A new algorithm for identifying loops in decompilation

Loop identification is an essential step of control flow analysis in decompilation. The Classical algorithm for identifying loops is Tarjan's interval-finding algorithm, which is restricted to reducible graphs. Havlak presents one extension of Tarjan's algorithm to deal with irreducible graphs, which constructs a loop-nesting forest for an arbitrary flow graph. There's evidence showing that the running time of this algorithm is quadratic in the worst-case, and not almost linear as claimed. Ramalingam presents an improved algorithm with low time complexity on arbitrary graphs, but it performs not quite well on "real" control flow graphs (CFG). We present a novel algorithm for identifying loops in arbitrary CFGs. Based on a more detailed exploration on properties of loops and depth-first search (DFS), this algorithm traverses a CFG only once based on DFS and collects all information needed on the fly. It runs in approximately linear time and does not use any complicated data structures such as Interval/Derived Sequence of Graphs (DSG) or UNION-FIND sets. To perform complexity analysis of the algorithm, we introduce a new concept called unstructuredness coefficient to describe the unstructuredness of CFGs, and we find that the unstructuredness coefficients of these executables are usually small (<1.5). Such "low-unstructuredness" property distinguishes these CFGs from general single-root connected directed graphs, and it offers an explanation why those algorithms existed perform not quite well on real-world cases. The new algorithm has been applied to 11526 CFGs in 6 typical binary executables on both Linux and Window platforms. Experimental result has validated our theoretical analysis and it shows that our algorithm runs 2-5 times faster than the Havlak-Tarjan algorithm, and 2-8 times faster than the Ramalingam-Havlak-Tarjan algorithm.