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Compiler Design: Difference between revisions
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The directed multigraph defined by interpreting basic blocks as vertices, and flow relationships as edges, yields its control flow graph (CFG). A start node exists for each CFG, corresponding to the basic block whose header is the first instruction of the program. | The directed multigraph defined by interpreting basic blocks as vertices, and flow relationships as edges, yields its control flow graph (CFG). A start node exists for each CFG, corresponding to the basic block whose header is the first instruction of the program. | ||
The antisymmetric, transitive, reflexive ''domination relation'' is | The antisymmetric, transitive, reflexive ''domination relation'' is defined on vertices of a CFG (and thus basic blocks of the underlying program). A vertex a ''dominates'' b (a <= b) if every path from the start node s to b passes through a. A vertex a ''properly dominates'' b (a < b) if a dominates and is not equal to b. A vertex a ''directly/immediately dominates'' b (a <d b) if a properly dominates b, and a dominates no vertex c that dominates b. This relation induces the [[Trees|dominator tree]], where nodes dominate all descendents in the tree. The start node s dominates all nodes, properly dominates all nodes but itself, and roots the dominator tree. | ||
* It should be obvious that a's preceding of b in the CFG is not necessary for even immediate dominance of b by a. | * It should be obvious that a's preceding of b in the CFG is not necessary for even immediate dominance of b by a. | ||
Loops can be discovered via domination analysis (it is important to note that this refers to loops in the generated code, not loop constructs of the source language, and furthermore that all possible loops will be found (ie, unstructured loops constructed from C <tt>goto</tt>s)). Discover all strongly-connected subgraphs (SCCs) of the CFG (subgraphs where, for each vertex, a path ('''not''' necessarily an edge) exists from that vertex to all other nodes of the subgraph); if a subgraph contains a node dominating all that subgraph's nodes, the subgraph is a loop. The trivial case is, of course, a statement which jumps to itself, ala the BASIC program <tt>10 GOTO 10</tt>. Implementation via [[Kosaraju's Algorithm]] is simple, with O(|V|+|E|) time complexity using graph encoding and O(N<sup>2</sup>) time complexity using adjacency matrices: | Loops can be discovered via domination analysis (it is important to note that this refers to loops in the generated code, not loop constructs of the source language, and furthermore that all possible loops will be found (ie, unstructured loops constructed from C <tt>goto</tt>s)). Discover all strongly-connected subgraphs (SCCs) of the CFG (subgraphs where, for each vertex, a path ('''not''' necessarily an edge) exists from that vertex to all other nodes of the subgraph); if a subgraph contains a node dominating all that subgraph's nodes, the subgraph is a loop. The trivial case is, of course, a statement which jumps to itself, ala the BASIC program <tt>10 GOTO 10</tt>. Implementation via [[Kosaraju's Algorithm]] is simple, with O(|V|+|E|) time complexity using graph encoding and O(N<sup>2</sup>) time complexity using adjacency matrices: | ||
