1//===- RootOrdering.h - Optimal root ordering ------------------*- C++ -*-===//
2//
3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4// See https://llvm.org/LICENSE.txt for license information.
5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6//
7//===----------------------------------------------------------------------===//
8//
9// This file contains definition for a cost graph over candidate roots and
10// an implementation of an algorithm to determine the optimal ordering over
11// these roots. Each edge in this graph indicates that the target root can be
12// connected (via a chain of positions) to the source root, and their cost
13// indicates the estimated cost of such traversal. The optimal root ordering
14// is then formulated as that of finding a spanning arborescence (i.e., a
15// directed spanning tree) of minimal weight.
16//
17//===----------------------------------------------------------------------===//
18
19#ifndef MLIR_LIB_CONVERSION_PDLTOPDLINTERP_ROOTORDERING_H_
20#define MLIR_LIB_CONVERSION_PDLTOPDLINTERP_ROOTORDERING_H_
21
22#include "mlir/IR/Value.h"
23#include "llvm/ADT/DenseMap.h"
24#include "llvm/ADT/SmallVector.h"
25#include <functional>
26#include <vector>
27
28namespace mlir {
29namespace pdl_to_pdl_interp {
30
31/// The information associated with an edge in the cost graph. Each node in
32/// the cost graph corresponds to a candidate root detected in the pdl.pattern,
33/// and each edge in the cost graph corresponds to connecting the two candidate
34/// roots via a chain of operations. The cost of an edge is the smallest number
35/// of upward traversals required to go from the source to the target root, and
36/// the connector is a `Value` in the intersection of the two subtrees rooted at
37/// the source and target root that results in that smallest number of upward
38/// traversals. Consider the following pattern with 3 roots op3, op4, and op5:
39///
40/// argA ---> op1 ---> op2 ---> op3 ---> res3
41/// ^ ^
42/// | |
43/// argB argC
44/// | |
45/// v v
46/// res4 <--- op4 op5 ---> res5
47/// ^ ^
48/// | |
49/// op6 op7
50///
51/// The cost of the edge op3 -> op4 is 1 (the upward traversal argB -> op4),
52/// with argB being the connector `Value` and similarly for op3 -> op5 (cost 1,
53/// connector argC). The cost of the edge op4 -> op3 is 3 (upward traversals
54/// argB -> op1 -> op2 -> op3, connector argB), while the cost of edge op5 ->
55/// op3 is 2 (uwpard traversals argC -> op2 -> op3). There are no edges between
56/// op4 and op5 in the cost graph, because the subtrees rooted at these two
57/// roots do not intersect. It is easy to see that the optimal root for this
58/// pattern is op3, resulting in the spanning arborescence op3 -> {op4, op5}.
59struct RootOrderingEntry {
60 /// The depth of the connector `Value` w.r.t. the target root.
61 ///
62 /// This is a pair where the first value is the additive cost (the depth of
63 /// the connector), and the second value is a priority for breaking ties
64 /// (with 0 being the highest). Typically, the priority is a unique edge ID.
65 std::pair<unsigned, unsigned> cost;
66
67 /// The connector value in the intersection of the two subtrees rooted at
68 /// the source and target root that results in that smallest depth w.r.t.
69 /// the target root.
70 Value connector;
71};
72
73/// A directed graph representing the cost of ordering the roots in the
74/// predicate tree. It is represented as an adjacency map, where the outer map
75/// is indexed by the target node, and the inner map is indexed by the source
76/// node. Each edge is associated with a cost and the underlying connector
77/// value.
78using RootOrderingGraph = DenseMap<Value, DenseMap<Value, RootOrderingEntry>>;
79
80/// The optimal branching algorithm solver. This solver accepts a graph and the
81/// root in its constructor, and is invoked via the solve() member function.
82/// This is a direct implementation of the Edmonds' algorithm, see
83/// https://en.wikipedia.org/wiki/Edmonds%27_algorithm. The worst-case
84/// computational complexity of this algorithm is O(N^3), for a single root.
85/// The PDL-to-PDLInterp lowering calls this N times (once for each candidate
86/// root), so the overall complexity root ordering is O(N^4). If needed, this
87/// could be reduced to O(N^3) with a more efficient algorithm. However, note
88/// that the underlying implementation is very efficient, and N in our
89/// instances tends to be very small (<10).
90class OptimalBranching {
91public:
92 /// A list of edges (child, parent).
93 using EdgeList = std::vector<std::pair<Value, Value>>;
94
95 /// Constructs the solver for the given graph and root value.
96 OptimalBranching(RootOrderingGraph graph, Value root);
97
98 /// Runs the Edmonds' algorithm for the current `graph`, returning the total
99 /// cost of the minimum-weight spanning arborescence (sum of the edge costs).
100 /// This function first determines the optimal local choice of the parents
101 /// and stores this choice in the `parents` mapping. If this choice results
102 /// in an acyclic graph, the function returns immediately. Otherwise, it
103 /// takes an arbitrary cycle, contracts it, and recurses on the new graph
104 /// (which is guaranteed to have fewer nodes than we began with). After we
105 /// return from recursion, we redirect the edges to/from the contracted node,
106 /// so the `parents` map contains a valid solution for the current graph.
107 unsigned solve();
108
109 /// Returns the computed parent map. This is the unique predecessor for each
110 /// node (root) in the optimal branching.
111 const DenseMap<Value, Value> &getRootOrderingParents() const {
112 return parents;
113 }
114
115 /// Returns the computed edges as visited in the preorder traversal.
116 /// The specified array determines the order for breaking any ties.
117 EdgeList preOrderTraversal(ArrayRef<Value> nodes) const;
118
119private:
120 /// The graph whose optimal branching we wish to determine.
121 RootOrderingGraph graph;
122
123 /// The root of the optimal branching.
124 Value root;
125
126 /// The computed parent mapping. This is the unique predecessor for each node
127 /// in the optimal branching. The keys of this map correspond to the keys of
128 /// the outer map of the input graph, and each value is one of the keys of
129 /// the inner map for this node. Also used as an intermediate (possibly
130 /// cyclical) result in the optimal branching algorithm.
131 DenseMap<Value, Value> parents;
132};
133
134} // namespace pdl_to_pdl_interp
135} // namespace mlir
136
137#endif // MLIR_CONVERSION_PDLTOPDLINTERP_ROOTORDERING_H_
138

source code of mlir/lib/Conversion/PDLToPDLInterp/RootOrdering.h