a smaller key k # ; see [21, 9]. We propose the following implementation of Phase I. We batch the current arcs with respect to their endpoints, that is, the priority queue contains all unlabeled vertices. For each vertex u # U , we maintain a list L(u) of all vertices v # S whose current arc enters u; the key of a vertex ....

J. R. Driscoll, H. N. Gabow, R. Shrairman, and R. E. Tarjan, Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation, Comm. ACM 31 (1988), pp. 1343--1354

....cost of insertion then rises to Theta(lg n) and extraction becomes the same (which may be an increase or a decrease) corm90] Mergeable heaps will have additional, useful properties if priorities are to be supported, as will become evident in x6.3.1. Fibonacci heaps [fred87] and relaxed heaps [dris88] permit insertion in constant amortized time and extraction in O(lg n) amortized (not amortized for relaxed heaps) time; relaxed heaps also have some advantages over Fibonacci heaps in parallel algorithms. However, direct support of prioritization will incur an extra cost regardless of the ....

James R. Driscoll, Harold N. Gabow, Ruth Shrairman, and Robert E. Tarjan. "Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation". Communications of the ACM, Vol. 31, No. 11, pp. 1343--1354, November 1988.

....a rough analysis indicates that this additional eld should allow the algorithm to run about twice as fast. Fibonacci and rank relaxed heaps (we did not implement run relaxed heaps, which su er from much larger overhead than rank relaxed heaps) o er the largest number of choices. Driscoll et al. [2], in their discussion of relaxed heaps, suggested storing in each node an array of child pointers of approximately log 2 jV j in length in order to reduce the large overhead associated with relaxed heaps; then, in order to keep the storage down to O(jV j) they suggested grouping approximately log ....

Driscoll, J.R., H.N. Gabow, R. Shrairman, and R.E. Tarjan, Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation, Comm. ACM 11 (1988), pp. 1343-1354. EMPIRICAL ASSESSMENT OF MST ALGORITHMS 17

....procedure immediately showed the asymptotic behavior and highlighted the relative eciency of each algorithm. Early in the implementation phase, we realized that Fibonacci heaps and relaxed heaps were not competitive. We then took a suggestion made in the original paper of Driscoll et al. [6] for implementing relaxed heaps: to group nodes into larger units so that changes in key would most often be resolved within a unit and not TOWARDS A DISCIPLINE OF EXPERIMENTAL ALGORITHMICS 11 require restructuring the heap. We then decided to implement this idea, which we called sacks, for other ....

Driscoll, J.R., Gabow, H.N., Shrairman, R., and Tarjan, R.E., \Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation," Commun. ACM 11 (1988), 1343-1354.

....optional. In existing priority queue implementations the approach is two fold. Most data structures require storing additional balance information associated with queue nodes in order to guarantee the worst case efficiency of individual operations (e.g. leftist trees [8] relaxed heaps [5], Brodal queues [3, 4] Others achieve good amortized performance by adjusting the structure during some operations rather than struggling to maintain balance constantly (skew heaps [12, 13] pairing heaps [6] Experiments indicate that the latter approach is more promising in practice [1, 7, ....

J. R. Driscoll, H. N. Gabow, R. Shrairman, R. E. Tarjan, Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation. Comm. ACM 31(1988), 1343-1354.

....indicates that this additional field allows the algorithm to run about twice as fast. Fibonacci and rank relaxed heaps (we did not implement run relaxed heaps, which can be seen to suffer from much larger overhead than rank relaxed heaps) offer the largest number of choices. Driscoll et al. [2], in their discussion of relaxed heaps, suggested storing in each node an array of child pointers of approximately log 2 jV j in length in order to reduce the large overhead associated with relaxed heaps; then, in order to keep the storage down to O(jV j) they suggested grouping approximately log ....

Driscoll, J.R., H.N. Gabow, R. Shrairman, and R.E. Tarjan, "Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation," Comm. ACM 11 (1988), pp. 1343--1354.

....shortest path problem has efficient sequential solutions, especially when G has nonnegative edge weights. In this case, the problem can be solved by Dijkstra s algorithm in O(m n log n) time using the Fibonacci heap or another priority queue data structure with the same resource bounds [3, 5]. If in addition G is planar, then the problem can be solved optimally in O(n) time [9] In the following we consider the single source shortest path problem in planar digraphs with nonnegative real edge weights. Despite much effort, no sublinear time, work optimal parallel algorithm has been ....

....graph, whose implementation is also sketched. The other two main subroutines used by our algorithm are: a) Dijkstra s sequential algorithm [1] We shall denote a call of the algorithm on a digraph H with source vertex s as Seq Dijkstra(s; H) b) A parallel version of Dijkstra s algorithm [3], applied to a digraph G 0 = V 0 ; E 0 ) and running in time O(m 0 =p n 0 log n 0 ) using p m 0 =n 0 EREW PRAM processors, where n 0 = jV 0 j and m 0 = jE 0 j. The parallelization of Dijkstra s algorithm is straightforward and obtained by doing distance label updates ....

J. Driscoll, H. Gabow, R. Shrairman, and R.E. Tarjan, Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation, Comm. of the ACM, 31(11):1343-1354, 1988.

....Heap with key k if i is not in Heap already, and changes the key of item i in Heap to k if i is in Heap already. We now verify that the algorithm does indeed maintain the invariants described above. Thus, we first need to show that all vertices satisfy the invariant whenever execution reaches line [8]. The precondition guarantees that all the initially inconsistent vertices must be in U. In lines [1] 7] the algorithm creates a heap out of all the initially inconsistent vertices in the graph, and simultaneously the value rhs (u) is properly defined for every inconsistent vertex u. Hence the ....

....initially inconsistent vertices must be in U. In lines [1] 7] the algorithm creates a heap out of all the initially inconsistent vertices in the graph, and simultaneously the value rhs (u) is properly defined for every inconsistent vertex u. Hence the invariant holds when execution reaches line [8] for the first time. The loop in lines [8] 31] processes and fixes the inconsistencies in the graph one by one, in increasing order of key value. An over consistent vertex u is processed (lines [11] 19] by updating d (u) to equal g u (d (x 1 ) d (x k ) the value of the right hand ....

[Article contains additional citation context not shown here]

Driscoll, J.R., Gabow, H.N., Shrairman, R., and Tarjan, R.E., "Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation," Communications of the ACM 31(11) pp. 1343-1354 (1988). - 28 -

....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 ....

James R. Driscoll, Harold N. Gabow, Ruth Shrairman, and Robert E. Tarjan. Relaxed heaps: An alternative to fibonacci heaps with applications to parallel computation. Communications of the ACM, 31(11):1343--1354, 1988.

....Finally, Sect. 7 summarizes the results and sketches some open problems and future improvements. Previous Work PRAM algorithms: There is no parallel O(n log n m) work PRAM algorithm with sublinear running time for general digraphs with non negative edge weights. The best O(n log n m) work solution [9] has running time O(n log n) All known algorithms with polylogarithmic execution time are work inefficient. O(log 2 n) time and O(n 3 (log log n= log n) 1=3 ) work for the algorithm in [11] An O(n) time algorithm requiring O( n m) log n) work was presented in [3] For special classes ....

....ffi o (v) where ffi o (v) minfc(v; w) v; w) 2 Eg; ffi o (v) is precomputed once and for all upon initialization. The Q i queues are used to efficiently derive the criterion of the OUT version indicating whether a node can be deleted in a phase. The queues are implemented as relaxed heaps [9] because they provide worst case running times: findMin, insert and decreaseKey are performed in O(1) time and delete deleteMin in O(log q) time where q denotes the local queue size. Let r be the number of delete phases which are needed, e.g. for the OUTvariant r = O( p n) whp. For the analysis ....

J. R. Driscoll, H. N. Gabow, R. Shrairman, and R. E. Tarjan. Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation. Communications of the ACM, 31(11):1343--1354, 1988.

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J. R. Driscoll, H. N. Gabow, R. Shrairman, and R. E. Tarjan. Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation. Communications of the ACM, 31(11):1343--1354, November 1988.

No context found.

Driscoll, J.R., Gabow, H.N., Shrairman, R., & Tarjan, R.E. (1988), `Relaxed Heaps: An Alternative to Fibonacci Heaps with Applications to Parallel Computation', Communications of the ACM, 31(11), pp. 1343--1354.

No context found.

J.R. Driscoll, H.N. Gabow, R. Shrairman, R.E. Tarjan. Relaxed Heaps: An Alternative to Fibonacci Heaps with Applications to Parallel Computation. Communications of the ACM, 31(11):1343-1354, 1988.

No context found.

James R. Driscoll, Harold N. Gabow, Ruth Shrairman, and Robert E. Tarjan. Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation. Comm. of the ACM, 31(11):1343--1354, 1988.

No context found.

J.R. Driscoll, H.N. Gabow, R. Shrairman, and R.E. Tarjan. Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation. Communications of the ACM, 31(11):1343-1354, 1988.

No context found.

Driscoll J.R., Gabow H.N., Shrairman R. and Tarjan R.E. Relaxed Heaps : An alternative to Fibonacci heaps with applications to parallel computation Communications of ACM, 31(11):1988.

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Driscoll, J.R., Gabow, H.N., Shrairman, R., and Tarjan, R.E., "Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation," Communications of the ACM 31(11) pp. 1343-1354 (1988). - 36 -

No context found.

J.R. Driscoll, H.N. Gabow, R. Shrairman, and R.E. Tarjan. "Relaxed heaps: An alternative to fibonacci heaps with applications to parallel computation". Comm. ACM, 31(11):1343--1354, Nov. 1988.

No context found.

J. R. Driscoll, H. N. Gabow, R. Shrairman, and R. E. Tarjan. Relaxed heaps: An alternative to fibonacci heaps with applications to parallel computation. Communications of the ACM, 31, 1988.

No context found.

Driscoll, J.R., Gabow, H.N., Shrairman, R., and Tarjan, R.E., "Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation," Communications of the ACM 31(11) pp. 1343-1354 (1988).

No context found.

Driscoll, J.R., Gabow, H.N., Shrairman, R., and Tarjan, R.E., "Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation," Communications of the ACM 31(11) pp. 1343-1354 (1988).

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