S. Alstrup, P. W. Lauridsen, and M. Thorup. Dominators in linear time. DIKU technical report, (35), 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....should be the technique of choice for computing dominators on cfgs. The dominance problem is an excellent example of the need to balance theory with practice. Ever since Lowry and Medlock s O(N )algorithm appeared in 1969 [23] researchers have steadily improved the time bound for this problem [7, 10, 17, 19, 22, 26, 29]. However, our results suggest that these improvements in asymptotic complexity may nothelponrealisticallyCopyright c sized examples, and that careful engineering makes the iterative scheme the clear method of choice. History Prosser introduced the notion of dominance in a 1959 paper on the ....

.... a set of data flow equations for the problem [4] Two years later, Allen and Lowry and Medlock do not give enough details to assess accurately the complexity of their algorithm, but Alstrup et al. claim that it has an asymptotic complexity of N ,whereN is the number of nodes in the graph [7]. Cocke showed aniterative algorithm for solving these equations and gave its complexity as O(N ) 5] In1975, Hecht and Ullman published an analysis of iterative algorithms using reverse postorder traversals. They showed that the dominance equations can be solved in linear time on reducible ....

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S. Alstrup, D. Harel, P. W. Lauridsen, and M. Thorup. Dominators in linear time. SIAM J. Comput., 28(6):2117--2132, June 1999.

....indicated the problem of code explosion as a consequence of operations being duplicated after code motions. A global scheduling method in which no duplicates are created at all is Dominator Path Scheduling (DPS) 68] This algorithm is based on faster algorithms to calculate dominators in CFGs [4]. Those algorithms calculate birth points of expressions: the highest place in the CFG where an expression is available. Since most instructions are not only expressions, but also write to some destination register, usage information has to be gathered as well, to guarantee that de nitions are ....

Alstrup, S., Harel, D., P.W., L., and Thorup, M. Dominators in linear time. Tech. Rep. 97/28, Datalogisk Institut, University of Copenhagen, 1997.

....the algorithms for constructing the conventional PDG slice and for constructing the post dominator tree. If interprocedural slicing is being used, this will require the construction of a System Dependence Graph 5 . Recently, a linear algorithm for constructing post dominator trees was proposed (Alstrup et al. 1997). The most commonly used algorithm (Horwitz et al. 1990) for (interprocedural) slice construction uses the System Dependence Graph (SDG) a generalisation of the Program Dependence Graph which caters for interprocedural slicing. The SDG slicing algorithm consists of three phases. 1. The ....

Alstrup, S., Harel, D., Lauridsen, P. W., and Thorup, M. (1997). Dominators in linear time. Technical Report DIKU 97/28, Department of Computer Science, University of Copenhagen.

....are lazy nodes. To linearise a dag means to turn it into a tree by introducing let bindings for shared subexpressions. If the dag did not contain any lazy nodes, let2 introduction would simply amount to nding the immediate dominator for each shared node, which can be achieved in linear time [2]. In the presence of conditionals, however, it may not be safe to let bind a shared subexpression at its immediate dominator, because such a let binding would force evaluation (as said, the underlying language is strict) which is illustrated in expression (1) if both a and b evaluate to true, ....

Stephen Alstrup, Dov Harel, Peter W. Lauridsen, and Mikkel Thorup. Dominators in linear time. SICOMP: SIAM Journal on Computing, 28, 1999.

....Ramalingam [14] improves upon the efficiency of these different algorithms. The concepts of domination and dominator tree have many uses in program analysis and optimization. The standard algorithm for computing dominators is Lengauer and Tarjan s [10] almost linear time algorithm. Alstrup et al. [3, 1] and Buchsbaum et al. 4] present linear time algorithms for this problem. The concept of the iterated dominance frontier was brought to prominence by the work of Cytron et al. 7] on the SSA form, a data structure with numerous applications in program optimization. The Cytron et al. algorithm ....

Stephen Alstrup, Dov Harel, Peter W. Lauridsen, and Mikkel Thorup. Dominators in linear time. SIAM J. Comput. To appear.

....many applications in program analysis and program optimization. The most well known and widely used algorithm for computing dominators is the almost linear time algorithm of Lengauer and Tarjan [11] Harel [9] claimed a linear time algorithm for constructing the dominator tree, but Alstrup et al. [3, 1] showed that Harel s algorithm was, in fact, non linear, and presented a modified algorithm that runs in linear time. More recently, Buchsbaum et al. 4] have presented a simpler linear time algorithm for constructing the dominator tree. They also present experimental results which show that, in ....

....we have presented new, almost linear time, algorithms for two graph theoretic problems, that of constructing the dominator tree of a graph and that of computing the iterated dominance frontier of a set of vertices in a graph. Though linear time algorithms are already known for these problems [1, 4, 13, 17], we believe that these new algorithms are interesting because of the approach we take In particular, we have utilized the above two applications as a vehicle for understanding the concepts of loops and loop nesting forests in graphs. We have shown how three previously defined loop nesting ....

Stephen Alstrup, Dov Harel, Peter W. Lauridsen, and Mikkel Thorup. Dominators in linear time. SIAM J. Comput. To appear.

....of immediate dominators (see Definition 16 or (Aho et al. 1986) and instrumentation variables. 2 dfwcetklu.tex; 14 04 2000; 9:10; p.31 32 Johann Blieberger Remark 8. Note that the dominator tree, a tree whose edges reflect the immediate dominance relation, can be constructed in linear time (see (Alstrup et al. 1996)) If we have symbolic expressions for all instrumentation variables b i , we can set up equations b i = X B j 2Succs(B i ) c ij b i = X B k 2Preds(B i ) c ki (5) where c ij is a symbolic counter assigned to edge e ij = B i ; B j ) Preds(B i ) and Succs(B i ) denote the set of ....

Alstrup, S., P. W. Lauridsen, and M. Thorup: 1996, `Dominators in Linear Time'. Technical Report TR DIKU 96-35, Department of Computer Science, University of Copenhagen.

....the hammock between two given nodes. Such queries are useful in software system analysis, to detect collections of systems with designated information choke points, e.g. to assess the impact of retiring legacy systems [5] We can solve such queries as follows. Let T 1 be the dominator tree [1, 6] of G, and let T 2 be the dominator tree of the reverse graph of G. The hammock between two nodes u and v in G is the intersection of the set of descendents of u in T 1 with the set of descendents of v in T 2 . By adding edges connecting corresponding nodes in T 1 and T 2 , this intersection is ....

S. Alstrup, D. Harel, P. W. Lauridsen, and M. Thorup. Dominators in linear time. SIAM J. Comp., 28(6):2117-32, 1999.

....is placed at the lowest common ancestor of its body s occurrences. Assume there are n nodes in the original graph. There exist some ingenious algorithms for finding the lowest common ancestors of all repeated subexpressions in time O(n #(n) where # is the inverse of the Ackermann function) [10, 1], or even O(n) 9] We are considering how these algorithms may be adapted to solve the above problem e#ciently the current implementation uses a naive algorithm whose worst case performance is quadratic. Some readers may wonder how we can end up with shared subexpressions that cannot be ....

Alstrup, Harel, Lauridsen, and Thorup. Dominators in linear time. SICOMP: SIAM Journal on Computing, 28, 1999.

.... algorithm for the decision problem: ESSNET: Given an essential net (of a linearly balanced IMLL sequent) is it correct Our approach uses a linear time algorithm for constructing the dominator tree of a flowgraph; several such algorithms have recently been obtained by various groups [1, 5, 9]. A flowgraph is just a directed graph such that every node of the graph is reachable from a fixed start node. Nodes of such graphs can be ordered by a notion of domination: we say that node v dominates w just in case every path from the start node that reaches w passes through v. The ordering has ....

....superlinear; it was discovered by Lengauer and Tarjan [15] in 1979 after a series of successive improvements by many authors. By using ingenious data structures, Harel [13] was able to reduce its complexity to linear time; his result was later refined by Alstrup, Harel, Lauridsen and Thorup in [1]. Although theoretically important, the later algorithms are not really practical because they depend on highly complex data structures (e.g. Q heaps due to Fredman and Willard [8] which are costly to use. Recently (in 1998) Buchsbaum, Kaplan, Rogers and Westbrook have introduced another ....

S. Alstrup, D. Harel, P. Lauridsen, and M. Thorup. Dominators in linear time. SIAM Journal on Computing, 28 (6):2117--2132, 1999.

....many applications in program analysis and program optimization. The most well known and widely used algorithm for computing dominators is the almost linear time algorithm of Lengauer and Tarjan [11] Harel [9] claimed a linear time algorithm for constructing the dominator tree, but Alstrup et al. [3, 1] showed that Harel s algorithm was, in fact, non linear, and presented a modified algorithm that runs in linear time. More recently, Buchsbaum et al. 4] have presented a simpler linear time algorithm for constructing the dominator tree. They also present experimental results which show that, in ....

....we have presented new, almost linear time, algorithms for two graph theoretic problems, that of constructing the dominator tree of a graph and that of computing the iterated dominance frontier of a set of vertices in a graph. Though linear time algorithms are already known for these problems [1, 4, 12, 18], we believe that these new algorithms are interesting because of the approach we take In particular, we have utilized the above two applications as a vehicle for understanding the concepts of loops and loop nesting forests in graphs. We have shown how three previously defined loop nesting ....

Stephen Alstrup, Dov Harel, Peter W. Lauridsen, and Mikkel Thorup. Dominators in linear time. SIAM J. Comput. To appear.

....the Ackermann function. Problem Previous Pointer Machine Bound Previous RAM Bound Off line LCAs O(pff(p; n) n) 1] O(n p) 16, 25] MST Verification O(mff(m; n) n) 27] O(n m) 9, 18] MST Construction O(mff(m; n) log ff(m; n) n) 7] O(n m) 13, 17] Dominators O(mff(m; n) n) 20] O(n m) [3, 15] level microtrees, and pointer based radix sort to the MST verification (and construction) and dominators problems. 2 Least Common Ancestors Let T = V; E) be a tree with root r, and let P V Theta V be a set of pairs of vertices of T . We wish to compute the least common ancestor lca(x; y) ....

....Then n Gamma 1 unions and m finds can be performed in O(mff(m; l) n) time. PROOF (SKETCH) Let the leaves of T be classed as special and all internal vertices classed as ordinary. Since unions occur bottom up, each non singleton set al..ways contains at least one leaf. 2 Alstrup et al. [3] prove a variant of Corollary 3.2, with the mff(m; l) term replaced by (l log l m) which suffices for their purposes. They derive the weaker result by processing long paths of unary vertices in T outside the standard set union data structure. We apply the standard set union data structure ....

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S. Alstrup, D. Harel, P. W. Lauridsen, and M. Tho9 rup. Dominators in linear time. Manuscript submitted, 1997.

....various results in compiler theory rely on the existence of a linear time dominators algorithm; Pingali and Bilardi [1997] give an example and further references. Harel [1985] claimed a linear time dominators algorithm, but careful examination of his abstract reveals problems with his arguments. Alstrup et al. 1997] detail some of the problems with Harel s approach and offer a linear time algorithm that employs powerful data structures based on bit manipulation to resolve these problems. While they achieve a lineartime dominators algorithm, their reliance on sophisticated data structures adds sufficient ....

....on bit manipulation to resolve these problems. While they achieve a lineartime dominators algorithm, their reliance on sophisticated data structures adds sufficient overhead to make any implementation impractical. We present a new linear time dominators algorithm, which is simpler than that of Alstrup et al. 1997]. Our algorithm requires no complicated data structures: we use only depth first search, the fast union find data structure [Tarjan and van Leeuwen 1984] topological sort, and memoization. We have implemented our algorithm, and we report experimental results, which show that, even with the extra ....

[Article contains additional citation context not shown here]

Alstrup, S., Harel, D., Lauridsen, P. W., and Thorup, M. 1997. Dominators in linear time. Manuscript available at ftp://ftp.diku.dk/pub/diku/users/stephen/dom.ps.

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S. Alstrup, P. W. Lauridsen, and M. Thorup. Dominators in linear time. DIKU technical report, (35), 1996.

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