Robert E. Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28:594--614, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....can be extended to data ow frameworks for which mop can be computed by considering paths containing at most k cycles. Again, the conditions only apply to distributive frameworks. Frameworks such as these are sometimes called fast frameworks. See for instance Tarjan s papers [21] 23] and [22]. More general conditions, not requiring distributivity, have also been considered by Graham and Wegman [4] and Rosen [15] These conditions do not yield an ecient algorithm for mop 4 ; but they do yield an ecient algorithm which computes a solution which is guaranteed to be between mfp and mop. ....

Robert Endre Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28(3):594-614, 1981.

....programming environments. They provide information about a program or environment without executing the code. A lot of theoretical results on data flow frameworks has been derived (cf. e.g. KU76, KU77] and a large number of algorithms has been developed (cf. e.g. AC76, GW76, HU77, Sre95, SGL98, Tar81a, Tar81b] See [MR90, RP86] for an overview. Worst Case Execution Time (WCET) analysis does not have such a long standing tradition (cf. e.g. CBW96, HS91, ITM90, NP93, Par93, PK89, PS97, Sha89] Designers of real time programming languages usually restrict language features in order to make it ....

....# SOL) Proof. Properties 3.3.1 3 are proved in [Pau88] A Gaussian Elimination Type algorithm can be used to solve sets of equations. For data flow analysis the special structure of flow graphs can be exploited to construct algorithms with improved time complexity. For example see [AC76, HU77, Tar81a, GW76, Sre95, SGL98] and [RP86] for an overview of the first four algorithms. For our purposes we use the algorithm of [Sre95, SGL98] Returning to our equations of Definition 3.4, we have to determine how to insert one equation into another, how to simplify our equations, and how to set up a ....

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Robert Endre Tarjan, Fast algorithms for solving path problems, J. ACM 28 (1981), no. 3, 594--614. 1, 10, 14

....absence of recursive calls, it is sufficient to proceed by a depth first search (DFS) traversal of the call graph propagating sideeffect information (read write sets) from callees to callers to complete the analysis. However in the presence of recursive calls, strongly connected components (SCC) Tar81] exist in the call graph and applying a simple DFS traversal can lead to erroneous results as will be shown below. In a graph, an SCC is the largest set of nodes that satisfy the following criteria: from each node in the SCC, it is possible to reach any other node in the SCC. In terms of the call ....

....nodes are now pointing to the same node, meaning that the two reference variables represented by both reference nodes in the graph might refer to the same object on the heap. The union (merging) of abstract location nodes is done using the fast set union find algorithm developed by Tarjan [Tar81] The algorithm works as follows: at the beginning of the analysis, each abstract heap location node is considered as a set, with the node as unique member. Each set has a unique representative called the equivalent class representative (ECR) which initially is the abstract heap location node ....

Robert E. Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28(3):594--614, July 1981.

....sum, concatenation and Kleene closure The answer to this question depends on the representation method. 3. How does the complexity of sum algorithms to solve problems about distances compare with similar algorithms using regular expression. 4. How to analyze the complexity of matrix algorithms ([11, 15, 14]) involving sums Some of these problems are quite dicult. In particular, question 1 must be answered (that is, a concrete representation method must be selected) before the complexity of the algorithms can be analyzed. In this paper we concentrate on high level algorithmic ideas without going ....

....algebra of sums is substantially more ecient. To the length of each edge we may either assign 1 or the nonnegative integer speci ed by some function f : E Z 0 The second observation is that we do not need to consider only the paths between two xed vertices. Well known matricial methods ([14], 7] which are used in a number of di erent situations such as the Floyd shortest path algorithm [5] Warshall s transitive closure computation [16] and Kleene s computation of a regular expression equivalent to an automaton can of course be applied to distance problems. 5.1 Examples Now ....

Robert Endre Tarjan. Fast algorithms for solving path problems. Journal of the Association for Computing Machinery, 28(3):594-614, 1981.

....The bulk of this work has been for sequential programs. The development of data flow frameworks by Kildall [Kil73] marked the beginning of a new era in compiler optimization. Setting data flow analysis on a sound formal foundation spurred a wealth of research into both generalizing, e.g. [KU77, CC77, GW76, Tar81] and specializing Kildall s result, e.g. KU76, WZ91, Tar81] This resulted in formalisms for specifying and classifying data flow analysis problems, and the development of algorithms for solving any problem in a given class. This meant that compiler developers no longer had to hand craft a ....

....development of data flow frameworks by Kildall [Kil73] marked the beginning of a new era in compiler optimization. Setting data flow analysis on a sound formal foundation spurred a wealth of research into both generalizing, e.g. KU77, CC77, GW76, Tar81] and specializing Kildall s result, e.g. [KU76, WZ91, Tar81]. This resulted in formalisms for specifying and classifying data flow analysis problems, and the development of algorithms for solving any problem in a given class. This meant that compiler developers no longer had to hand craft a system of equations and a solver for an analysis problem. Instead, ....

Robert Endre Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28(3):594--614, jul 1981.

....by finding the strongly connected components of the call graph [5] and treat them as one virtual node. Methods included in the same component will have the same purity, as they may at some time call any of the others. Finding the strongly connected components has a well known linear time solution [11]. By using these virtual nodes, we turn the call graph into a directed acyclic graph, where a simple depth first traversal can be used to find the purities. The example in Figure 4 will then have bar( baz( and qux( as one virtual node in the callgraph, called from main( All the methods ....

Robert Endre Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28(3):594-- 614, July 1981.

....is not aliased to p or r. Our method thereby improves the precision of alias information for dynamically allocated objects. A more detailed description is provided in Section 6. 3 Intraprocedural Alias Computation The intraprocedural alias computation can be formulated as a data flow framework [20, 22], in which the solution at a given program point is related to the solution at other points. A control flow graph is a directed graph CFG = NCFG ; ECFG ; Entry; Exit . The nodes NCFG are the statements of a procedure and two additional nodes, Entry and Exit. The edges ECFG represent transfers ....

Robert Tarjan. Fast algorithms for solving path problems. Journal of the Association for Computing Machinery, 28(3):594--614, 1981.

....cycle free. We can then obtain a topsort order among the siblings in children(u) Irreducible graphs can also be handled by computing a path sequence for each dominator strong components. Due to limited space, we cannot detail how to compute the path sequence. Interested readers should refer to [Tar81a] The two existing GSA algorithms only handle reducible graph, but the algorithm here can be extended to handle irreducible graph. In the merge phase, the algorithm follows the topsort order and computes for each child of u a path expression GP (v) representing all the gating paths of v. In the ....

....we can set R(v) to in the UPDATE operation to represent the unconditional reach and, in this way, to simplify the path expressions for subsequent calls to EV AL. This algorithm is a variant of Tarjan s fast algorithm for solving path problems using dominator strong components decomposition [Tar81a] Its correctness can be derived from the following Lemma, which we quote without proof here. We will work through an example to illustrate the algorithm. Lemma 7. Tar81a] ffl For edges e = w; v) in CFG such that w 6= u, the corresponding path expression in the ListP (v) computed by the ....

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Robert Endre Tarjan. Fast algorithm for solving path problems. Journal of the ACM, 28(3):594--614, July 1981.

....in children(u) Irreducible graphs can also be handled by computing a path sequence for each dominator strong components. Whereas the two existing GSA algorithms handle only reducible graph, the algorithm here can be extended to handle irreducible graphs. Interested readers should refer to [Tar81a] for details on building path expressions for irreducible graph. In the merge phase, the algorithm follows the topsort order and computes for each child of u a path expression GP (v) representing all the gating paths of v. During the processes, if there is a p(v; v) in ListP (v) indicating that ....

....we can set R(v) to in the UPDATE operation to represent the unconditional reach and, in this way, simplify the path expressions for subsequent calls to EVAL. This algorithm is a variant of Tarjan s fast algorithm for solving path problems using dominator strong components decomposition [Tar81a] Its correctness can be derived from the following Lemma, which we quote without proof here. We will work through an example to illustrate the algorithm. Lemma 3.7 [Tar81a] ffl For edges e = w; v) in CFG such that w 6= u, the corresponding path expression in the ListP (v) computed by the ....

[Article contains additional citation context not shown here]

Robert Endre Tarjan. Fast algorithm for solving path problems. Journal of the ACM, 28(3):594--614, July 1981.

....in a loop nest, the nesting structure of each portion which is reducible separated by irreducible regions. Finally we analyze the complexity of the algorithm. When a flow graph is reducible, our algorithm has the same time complexity T 1 = O(jEj Theta ff(jEj; jN j) as Tarjan s approach [Tar81] The worst case time complexity of the algorithm occurs when it needs to find irreducible loops at every level. Assume k is the number of levels in the DJ graph. Then the time complexity of the algorithm is T 1 O(k Theta jEj) since finding strongly connected components needs O(jEj) time. We ....

Robert E. Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28(3):594--614, July 1981.

....The bulk of this work has been for sequential programs. The development of data flow frameworks by Kildall [Kil73] marked the beginning of a new era in compiler optimization. Setting data flow analysis on a sound formal foundation spurred a wealth of research into both generalizing, e.g. [KU77, CC77, GW76, Tar81] and specializing Kildall s result, e.g. KU76, WZ91, Tar81] This resulted in formalisms for specifying and classifying data flow analysis problems, and the development of algorithms for solving any problem in a given class. This meant that compiler developers no longer had to hand craft a ....

....development of data flow frameworks by Kildall [Kil73] marked the beginning of a new era in compiler optimization. Setting data flow analysis on a sound formal foundation spurred a wealth of research into both generalizing, e.g. KU77, CC77, GW76, Tar81] and specializing Kildall s result, e.g. [KU76, WZ91, Tar81]. This resulted in formalisms for specifying and classifying data flow analysis problems, and the development of algorithms for solving any problem in a given class. This meant that compiler developers no longer had to hand craft a system of equations and a solver for an analysis problem. Instead, ....

Robert Endre Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28(3):594-- 614, July 1981.

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Robert E. Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28:594--614, 1981.

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Robert E. Tarjan. Fast algorithms for solving path problems. Journal of the ACM, 28(3):594-614, July 1981.

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Robert Endre Tarjan. Fast algorithm for solving path problems. Journal of the ACM, 28(3):594--614, July 1981.

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