#### lospre in linear time

##### Philipp Klaus Krause

Lifetime-optimal speculative partial redundancy elimination (lospre) is the most advanced currently known redundancy elimination technique. It subsumes many previously known approaches, such as common subexpression elimination, global common subexpression elimination, and loop-invariant code motion. However, previously known lospre algorithms have high time complexity; faster but less powerful approaches have been used and developed further instead. We present a simple linear-time algorithm for lospre for structured programs that can also handle some more general scenarios compared to previous approaches. We prove that our approach is optimal and that the runtime is linear in the number of nodes in the control-flow graph. The condition on programs of being structured is automatically true for many programming languages and for others, such as C, is equivalent to a bound on the number of goto labels per function. An implementation in a mainstream C compiler demonstrates the practical feasibility of our approach. Our approach is based on graph-structure theory and uses tree-decompositions. We also show that, for structured programs, the runtime of deterministic implementations of the previously known MC-PRE and MC-SSAPRE algorithms can be bounded by $O(n^{2.5})$, improving the previous bounds of $O(n^3)$.

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