## A New, Simpler Linear-Time Dominators Algorithm (1998)

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Venue: | ACM Transactions on Programming Languages and Systems |

Citations: | 14 - 4 self |

### BibTeX

@ARTICLE{Buchsbaum98anew,,

author = {Adam L. Buchsbaum and Haim Kaplan and Anne Rogers and Jeffery R. Westbrook},

title = {A New, Simpler Linear-Time Dominators Algorithm},

journal = {ACM Transactions on Programming Languages and Systems},

year = {1998},

volume = {20},

pages = {1265--1296}

}

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### Abstract

this article is organized as follows. Section 2 outlines Lengauer and Tarjan's approach. Section 3 gives a broad overview of our algorithm and dierentiates it from previous work. Section 4 presents our algorithm in detail, and Section 5 analyzes its running time. Section 6 presents our new path-compression result, on which the analysis relies. Section 7 describes our implementation, and Section 8 reports experimental results. We conclude in Section 9

### Citations

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Citation Context ...ly assigning each vertex in each SCC the minimum label among all the vertices in the SCC, and then pushing the labels through the SCCs in topological order. Computing SCCs can be done in linear time [=-=Tarjan 1972-=-], as can the topological processing of the SCCs. Step (3) links root(T ) once for each trivial microtree T . Thus, the time to compute the pxdoms, summed over all the microtrees, is O(m + n) plus the... |

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Citation Context ...2; Lorry and Medlock 1969]. Finding immediate dominators in a flowgraph is an elegant problem in graph theory, with applications in global flow analysis and program optimization [Aho and Ullman 1972; =-=Cytron et al. 1991-=-; Ferrante et al. 1987; Lorry and Medlock 1969]. Lorry and Medlock [1969] introduced an O(n4)-time algorithm, where n = |V | and m = |A|, to find all the immediate dominators in a flowgraph. Successiv... |

864 | The program dependence graph and its use in optimization
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Citation Context ... 1969]. Finding immediate dominators in a flowgraph is an elegant problem in graph theory, with applications in global flow analysis and program optimization [Aho and Ullman 1972; Cytron et al. 1991; =-=Ferrante et al. 1987-=-; Lorry and Medlock 1969]. Lorry and Medlock [1969] introduced an O(n4)-time algorithm, where n = |V | and m = |A|, to find all the immediate dominators in a flowgraph. Successive improveSome of this ... |

210 |
The Theory of Parsing
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(Show Context)
Citation Context ...mmediate dominator (idom) of v, denoted w = idom(v), if (1) w dominates v and (2) every other vertex x that dominates v also dominates w. Every vertex in a flowgraph has a unique immediate dominator [=-=Aho and Ullman 1972-=-; Lorry and Medlock 1969]. Finding immediate dominators in a flowgraph is an elegant problem in graph theory, with applications in global flow analysis and program optimization [Aho and Ullman 1972; C... |

161 |
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
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Citation Context ...a method to restrict the set of such outside dominator candidates. Alstrup et al. [1997] demonstrate deficiencies in Harel’s arguments and correct these problems, using Fredman and Willard’s Q-heaps [=-=Fredman and Willard 1994-=-] to manage the microtrees. We apply the LT algorithm just to the upper portion, D′, of D. We combine our partitioning scheme with a new path compression result to show that the LT algorithm runs in l... |

158 |
A linear-time algorithm for a special case of disjoint set union
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Citation Context ...ing an upper subtree, D′, of singleton microtrees as we do. They then perform links and evals on D′ using two novel data structures, as well as the Gabow-Tarjan linear-time disjoint set union result [=-=Gabow and Tarjan 1985-=-] and transformations to D′. Both algorithms use precomputed tables to process evals on the internal microtree vertices. This approach requires information regarding which vertices outside microtrees ... |

154 | A fast algorithm for finding dominators in a flowgraph. ACMTrans. Program. Languages Syst
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Citation Context .... Section 7 describes our implementation, and Section 8 reports experimental results. We conclude in Section 9. 2. THE LENGAUER-TARJAN ALGORITHM Here we outline the Lengauer and Tarjan (LT) approach [=-=Lengauer and Tarjan 1979-=-] at a high level, to provide some details needed by our algorithm. Appel [1998] provides a thorough description of the LT algorithm. Let G = (V,A, r) be an input flowgraph with n vertices and m arcs.... |

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Citation Context ...5054 16384 40959 90110 32768 81919 180222 2097152 5242879 11534334 with their sizes. The ATIS, NAB, and PW graphs are derived from weighted finite-state automata used in automatic speech recognition [=-=Pereira and Riley 1997-=-; Pereira et al. 1994] by removing weights, labels, and multiple arcs. The Phone graph represents telephone calling patterns. The augmented binary graphs (AB1 and AB2) were generated synthetically by ... |

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Citation Context ... Table II contains average running times for LT and for BKRW with microtree sizes of two and three. Figure 15 displays a scatter plot in which each 3Machine SUIF is an extension of the SUIF compiler [=-=Amarasinghe et al. 1995-=-] from Stanford. We used Machine SUIF version 1.1.2. A New, Simpler Linear-Time Dominators Algorithm · 27 Table I. Graph Sizes, Averaged Over the Flowgraphs in Each Benchmark, for the SPEC 95 Flowgrap... |

111 | Worst-case analysis of set union algorithms - Tarjan, Leeuwen - 1984 |

89 | 1979a. Applications of path compression on balanced trees - Tarjan |

79 | A class of algorithms which require nonlinear time to maintain disjoint sets - Tarjan - 1979 |

52 |
Finding dominators in directed graphs
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Citation Context ...ires prior specific permission and/or a fee. c© 1999 ACM 0164-0925/99/0100-0111 $00.75 2 · Adam L. Buchsbaum et al. ments to this time bound were achieved [Aho and Ullman 1972; Purdom and Moore 1972; =-=Tarjan 1974-=-], culminating in Lengauer and Tarjan’s [1979] O(mα(m,n))-time algorithm; α is the standard functional inverse of the Ackermann function and grows extremely slowly with m and n [Tarjan and van Leeuwen... |

49 | Weighted rational transductions and their application to human language processing
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Citation Context ...2768 81919 180222 2097152 5242879 11534334 with their sizes. The ATIS, NAB, and PW graphs are derived from weighted finite-state automata used in automatic speech recognition [Pereira and Riley 1997; =-=Pereira et al. 1994-=-] by removing weights, labels, and multiple arcs. The Phone graph represents telephone calling patterns. The augmented binary graphs (AB1 and AB2) were generated synthetically by building a binary tre... |

38 |
A linear time algorithm for finding dominators in flow graphs and related problems
- Harel
- 1985
(Show Context)
Citation Context ...n contain more than one vertex, and performs links and evals as described by Lengauer and Tarjan [1979] on the tree D′ induced by the microtree roots. Alstrup et al. [1997] simplify Harel’s approach [=-=Harel 1985-=-] by restricting nonsingleton microtrees to the bottom of D, leaving an upper subtree, D′, of singleton microtrees as we do. They then perform links and evals on D′ using two novel data structures, as... |

37 |
Object code optimization
- Lorry, Medlock
- 1969
(Show Context)
Citation Context ...dom) of v, denoted w = idom(v), if (1) w dominates v and (2) every other vertex x that dominates v also dominates w. Every vertex in a flowgraph has a unique immediate dominator [Aho and Ullman 1972; =-=Lorry and Medlock 1969-=-]. Finding immediate dominators in a flowgraph is an elegant problem in graph theory, with applications in global flow analysis and program optimization [Aho and Ullman 1972; Cytron et al. 1991; Ferra... |

29 | Linear-time pointer-machine algorithms for least common ancestors, MST verification, and dominators
- Buchsbaum, Kaplan, et al.
- 1998
(Show Context)
Citation Context ...trees, as in Harel’s approach to dominators, our path-compression result allows microtree processing to be restricted to the “bottom” of a tree traversal of the graph. We have applied this technique [=-=Buchsbaum et al. 1998-=-] to simplify previous linear-time algorithms for least common ancestors, minimum spanning tree (MST) verification, and randomized MST construction. We also show [Buchsbaum et al. 1998] how to apply o... |

17 | Data-structural bootstrapping, linear path compression, and catenable heap-ordered double-ended queues
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(Show Context)
Citation Context ... union and find operations in O(m + n) time. We do not require advance knowledge of the unions themselves, only that their order be constrained. Other results on improved bounds for path compression [=-=Buchsbaum et al. 1995-=-; Loebl and Nešetřil 1997; Lucas 1990] generally restrict the order in which finds, not unions, are performed. Of the n vertices, designate l to be special and the remainder n− l to be ordinary. The... |

16 | Optimal control dependence computation and the Roman chariots problem - Pingali, Bilardi - 1997 |

10 | Linearity and unprovability of set union problem strategies I. Linearity of strong postorder - Loebl, Nešetřil - 1997 |

10 |
Postorder disjoint set union is linear
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(Show Context)
Citation Context ... not require advance knowledge of the unions themselves, only that their order be constrained. Other results on improved bounds for path compression [Buchsbaum et al. 1995; Loebl and Nešetřil 1997; =-=Lucas 1990-=-] generally restrict the order in which finds, not unions, are performed. Of the n vertices, designate l to be special and the remainder n− l to be ordinary. The following theorem shows that by requir... |

9 | Optimal parallel verification of minimum spanning trees in logarithmic time. Algorithmica 17, 11–8. 32 · Adam L. Buchsbaum et al - Dixon, Tarjan - 1997 |

7 |
The SPEC95 Benchmark Suite. The Standard Performance Evaluation
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(Show Context)
Citation Context ...implementation of the LT algorithm derived from their paper [Lengauer and Tarjan 1979]. We performed two sets of experiments. The first set used flowgraphs collected from the SPEC 95 benchmark suite [=-=SPEC 1995-=-], using the CFG library from the Machine SUIF compiler [Holloway and Young 1997] from Harvard.3 (Six files from the integer suite could not be compiled by Machine SUIF v. 1.1.2 and are omitted from t... |

6 |
Algorithm 430: Immediate predominators in a directed graph
- Purdom, Moore
- 1972
(Show Context)
Citation Context ...istribute to lists requires prior specific permission and/or a fee. c© 1999 ACM 0164-0925/99/0100-0111 $00.75 2 · Adam L. Buchsbaum et al. ments to this time bound were achieved [Aho and Ullman 1972; =-=Purdom and Moore 1972-=-; Tarjan 1974], culminating in Lengauer and Tarjan’s [1979] O(mα(m,n))-time algorithm; α is the standard functional inverse of the Ackermann function and grows extremely slowly with m and n [Tarjan an... |

6 | The flow analysis and transformation libraries of Machine SUIF
- Holloway, Young
- 1997
(Show Context)
Citation Context ...uer and Tarjan 1979]. We performed two sets of experiments. The first set used flowgraphs collected from the SPEC 95 benchmark suite [SPEC 1995], using the CFG library from the Machine SUIF compiler [=-=Holloway and Young 1997-=-] from Harvard.3 (Six files from the integer suite could not be compiled by Machine SUIF v. 1.1.2 and are omitted from the data.) The second set used some large graphs collected from our Lab. We perfo... |

1 | Dominators in linear time. Manuscript available at ftp://ftp.diku.dk/pub/diku/users/stephen/dom.ps - Alstrup, Harel, et al. - 1997 |