Previously, we defined a struct named `RootOrderingCost`, which stored the cost (a pair consisting of the depth of the connector and a tie breaking ID), as well as the connector itself. This created some confusion, because we would sometimes write, e.g., `cost.cost.first` (the first `cost` referring to the struct, the second one referring to the `cost` field, and `first` referring to the depth). In order to address this confusion, here we rename `RootOrderingCost` to `RootOrderingEntry` (keeping the fields and their names as-is).
This clarification exposed non-determinism in the optimal branching algorithm. When choosing the best local parent, we were previuosly only considering its depth (`cost.first`) and not the tie-breaking ID (`cost.second`). This led to non-deterministic choice of the parent when multiple potential parents had the same depth. The solution is to compare both the depth and the tie-breaking ID.
Testing: Rely on existing unit tests. Non-detgerminism is hard to unit-test.
Reviewed By: rriddle, Mogball
Differential Revision: https://reviews.llvm.org/D116079
This is commit 3 of 4 for the multi-root matching in PDL, discussed in https://llvm.discourse.group/t/rfc-multi-root-pdl-patterns-for-kernel-matching/4148 (topic flagged for review).
We form a graph over the specified roots, provided in `pdl.rewrite`, where two roots are connected by a directed edge if the target root can be connected (via a chain of operations) in the underlying pattern to the source root. We place a restriction that the path connecting the two candidate roots must only contain the nodes in the subgraphs underneath these two roots. The cost of an edge is the smallest number of upward traversals (edges) required to go from the source to the target root, and the connector is a `Value` in the intersection of the two subtrees rooted at the source and target root that results in that smallest number of such upward traversals. Optimal root ordering is then formulated as the problem of finding a spanning arborescence (i.e., a directed spanning tree) of minimal weight.
In order to determine the spanning arborescence (directed spanning tree) of minimum weight, we use the [Edmonds' algorithm](https://en.wikipedia.org/wiki/Edmonds%27_algorithm). The worst-case computational complexity of this algorithm is O(_N_^3) for a single root, where _N_ is the number of specified roots. The `pdl`-to-`pdl_interp` lowering calls this algorithm as a subroutine _N_ times (once for each candidate root), so the overall complexity of root ordering is O(_N_^4). If needed, this complexity could be reduced to O(_N_^3) with a more efficient algorithm. However, note that the underlying implementation is very efficient, and _N_ in our instances tends to be very small (<10). Therefore, we believe that the proposed (asymptotically suboptimal) implementation will suffice for now.
Testing: a unit test of the algorithm
Reviewed By: rriddle
Differential Revision: https://reviews.llvm.org/D108549
Stanislav Funiak