Reducible Flow Graphs Research Articles

An abstract interpretation for SPMD divergence on reducible control flow graphs

Vectorizing compilers employ divergence analysis to detect at which program point a specific variable is uniform, i.e. has the same value on all SPMD threads that execute this program point. They exploit uniformity to retain branching to counter branch divergence and defer computations to scalar processor units. Divergence is a hyper-property and is closely related to non-interference and binding time. There exist several divergence, binding time, and non-interference analyses already but they either sacrifice precision or make significant restrictions to the syntactical structure of the program in order to achieve soundness. In this paper, we present the first abstract interpretation for uniformity that is general enough to be applicable to reducible CFGs and, at the same time, more precise than other analyses that achieve at least the same generality. Our analysis comes with a correctness proof that is to a large part mechanized in Coq. Our experimental evaluation shows that the compile time and the precision of our analysis is on par with LLVM's default divergence analysis that is only sound on more restricted CFGs. At the same time, our analysis is faster and achieves better precision than a state-of-the-art non-interference analysis that is sound and at least as general as our analysis.

On feedback vertex set in reducible flow hypergraphs

A directed hypergraph H = (V, A) is a finite set of vertices V and a set of hyper-arcs A, where each hyper-arc is an ordered pair of nonempty subsets of vertices. A flow hypergraph H = (V, A, s) is a triple, such that (V, A) is a directed hypergraph, s e V is a distinguished vertex such that s reaches every vertex of V. Reducible flow hypergraphs are a generalization of Hecht and Ullman’s reducible flowgraphs. The feedback vertex set (fvs) decision problem has a directed hypergraph H and an integer k ≥ 0 as input and the question is whether there is V'⊆V, |V' |≤k such that H\\V' is an acyclic directed hypergraph. It is known that fvs is polynomial time solvable for reducible flowgraphs. In this article we prove that fvs is NP-complete for reducible flow hypergraphs showing a reduction from 3-satisfiability problem with at most 3 occurrences per variable (3sat3-). We exhibit a polynomial-time ∆-approximation for fvs in reducible flow hypergraphs, where ∆ is the maximum number of hyper-arcs adjacent to a vertex of H.

A [formula omitted]–[formula omitted] relation in flowgraphs and some applications

A flowgraph G=(V,A,s) is a digraph such that there is a dipath from s to u for every vertex u in G. A vertex w dominates a vertex u if and only if all dipaths in G from s to u pass through w. The vertex s dominates trivially any vertex in G. In this work, we define two sets called dominator cover and junction partition. A dominator cover in G is a set of vertices that includes s and non-trivial dominators for all vertices in G. A junction partition B={B1,…,Bk} is a partition of V which satisfies the following property: if vertex u belongs to Bi and vertex v belongs to Bj (i≠j), then there are internally vertex-disjoint dipaths from s to u and from s to v, for short, s is a junction of u and v. The first part of this work shows that, in flowgraphs, the minimum size of a dominator cover is equal to the maximum size of a junction partition. The second part describes some applications for this relation, such as, a new correctness proof of an algorithm that finds a maximum junction partition in reducible flowgraphs; and good time and space complexity algorithms for problems related to junctions and nearest common ancestors in acyclic digraphs.

Descending chains and narrowing on template abstract domains

A static analysis by abstract interpretation is typically composed of an ascending phase followed by a descending one. The descending phase is used to improve the precision of the analysis after that a post-fixpoint has been reached. Termination is often guaranteed by using narrowing operators, especially on numerical domains which are generally endowed with infinite descending chains. Under the hypothesis of dealing with reducible flow graphs, we provide an abstract semantics which improves the analysis precision and we show that, for a large class of numerical abstract domains over integer variables (such as intervals, octagons, template parallelotopes and template polyhedra), infinite descending chains cannot arise and we can safely omit narrowing. The abstract semantics is a slight variation of the standard one and can be easily implemented. We also provide an acceleration procedure which ensures termination of the descending phase without narrowing even with non-reducible graphs. Finally, we propose a new family of weak narrowing operators for real variables which improve the analysis precision.

Packing feedback arc sets in reducible flow graphs

In this paper we establish a min---max relation in arc-weighted reducible flow graphs. In particular, we prove that the maximum cardinality of feedback arc set packings equals the minimum total weight of cycles. We also present an $$O(n^2 m)$$O(n2m) algorithm for finding a maximum feedback arc set packing in reducible flow graphs.

A symbolic analysis framework for static analysis of imperative programming languages

We present a generic symbolic analysis framework for imperative programming languages. Our framework is capable of computing all valid variable bindings of a program at given program points. This information is invaluable for domain-specific static program analyses such as memory leak detection, program parallelization, and the detection of superfluous bound checks, variable aliases and task deadlocks.We employ path expression algebra to model the control flow information of programs. A homomorphism maps path expressions into the symbolic domain. At the center of the symbolic domain is a compact algebraic structure called supercontext. A supercontext contains the complete control and data flow analysis information valid at a given program point.Our approach to compute supercontexts is based purely on algebra and is fully automated. This novel representation of program semantics closes the gap between program analysis and computer algebra systems, which makes supercontexts an ideal symbolic intermediate representation for all domain-specific static program analyses.Our approach is more general than existing methods because it can derive solutions for arbitrary (even intra-loop and nested loop) nodes of reducible and irreducible control flow graphs. We prove the correctness of our symbolic analysis method. Our experimental results show that the problem sizes arising from real-world applications such as the SPEC95 benchmark suite are tractable for our symbolic analysis framework.

Flow hypergraph reducibility

Reducible flowgraphs were first defined by Allen in terms of intervals; another definition based on two flowgraph transformations was presented by Hecht and Ullman. In this paper, we extend the notion of reducibility to directed hypergraphs, proving that the interval and the transformation approaches preserve the equivalence when applied to this family.

Flow Hypergraph Reducibility

Abstract Reducible flow graphs were first defined by Hecht and Ullman in terms of intervals; another definition, based on two flow graph transformations, was also presented. In this paper, we extend the notion of reducibility to directed hypergraphs, proving that the interval and the transformation approaches are still equivalent when applied to this family.

An Efficient Algorithm for Finding Maximum Cycle Packings in Reducible Flow Graphs

Reducible flow graphs occur naturally in connection with flowcharts of computer programs and are used extensively for code optimization and global data flow analysis. In this paper we present an O(n2 log(n2/m)) algorithm for finding a maximum cycle packing in any weighted reducible flow graph with n vertices and m arcs; our algorithm heavily relies on Ramachandran's earlier work concerning reducible flow graphs.

Solving problems for maximal reducible flowgraphs

In this paper, the family of Maximal Reducible Flowgraphs (MRFs) is recursively defined, based on a decomposition theorem. A one-to-one association between MRFs and extended binary trees allows to deduce some numerical properties of the family. Hamiltonian problems, testing isomorphism and finding a minimum cardinality feedback arc set are efficiently solved for MRFs. The results concerning hamiltonian paths and cycles also hold for reducible flowgraphs.

경로 압축을 이용한 DJ 그래프의 지연 감축 알고리즘

효과적이고 정확한 데이터 흐름 문제 분석은 흐름그래프와 지배자 트리 그리고 DJ 그래프를 사용한다. 데이터 흐름 문제 해결은 흐름 그래프를 안전하게 지배자 트리로 감축하는 것이다. 흐름 그래프는 파스 트리를 대신하고, DJ 그래프는 감축 가능하거나 감축이 불가능한 흐름 그래프를 지배자 트리로 정확하게 감축하는데 이용된다. 본 연구에서는 Tarjan의 경로 압축 알고리즘을 이용하기 위하여 Top 노드 찾기 알고리즘을 제시하고 기존의 지연감축 알고리즘을 경로압축을 이용하여 개선한다. 경로압축을 이용한 지연감축 알고리즘은 DJ 그래프를 지연 감축하면서 노드를 끌어올려 지배자 트리의 경로를 압축시킨다. 실제로 제안된 알고리즘은 22% 정도 노드들을 끌어올렸고, 20% 정도 경로를 압축시켰다. 압축된 지배자 트리는 효과적인 데이터 흐름 분석을 가능하게 하고, 코드 최적화 과정의 노드 끌어올리기 효과를 가져와 코드 최적화 과정의 복잡도를 개선하는 효과를 가져온다. The effective and accurate data flow problem analysis uses the dominator tree and DJ graphs. The data flow problem solving is to safely reduce the flow graph to the dominator tree. The flow graph replaces a parse tree and used to accurately reduce either reducible or irreducible flow graph to the dominator tree. In this paper, in order to utilize Tarian's path compress algorithm, the Top node finding algorithm is suggested and the existing delay reduction algorithm is improved using Path compression. The delayed reduction a1gorithm using path compression actually compresses the pathway of the dominator tree by hoisting the node while reducing to delay the DJ graph. Realty, the suggested algorithm had hoisted nodes in 22% and had compressed path in 20%. The compressed dominator tree makes it possible to analyze the effective data flow analysis and brings the improved effect for the complexity of code optimization process with the node hoisting effect of code optimization process.

Solving Problems for Maximal Reducible Flowgraphs

In this paper, the family of Maximal Reducible Flowgraphs (MRFs) is recursively defined, based on a decomposition theorem. A one-to-one association between MRFs and extended binary trees allows to deduce some numerical properties of the family. Hamiltonian problems, testing isomorphism and finding a minimum cardinality feedback arc set are efficiently solved for MRFs. The results concerning hamiltonian paths and cycles also hold for reducible flowgraphs.

Making graphs reducible with controlled node splitting

Several compiler optimizations, such as data flow analysis, the exploitation of instruction-level parallelism (ILP), loop transformations, and memory disambiguation, require programs with reducible control flow graphs. However, not all programs satisfy this property. A new method for transforming irreducible control flow graphs to reducible control flow graphs, called Controlled Node Splitting (CNS), is presented. CNS duplicates nodes of the control flow graph to obtain reducible control flow graphs. CNS results in a minimum number of splits and a minimum number of duplicates. Since the computation time to find the optimal split sequence is large, a heuristic has been developed. The results of this heuristic are close to the optimum. Straightforward application of node splitting resulted in an average code size increase of 235% per procedure of our benchmark programs. CNS with the heuristic limits this increase to only 3%. The impact on the total code size of the complete programs is 13.6% for a straightforward application of node splitting. However, when CNS is used, with the heuristic the average growth in code size of a complete program dramatically reduces to 0.2%

Parallel Algorithms for Reducible Flow Graphs

We present parallel NC algorithms for recognizing a reducible flow graph (rfg) and for finding dominators, minimum feedback vertex sets, and a depth first search tree in an rfg. On ann-node rfg, all of these algorithms run inpolylog(n) time usingM(n) processors, whereM(n) is the number of processors needed to multiply twon×nmatrices in polylog time. We show that finding a minimum feedback vertex set in vertex-weighted rfg's or finding a minimum feedback arc set in arc-weighted rfg's is P-complete. For arc or vertex weights in unary, we present RNC algorithms for these problems and show that the problems are in NC if and only if the problem of finding a minimum cut in a flow network is in NC.

Incremental computation of dominator trees

article Free Access Share on Incremental computation of dominator trees Authors: Vugranam C. Sreedhar Hewlett-Packard Company, Cupertino, CA Hewlett-Packard Company, Cupertino, CAView Profile , Guang R. Gao McGill Univ., Montre´al, P.Q., Canada McGill Univ., Montre´al, P.Q., CanadaView Profile , Yong-Fong Lee Intel Corporation, Santa Clara, CA Intel Corporation, Santa Clara, CAView Profile Authors Info & Claims ACM Transactions on Programming Languages and SystemsVolume 19Issue 2March 1997 pp 239–252https://doi.org/10.1145/244795.244799Published:03 March 1997Publication History 14citation600DownloadsMetricsTotal Citations14Total Downloads600Last 12 Months40Last 6 weeks5 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Incremental computation of dominator trees

Data flow analysis based on an incremental approach may require that the dominator tree be correctly maintained at all times. Previous solutions to the problem of incrementally maintaining dominator trees were restricted to reducible flowgraphs. In this paper we present a new algorithm for incrementally maintaining the dominator tree of an arbitrary flowgraph, either reducible or irreducible, based on a program representation called the DJ-graph. For the case where an edge is inserted, our algorithm is also faster than previous approaches (in the worst case). For the deletion case, our algorithm is likely to run fast on the average cases.

PARALLEL DOMINATOR COMPUTATION ON A RAP

ABSTRACT Two constant time algorithms, which are based on the reflexive transitive closure of a directed graph, are proposed for computing dominators and dominator tree of a flow graph respectively. The parallel computation model used is a reconfigurable array of processors. A reconfigurable array of processors is defined to be an array of processors connected to a reconfigurable bus system whose configuration can be dynamically changed. Other applications that are based on the proposed algorithms are also solved in a constant time respectively. These problems include finding the back edges in a flow graph, recognizing the acyclic directed graph, recognizing the reducible flow graph, finding the natural loops, and finding the inner loop.

Efficient parallel algorithms for path problems in directed graphs

In this paper we describe a technique for finding efficient parallel algorithms for problems on directed graphs that involve checking the existence of certain kinds of paths in the graph. This technique provides efficient algorithms for finding dominators in flow graphs, performing interval and loop analysis on reducible flow graphs, and finding the feedback vertices of a digraph. Each of these algorithms takesO(log2 n) time using the same number of processors needed for fast matrix multiplication. All of these bounds are for an EREW PRAM.

A Minimax Arc Theorem for Reducible Flow Graphs

A conjecture of Frank and Gyarfas is established by proving that the cardinality of a minimum feedback arc set in a reducible flow graph is equal to the cardinality of a maximum collection of arc disjoint cycles.

On locating minimum feedback vertex sets

In this paper we study the problem of locating minimum feedback vertex sets in directed graphs. First, we introduce three new transformation-based classes of graphs for which minimum feedback vertex sets can be computed in polynomial time. Second, we delineate an inclusion hierarchy among all of the classes of graphs for which polynomial time feedback vertex set algorithms presently exist. Among the classes of graphs included in the hierarchy are: the reducible flow graphs, the cyclically reducible graphs, the three transformation-based classes that we introduce, and an infinite sequence of classes based on an algorithm of Smith and Walford for finding minimum feedback vertex sets in arbitrary graphs. It follows from our results that one of our new classes, as well as each “Smith/Walford” class, properly includes both the reducible flow graphs and the cyclically reducible graphs. The results presented here serve to unify and focus the work on locating minimum feedback vertex sets in polynomial time.