IRStrahler

IRStrahler is a metric used in compiler analysis to assign Strahler numbers to nodes in an intermediate representation (IR) graph. It generalizes the classic Strahler number, which originated for trees, to directed graphs that model data dependencies in an IR. The resulting numbers provide a compact summary of the structural complexity of expressions and can help understand resource requirements and optimization opportunities.

Definition and computation: For a node v, if v has no predecessors, its Strahler number S(v) is 1. Let Pred(v) be the set of immediate predecessors of v. Compute m as the maximum Strahler number among the predecessors. If the number of predecessors with S equal to m is at least two, then S(v) = m + 1; otherwise S(v) = m. In cyclic IRs, the common approach is to compute fixed points by breaking cycles or condensing strongly connected components and then propagating values in a topological fashion.

In practice, IRStrahler is often computed on the IR’s data-flow graph, and in SSA form phi nodes are considered by aggregating incoming values from multiple predecessors. The resulting numbers reflect the maximum “level” of nested dependency that must be realized when evaluating expressions.

Applications and limitations: IRStrahler helps estimate register pressure and informs scheduling and allocation decisions, guiding optimizations such as register reuse and instruction ordering. It is an analytical proxy rather than a perfect predictor; actual runtime resource usage depends on the target architecture, memory operations, and the chosen scheduling strategy. The metric is sensitive to the granularity of the IR and the particular graph (data-flow vs control-flow) used for analysis.