B. M. E. Moret, H. D. Shapiro. An empirical assessment of algorithms for constructing a minimum spanning tree. DIMACS Monographs in Discrete Mathematics and Theoretical Computer Science 15 (1994), pp. 99-117.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(see [4] All algorithms tested share the same implementation of basic data structures (graphs, heaps, stacks) We opted for using binary heaps to implement priority queues. Although they are asymptotically less ecient, their performance is competitive in practice with that of Fibonacci heaps [7]. With binary heaps, the worst cases of our algorithms are slightly di erent from those mentioned in previous sections. For DNH Prim, DNH Bor uvka, and DNHz, the worst case is O(jEj log jV j) for Bor uvka, O(jEj log jT j log jV j) and for all versions of Prim and Kruskal, O(jT jjEj log jV j) ....

B. M. E. Moret and H. D. Shapiro. An empirical assessment of algorithms for constructing a minimum spanning tree. DIMACS Monographs in Discrete Mathematics and Theoretical Computer Science, 15:99-117, 1994.

....BSP algorithms for the MST problem requiring O(log p) supersteps, and proved a lower bound of min(n;m=p) on the per processor communication volume for MST algorithms. They did not provide any implementation results for their algorithms. There are only a few empirical investigations of sequential [16, 3, 23, 24], fine grained parallel [3] and coarsegrained parallel [18, 11] MST algorithms. The work in [11] focuses on variants of list ranking as applied to MST computations. It uses W(logn) supersteps and its overall running time is not analyzed. The algorithm tested in [18] is only partially documented ....

B. M. E. Moret and H. D. Shapiro. An empirical assessment of algorithms for constructing a minimum spanning tree. In Computational Support for Discrete Mathem., volume 15 of DIMACS Series in Discrete Mathem. and Theoretical Comp. Science, pages 99--117. American Mathematical Society, 1994.

....desire to understand how di erent algorithms perform in practice has led to a recent increase in the experimental study of algorithms. There have been a number of experimental studies of graph algorithms which focus on important problems such as shortest paths [9, 5] minimum spanning trees (MST) [17], network ow and matching [1, 4, 10, 19] and min cut algorithms [6] These experiments provide valuable insight into the performance of di erent algorithms and can suggest new algorithmic choices. The authors of these studies reasonably spend most of their e ort on the higher level algorithm ....

....Related Work Because of its importance, compiler writers have spent considerable e ort on generating code with good locality [3] however substantial additional improvements can be gained by proper algorithm design. Moret and Shapiro discuss cache e ects on graph algorithms in their MST paper [17]. They comment that data caching and performance is a ected by the method used to store a graph. They attempt to normalize for machine e ects by using running times relative to the time needed to scan the adjacency structure. More recently, several researchers have focused on designing algorithms ....

B. Moret and H. Shapiro. An Empirical Assessment of Algorithms for Constructing a Minimum spanning tree. DIMACS Series in Discrete Math and Theoretical CS, vol. 15, 99-117, 1994.

....node x and an interval I , and visits all vertices in the subgraph corresponding to x whose d values lie in I . 3 4 Design Choices 4.1 Dijkstra s Algorithm We use a pairing heap [F 86] to implement the priority queue in Dijkstra s algorithm. We made this choice based on the results reported in [MS94] for minimum spanning tree (MST) algorithms. In that experiment the pairing heap was found to be superior to the Fibonacci heap (the choice for the theoretical bound) as well as d ary heaps, relaxed heaps and splay heaps in implementations of the Prim Dijkstra MST algorithm. 3 Since the ....

....nd and sort the MST edges, we chose to use Kruskal s MST algorithm, which runs in O(m log n) time but does both of these tasks in one pass. Some of our data on larger and denser graphs suggests that it may be better to use the Prim Dijkstra MST algorithm, which is empirically faster than Kruskal s [MS94], followed by a step to sort only the MST edges. 2. Updating D values: In [PR02] the D value of an internal CH node is de ned to be the minimum D value over its descendant leaves. As leaf D values change, the internal D values must be updated. Rather than use Gabow s near linear time data ....

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B. M. E. Moret, H. D. Shapiro. An empirical assessment of algorithms for constructing a minimum spanning tree. In DIMACS Series on Discrete Math. and Theor. CS, 1994.

....node x and an interval I , and visits all vertices in the subgraph corresponding to x whose d values lie in I . 4 Design Choices 4.1 Dijkstra s Algorithm We use a pairing heap [F 86] to implement the priority queue in Dijkstra s algorithm. We made this choice based on the results reported in [MS94] for minimum spanning tree (MST) algorithms. In that experiment the pairing heap was found to be superior to the Fibonacci heap (the choice for the theoretical bound) as well as d ary heaps, relaxed heaps and splay heaps in implementations of the Prim Dijkstra MST algorithm. 3 Since the ....

....nd and sort the MST edges, we chose to use Kruskal s MST algorithm, which runs in O(m log n) time but does both of these tasks in one pass. Some of our data on larger and denser graphs suggests that it may be better to use the Prim Dijkstra MST algorithm, which is empirically faster than Kruskal s [MS94], followed by a step to sort only the MST edges. 2. Updating D values: In [PR02] the D value of an internal CH node is de ned to be the minimum D value over its descendant leaves. As leaf D values change, the internal D values must be updated. Rather than use Gabow s near linear time data ....

[Article contains additional citation context not shown here]

B. M. E. Moret, H. D. Shapiro. An empirical assessment of algorithms for constructing a minimum spanning tree. In DIMACS Series on Discrete Math. and Theor. CS, 1994.

....instances, far beyond the needs of any reasonable practical application. This means that, due to the high constants hidden in the analysis, theoretical bounds may fail to describe the behavior of algorithms on many instances of practical interest. As a typical example, the experimental study in [27] shows that the minimum spanning tree algorithm of Fredman and Tarjan improves in practice upon the classical Prim s algorithm only for huge dense graphs with more than one million nodes. The situation may be even worse: constants hidden in the asymptotic time bounds may be so large as to ....

B.M.E. Moret and H.D. Shapiro. An empirical assessment of algorithms for constructing a minimal spanning tree. Computational Support for Discrete Mathematics N. Dean and G. Shannon eds. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 15:99--117, 1994.

.... area of external memory algorithms, mostly based on a model due to [Aggarwal and Vitter 1988] However, there is another important optimisation which is required for optimum performance, namely minimising misses in the translation lookaside bu er (TLB) Some papers from the early 90 s (see e.g. [Moret and Shapiro 1994]) have noted the importance of minimising TLB misses, but the topic has seemingly been neglected since then. The TLB is used to support virtual memory in multi processing operating systems [Hennessy and Patterson 1996] Although there may be several active processes in the system, virtual memory ....

Moret, B. and Shapiro, H. 1994. An empirical assessment of algorithms for constructing a minimum spanning tree. In ? , Volume 15 of DIMACS Monographs in Discrete Mathematics and Theoretical Computer Science, pp. 99-117.

....The desire to understand how different algorithms perform in practice has led to a recent increase in the experimental study of algorithms. There have been a number of experimental studies of graph algorithms which focus on important problems such as shortest paths [13, 9] minimum spanning trees [21], network flow and matching [5, 8, 15, 23] and min cut algorithms [10] These experiments provide valuable insight into the performance of different algorithms and can suggest new algorithmic choices. The authors of these studies reasonably spend most of their effort on the higher level algorithm ....

....on algorithm design and analysis for memory hierarchies [1, 2, 3, 4] but most of this prior work assumes greater control over movement between the memory levels than exists in most current systems. Moret and Shapiro discuss cache effects on graph algorithms in their minimum spanning tree paper [21]. They comment that data caching and performance is affected by the method used to store a graph. Thus they attempt to normalize for machine effects by using running times relative to the time needed to scan the adjacency structure. More recently several researchers have focused on designing ....

B. Moret and H. Shapiro. An Empirical Assessment of Algorithms for Constructing a Minimum spanning tree. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 15, pp. 99-117, 1994.

.... studies have been presented in [7, 12, 21, 22, 29, 32, 34, 40] Also embedded with the proposal of new implementations we can find analyses and comparisons of PQs in [3, 4, 5, 20, 33, 35] In addition, in the context of other applications for PQs we can find comparative empirical studies in [19, 20, 25, 30]. Thus the empirical studies have been used in the past as the only practical tool to decide which PQ implementation is better under specific applications. In this case, the asymptotic performance bounds associated with every PQ do not help us so much since they usually hide significant constant ....

....in [28] In this report we extend this last work including multi list based PQs and additional tree based PQs. So the results presented here extend those in [27, 28] including new PQs (some of them with no test reported in the literature) and complement the empirical work presented so far in [3, 4, 5, 7, 12, 20, 21, 22, 29, 30, 32, 33, 35, 34, 40] using a fairly different random event generator for the work load applied to the PQs. 2.1 Variant to the Hold Model The model that we use in this work views the system as composed of N active objects where each active object has only one event E scheduled in the PQ and the rest in their local ....

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B.M.E Moret and H.D. Shapiro. "An empirical assessment of algorithms for constructing a minimum spanning tree". DIMACS Series in Discrete Mathematics and Theorical Computer Science, 15:99--117, June 1994.

....of PQs have been so far presented E mail: Mauricio.Marin comlab.oxford.ac. uk y Wolson Building, Parks Road, Oxford OX1 3QD, England, UK in [6, 10, 22, 24, 31, 32, 38, 39, 41, 47] Also, embedded with the proposal of new PQs and PES implementations, we can find analyses and comparisons in [3, 4, 5, 21, 20, 26, 33, 40, 42]. Most of these studies show that no single PQ implementation is the best for all cases and applications, cf. 22, 31, 33, 38, 42] This paper introduces a tournament based complete binary tree (CBT) as a basic data structure to implement PQs and thereby the PES. Using a variant of the empirical ....

.... in [6, 10, 22, 24, 31, 32, 38, 39, 41, 47] Also, embedded with the proposal of new PQs and PES implementations, we can find analyses and comparisons in [3, 4, 5, 21, 20, 26, 33, 40, 42] Most of these studies show that no single PQ implementation is the best for all cases and applications, cf. [22, 31, 33, 38, 42]. This paper introduces a tournament based complete binary tree (CBT) as a basic data structure to implement PQs and thereby the PES. Using a variant of the empirical hold model [5, 22, 24, 32, 40, 42, 41] which includes deletion of arbitrary items, we show that the CBT outperforms most of the ....

B.M.E Moret and H.D. Shapiro. "An empirical assessment of algorithms for constructing a minimum spanning tree". DIMACS Series in Discrete Mathematics and Theorical Computer Science, 15:99--117, June 1994.

....The desire to understand how different algorithms perform in practice has led to a recent increase in the experimental study of algorithms. There have been a number of experimental studies of graph algorithms which focus on important problems such as shortest paths [9, 5] minimum spanning trees [17], network flow and matching [1, 4, 10, 19] and min cut algorithms [6] These experiments provide valuable insight into the performance of different algorithms and can suggest new algorithmic choices. The authors of these studies reasonably spend most of their effort on the higher level algorithm ....

....Related Work Because of its importance, compiler writers have spent considerable effort on generating code with good locality [3] however substantial additional improvements can be gained by proper algorithm design. Moret and Shapiro discuss cache effects on graph algorithms in their MST paper [17]. They comment that data caching and performance is affected by the method used to store a graph. They attempt to normalize for machine effects by using running times relative to the time needed to scan the adjacency structure. More recently, several researchers have focused on designing ....

B. Moret and H. Shapiro. An Empirical Assessment of Algorithms for Constructing a Minimum spanning tree. DIMACS Series in Discrete Math and Theoretical CS, vol. 15, 99-117, 1994.

....of edges. Algorithms for the MST problem have been implemented and studied extensively with various refinements and novel data structures, for graphs of different orders and densities (see Chapter 13 in [3] Some recent papers have examined theoretical versus practical behavior of MST algorithms [15, 36, 48]. Algorithms for the MST problem have also been proposed for parallel machines in the past 20 years, and more recently, implemented [21, 26, 40] However, in many real life network optimization situations the MST is often required to satisfy some additional constraint. This additional constraint ....

B. M. E. Moret and H. D. Shapiro. An empirical assessment of algorithms for constructing a minimum spanning tree. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 15:99--117, 1994.

....PQ allows retrieval of the smallest time event and efficient insertion of new events (see [7] In the past, several event management strategies have been devised, analyzed, compared, and used in commercial software. Among them we have algorithms based on linked lists [2, 19, 8, 10, 22] and trees [12, 9, 3, 6, 19, 26, 24, 10, 25, 7, 14, 20]. However, systems with many active passive objects need special strategies with a different approach for event management which is very clear in molecular fluid simulation [21, 15, 23, 13, 17] In this paper we generalize a recently introduced PQ which we call Local Minima (LM) Although LM ....

....insight into the performance of a number of general purpose priority queues that might be used as a part of the LM data structure [12, 9, 3, 26, 24, 25, 7] We selected the priority queues reported in the literature as the most efficient ones. So we complement the empirical work presented in [6, 19, 10, 14, 20] showing that an unusual adaptation of the complete binary tree (CBT) 12] is the best structure in this new class of applications for priority queues. A preliminary version of part of this paper was presented in [1] The organization of the paper is as follows. Section 2 gives preliminary ....

B.M.E Moret and H.D. Shapiro, "An Empirical Assessment of Algorithms for Constructing a Minimum Spanning Tree", DIMACS Series in Discrete Mathematics and Theorical Computer Science, V.15 (June 1994).

....time is O( E #( E , V ) where #(m, n) is given by min i log (i) n # m n , so that, in particular, #(n, n) is just log # n. This bound is much better for dense graphs than that of Prim s algorithm, which is O( E log V ) when implemented with binary heaps, but experimentation [MS94] verifies that the crossover point occurs for dense graphs with well over a million vertices and thus hundreds of millions of edges beyond the size of any reasonable data set. In another facet of the same problem, the constants hidden in the asymptotic analysis may prevent any practical ....

.... of pairing heaps [SV87] An early experimental study on a large scale was that of Moret and Shapiro on sorting algorithms [MS91a] Chapter 8) itself inspired by the work of Knuth in his Volume III [Knu98] followed by that of the same authors on algorithms for constructing minimum spanning trees [MS94] In 1991, David Johnson and others initiated the very successful DIMACS Computational Challenges, the first of which [JM93] focused on network flow and shortest path algorithms, indirectly giving rise to several modern, thorough studies, by Cherkassky et al. on shortest paths [CGR96] by ....

B. M. E. Moret and H. D. Shapiro, An empirical assessment of algorithms for constructing a minimal spanning tree, DIMACS Series in Discrete Mathematics and Theoretical Computer Science (N. Dean and G. Shannon, eds.), vol. 15, Amer. Math. Soc., 1994, pp. 99--117.

.... on the behavior of implementations [1, 11, 16 18, 27, 29] ever more e#cient implementation of network flow algorithms [7, 8, 13] and the characterization of the behavior of everyday algorithms and data structures such as priority queues [15, 32] shortest paths [6] minimum spanning trees [22], and sorting [21] More references can be found in [20] as well as by going to the web site for the ACM Journal of Experimental Algorithmics at www.jea.acm.org. High Performance Algorithm Engineering High Performance Algorithm Engineering focuses on one of the many facets of algorithm ....

Moret, B.M.E., & Shapiro, H.D., "An empirical assessment of algorithms for constructing a minimal spanning tree," in Computational Support for Discrete Mathematics, N. Dean & G. Shannon, eds., DIMACS Monographs in Discrete Math. & Theor. Comput. Sci. 15 (1994), 99--117.

....trees. Its asymptotic worst case running time is O(jEj (jEj; jV j) where (m; n) is given by minfi j log (i) n m=ng, so that, in particular, n; n) is just log n. This bound is much better for dense graphs than that of Prim s algorithm, which is O(jEj log jV j) but experimentation [20] veri es that the crossover point occurs for dense graphs with well over a million vertices beyond the size of any reasonable data set. The worst case behavior may be restricted to a very small subset of instances and thus not at all characteristic of instances encountered in practice. A ....

....of developing implementations for all successive layers. Moreover, the paper andpencil algorithms often ignore issues critical in making implementations ecient (from elementary ideas such as the use of sentinels to more elaborate ones such as the use of sacks in sophisticated priority queues [20]) the implementer will have to resolve these issues on the y, possibly with very poor results. 4 BERNARD M.E. MORET What can we do to improve this situation There is no reason to abandon asymptotic worst case analysis: it has served the community very well for over thirty years and led to ....

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Moret, B.M.E., and Shapiro, H.D., \An empirical assessment of algorithms for constructing a minimal spanning tree," in Computational Support for Discrete Mathematics, N. Dean and G. Shannon, eds., DIMACS Series in Discrete Mathematics and Theoretical Computer Science 15 (1994), 99-117.

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B. M. E. Moret, H. D. Shapiro. An empirical assessment of algorithms for constructing a minimum spanning tree. DIMACS Monographs in Discrete Mathematics and Theoretical Computer Science 15 (1994), pp. 99-117.

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