
In
computer science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, a control-flow graph (CFG) is a
representation, using
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
notation, of all paths that might be traversed through a
program during its
execution
Capital punishment, also known as the death penalty and formerly called judicial homicide, is the state-sanctioned killing of a person as punishment for actual or supposed misconduct. The sentence ordering that an offender be punished in ...
. The control-flow graph was conceived by
Frances E. Allen, who noted that
Reese T. Prosser used
boolean connectivity matrices for flow analysis before.
The CFG is essential to many
compiler optimization
An optimizing compiler is a compiler designed to generate code that is optimized in aspects such as minimizing program execution time, memory usage, storage size, and power consumption. Optimization is generally implemented as a sequence of op ...
s and
static-analysis tools.
Definition
In a control-flow graph each
node
In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).
Node may refer to:
In mathematics
* Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two or more curves, lines ...
in the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
represents a
basic block
In compiler construction, a basic block is a straight-line code sequence with no branches in except to the entry and no branches out except at the exit. This restricted form makes a basic block highly amenable to analysis. Compilers usually decom ...
, i.e. a straight-line sequence of code with a single entry point and a single exit point, where no branches or jumps occur within the block. Basic blocks start with jump targets and end with jumps or branch instructions. Directed
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
s are used to represent jumps in the
control flow
In computer science, control flow (or flow of control) is the order in which individual statements, instructions or function calls of an imperative program are executed or evaluated. The emphasis on explicit control flow distinguishes an '' ...
. There are, in most presentations, two specially designated blocks: the ''entry block'', through which control enters into the flow graph, and the ''exit block'', through which all control flow leaves.
Because of its construction procedure, in a CFG, every edge A→B has the property that:
:
outdegree(A) > 1 or indegree(B) > 1 (or both).
The CFG can thus be obtained, at least conceptually, by starting from the program's (full) flow graph—i.e. the graph in which every node represents an individual instruction—and performing an
edge contraction
In graph theory, an edge contraction is an operation that removes an edge from a graph while simultaneously merging the two vertices that it previously joined. Edge contraction is a fundamental operation in the theory of graph minors. Vertex id ...
for every edge that falsifies the predicate above, i.e. contracting every edge whose source has a single exit and whose destination has a single entry. This contraction-based algorithm is of no practical importance, except as a visualization aid for understanding the CFG construction, because the CFG can be more efficiently constructed directly from the program by
scanning it for basic blocks.
Example
Consider the following fragment of code:
0: (A) t0 = read_num
1: (A) if t0 mod 2 0
2: (B) print t0 + " is even."
3: (B) goto 5
4: (C) print t0 + " is odd."
5: (D) end program
In the above, we have 4 basic blocks: A from 0 to 1, B from 2 to 3, C at 4 and D at 5. In particular, in this case, A is the "entry block", D the "exit block" and lines 4 and 5 are jump targets. A graph for this fragment has edges from A to B, A to C, B to D and C to D.
Reachability
Reachability
In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s can reach a vertex t (and t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with s a ...
is a graph property useful in optimization.
If a subgraph is not connected from the subgraph containing the entry block, that subgraph is unreachable during any execution, and so is
unreachable code
In computer programming, unreachable code is part of the source code of a program which can never be executed because there exists no control flow path to the code from the rest of the program.
Unreachable code is sometimes also called ''dead code ...
; under normal conditions it can be safely removed.
If the exit block is unreachable from the entry block, an
infinite loop
In computer programming, an infinite loop (or endless loop) is a sequence of instructions that, as written, will continue endlessly, unless an external intervention occurs, such as turning off power via a switch or pulling a plug. It may be inte ...
may exist. Not all infinite loops are detectable, see
Halting problem
In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for ...
. A halting order may also exist there.
Unreachable code and infinite loops are possible even if the programmer does not explicitly code them: optimizations like
constant propagation and
constant folding
Constant folding and constant propagation are related compiler optimizations used by many modern compilers. An advanced form of constant propagation known as sparse conditional constant propagation can more accurately propagate constants and sim ...
followed by
jump threading can collapse multiple basic blocks into one, cause edges to be removed from a CFG, etc., thus possibly disconnecting parts of the graph.
Domination relationship
A block M ''
dominates'' a block N if every path from the entry that reaches block N has to pass through block M. The entry block dominates all blocks.
In the reverse direction, block M ''postdominates'' block N if every path from N to the exit has to pass through block M. The exit block postdominates all blocks.
It is said that a block M ''immediately dominates'' block N if M dominates N, and there is no intervening block P such that M dominates P and P dominates N. In other words, M is the last dominator on all paths from entry to N. Each block has a unique immediate dominator.
Similarly, there is a notion of ''immediate postdominator'', analogous to ''immediate dominator''.
The
''dominator tree'' is an ancillary data structure depicting the dominator relationships. There is an arc from Block M to Block N if M is an immediate dominator of N. This graph is a tree, since each block has a unique immediate dominator. This tree is rooted at the entry block. The dominator tree can be calculated efficiently using
Lengauer–Tarjan's algorithm.
A ''postdominator tree'' is analogous to the ''dominator tree''. This tree is rooted at the exit block.
Special edges
A ''back edge'' is an edge that points to a block that has already been met during a depth-first (
DFS) traversal of the graph. Back edges are typical of loops.
A ''critical edge'' is an edge which is neither the only edge leaving its source block, nor the only edge entering its destination block. These edges must be ''split'': a new block must be created in the middle of the edge, in order to insert computations on the edge without affecting any other edges.
An ''abnormal edge'' is an edge whose destination is unknown.
Exception handling
In computing and computer programming, exception handling is the process of responding to the occurrence of ''exceptions'' – anomalous or exceptional conditions requiring special processing – during the execution of a program. In general, an ...
constructs can produce them. These edges tend to inhibit optimization.
An ''impossible edge'' (also known as a ''fake edge'') is an edge which has been added to the graph solely to preserve the property that the exit block postdominates all blocks. It cannot ever be traversed.
Loop management
A ''loop header'' (sometimes called the ''entry point'' of the loop) is a dominator that is the target of a loop-forming back edge. The loop header dominates all blocks in the loop body. A block may be a loop header for more than one loop. A loop may have multiple entry points, in which case it has no "loop header".
Suppose block M is a dominator with several incoming edges, some of them being back edges (so M is a loop header). It is advantageous to several optimization passes to break M up into two blocks M
pre and M
loop. The contents of M and back edges are moved to M
loop, the rest of the edges are moved to point into M
pre, and a new edge from M
pre to M
loop is inserted (so that M
pre is the immediate dominator of M
loop). In the beginning, M
pre would be empty, but passes like
loop-invariant code motion
In computer programming, loop-invariant code consists of statements or expressions (in an imperative programming, imperative programming language) that can be moved outside the body of a loop without affecting the semantics of the program. Loop-i ...
could populate it. M
pre is called the ''loop pre-header'', and M
loop would be the loop header.
Reducibility
A reducible CFG is one with edges that can be partitioned into two disjoint sets: forward edges, and back edges, such that:
* Forward edges form a
directed acyclic graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ...
with all nodes reachable from the entry node.
* For all back edges (A, B), node B
dominates node A.
Structured programming Structured programming is a programming paradigm aimed at improving the clarity, quality, and development time of a computer program by making specific disciplined use of the structured control flow constructs of selection ( if/then/else) and repet ...
languages are often designed such that all CFGs they produce are reducible, and common structured programming statements such as IF, FOR, WHILE, BREAK, and CONTINUE produce reducible graphs. To produce irreducible graphs, statements such as
GOTO are needed. Irreducible graphs may also be produced by some compiler optimizations.
Loop connectedness
The loop connectedness of a CFG is defined with respect to a given
depth-first search
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible al ...
tree (DFST) of the CFG. This DFST should be rooted at the start node and cover every node of the CFG.
Edges in the CFG which run from a node to one of its DFST ancestors (including itself) are called back edges.
The loop connectedness is the largest number of back edges found in any cycle-free path of the CFG. In a reducible CFG, the loop connectedness is independent of the DFST chosen.
Loop connectedness has been used to reason about the time complexity of
data-flow analysis
Data-flow analysis is a technique for gathering information about the possible set of values calculated at various points in a computer program. It forms the foundation for a wide variety of compiler optimizations and program verification techn ...
.
Inter-procedural control-flow graph
While control-flow graphs represent the control flow of a single procedure, inter-procedural control-flow graphs represent the control flow of whole programs.
See also
*
Abstract syntax tree
An abstract syntax tree (AST) is a data structure used in computer science to represent the structure of a program or code snippet. It is a tree representation of the abstract syntactic structure of text (often source code) written in a formal ...
*
Flowchart
A flowchart is a type of diagram that represents a workflow or process. A flowchart can also be defined as a diagrammatic representation of an algorithm, a step-by-step approach to solving a task.
The flowchart shows the steps as boxes of v ...
*
Control-flow diagram
*
Control-flow analysis
In computer science, control-flow analysis (CFA) is a static code analysis, static-code-analysis technique for determining the control flow of a program. The control flow is expressed as a control-flow graph (CFG). For both functional programming ...
*
Data-flow analysis
Data-flow analysis is a technique for gathering information about the possible set of values calculated at various points in a computer program. It forms the foundation for a wide variety of compiler optimizations and program verification techn ...
*
Interval (graph theory)
*
Program dependence graph
*
Cyclomatic complexity
*
Static single assignment
*
Compiler construction
In computing, a compiler is a computer program that Translator (computing), translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primaril ...
*
Intermediate representation
An intermediate representation (IR) is the data structure or code used internally by a compiler or virtual machine to represent source code. An IR is designed to be conducive to further processing, such as optimization and translation. A "good" ...
References
External links
The Machine-SUIF Control Flow Graph Library*Paper
by
Zdeněk Dvořák ''et al.''
;Examples
{{Webarchive, url=https://web.archive.org/web/20110825221459/http://compilers.cs.ucla.edu/avrora/cfg.html , date=2011-08-25
Compiler construction
*
Application-specific graphs
Modeling languages