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1. Flow Graph Theory. Depth-First Ordering. Efficiency of Iterative Algorithms. Reducible Flow Graphs Example: Non-Tree Edges. 1. 3. 5. 2. 4. Retreating. Forward 15. Back Edges. ?An edge is a back edge if its head dominates its tail. ?Theorem: Every back edge is a retreating edge in every DFST of every flow graph.
Most of the execution time is spent in loops - the 90/10 law, which states that 90% of the time is spent in 10% of the code, and only 10% of the time in the remaining 90% of the code. Loop is a graph theoretical term: ? dominance, dominator and postdominators – relations between nodes on the CFG. ? head, back edge
Control Flow Graphs. Nodes. Statements or Basic Blocks. (Maximal sequence of code with branching only allowed at end). Edges. Possible transfer of control. Example: if P then S1 else S2. S3. P. S1. S2. S3. CFG. P predecessor of S1 and S2. S1, S2 sucessors of P. Finding Basic Blocks. Identify Headers. The first
Control Flow Graph: An Example int evensum(int i). { int sum = 0; while (i <= 10) { if (i/2 == 0) sum = sum + i; i++;. } return sum;. } F sum = 0; i <= 10 i/2 == 0 sum = sum + i; i++;. T. T. F entry exit return sum;
The first section of the paper, "Basic Concepts," is primarily a catalog of relevant information about directed graphs; similar information can be found in any introductory material on the subject. (Reference [2], for example, covers this material.) The use of directed graphs to express control flow relationships is also given.
14 Apr 2011 Running example. 1. A = 4. 2 t1 = A * B. 3 L1: t2 = t1 / C. 4 if t2 < W goto L2. 5. M = t1 * k. 6 t3 = M + I. 7 L2: H = I. 8. M = t3 - H. 9 if t3 ? 0 goto L3. 10 Control flow graphs. • Divides statements into basic blocks. • Basic block: a maximal sequence of statements I0, I1, I2, , In such that if Ij and Ij+1 are two
21 Jun 2011 Regarding a program, the flow graph is a representation of its possible control flows. More precisely, nodes correspond to statements and decisions and edges represent the possible flow of control between statements. For example, in C-functions, statements are expressions with a semi-colon such as
14 Jan 2010 the only exit node of the control flow graph. Note that the control flow graph Gf is a graph where each node (except nin and nout) corresponds to one statement in the func- tion f. Figure 1 shows an example where we labelled the nodes with the types of the statements or with “in", “out" for better readability.
(3) x = y + z;. (4) z = 1;. (5) z = x;. • Need to know whether values assigned to x at (1) is never used later (i.e., x is dead at statement (1)). – Obvious for this simple example (with no control flow). – Not obvious for complex flow of control. Page 2. 2. Dead Variable Example. • Add control flow to example: x = y + 1; y = 2 * z;.
(5) i := i + 1. (6) t2 := 4 * i. (7) t3 := a[t2]. (8) if t3 < v goto (5). (9) j := j – 1. (10) t4 := 4 * j. (11) t5 := a[t4]. (12) if t5 > v goto (9). (13) if i >= j goto (23). (14) t6 := 4*i. (15) x := a[t6] Basic Block Example. • How many basic blocks in this code fragment? • What are they? Control Flow Analysis. 6. Basic Block Example. (1) i := m – 1.
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