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Dataflow analysis is most usefully performed into and out of ''regions'', subsets of the nodes such that a ''header'' exists which dominates all nodes in the region, and all edges between nodes in the region are themselves in the region. A loop is a region which is strongly connected, where all back-edges to the header are themselves within the region '''FIXME -- unclear'''. | Dataflow analysis is most usefully performed into and out of ''regions'', subsets of the nodes such that a ''header'' exists which dominates all nodes in the region, and all edges between nodes in the region are themselves in the region. A loop is a region which is strongly connected, where all back-edges to the header are themselves within the region '''FIXME -- unclear'''. | ||

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> | 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>gotos</tt>)). 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: | ||

* Perform a recursive depth-first traversal of the graph starting from s. Each time you return, add that node onto an auxiliary vector. Upon the traversal's completion, this vector sorts the nodes topologically. | * Perform a recursive depth-first traversal of the graph starting from s. Each time you return, add that node onto an auxiliary vector. Upon the traversal's completion, this vector sorts the nodes topologically. | ||

* Until the the vector is empty, use the last node of the vector to begin traversing the transpose graph. Remove the path from the vector; these paths partition the graph into SCCs. | * Until the the vector is empty, use the last node of the vector to begin traversing the transpose graph. Remove the path from the vector; these paths partition the graph into SCCs. |