SIMD and Graphs: Partitioning Graphs into data-dependency levels - Building


This post is a new episode in a miniseries focused on SIMD instructions. This post focuses on fusing common single operations across multiple data. If you haven't read the first post, I recommend reading the first post on graph matching first. 1. Parse the sympy graph into a subset of pattern matches with partial orderings of patterns that contain (depend upon) other patterns 2. Perform common subexpression elimination to deduplicate computations 3. Run the Coffman-Graham algorithm to get a bounded nearly optimal allocation of the matches (This post) 4. Edit the SIMD compute node in place of matching patterns

SIMD (Single Instruction Multiple Data) looks to speed up performing the same calculation across multiple sets of data. Now that we have operations that match the SIMD operation, how do we know which nodes in the graph we can combine? We need to respect data dependencies to make sure that we don't try to do two operations that depend on each other or perform a computation before its inputs are ready.

Skip to a solution


A brief note on the missteps: These are avenues that I didn't explore or bugs that I ran across. While not immediately helpful, they may form a bridge to future ideas in the series.

A bug when it comes to matching

If the matching provided by the end user is too strict, it can result in an empty set of "ordered nodes". From there, there is no ordering between the following nodes added to the list. The symptom of this is to add nodes to levels in the wrong order and miss dependencies. The easiest place to check for this is at the end of the setup by asserting that there are ordered nodes before trying to add in nodes with dependencies. This was tricky to find because it originally showed up in a way that made it look like the dependency checking was not working.

Graph uniqueness

The algorithm currently doesn't interface well with common subexpression elimination. This would lead to nodes being referred to through multiple paths in the graph and cause the node identification via path to break down.

Crossover between SIMD operations

This algorithm focuses on introducing operations for a single type of SIMD operation. Further design will be needed for choosing between multiple operations. For example, some SIMD operations perform an add and multiply and different SIMD operations operate at different precisions.

A Solution

Github PR

The algorithm of choice for this kind of bundling is the Coffman-Graham algorithm. It's original intent is to schedule minimal-length schedules across multiple processors for a known finite number of processors; however, we can bend it slightly to adapt it to SIMD algorithms where we have multiple processor lanes as long as the data is independent. Even better, the algorithm is demonstrated to be optimal.

The Algorithm

The Coffman-Graham algorithm breaks down into two phases: 1. Topological Sort 2. Leveling

Topological Sort

In the first phase, nodes in the compute graph are ordered based on their dependencies (topological sort) with a special lexicographical ordering: Take nodes that had their dependencies recently added to the sorted ordering before nodes with less recent dependencies. Intuitively, this has the effect of performing a depth-first ordering of the data dependencies to order longer chains earlier and bundle together chains of dependencies.

ordered_nodes, unordered_nodes = setup_coffman_graham(graph, matcher)
assert len(ordered_nodes) > 0

def coffman_graham_filter_criteria(node):
    path_to_expr, expr = node

    for idx in range(len(expr.args)):
        if tuple(path_to_expr + (idx,)) in unordered_nodes:
            # Don't try to insert nodes that depend on unordered nodes
            return False

    # Node only depends on ordered nodes
    return True

def coffman_graham_sort_criteria(node):
    path_to_expr, expr = node
    args_depth_from_end = []

    for arg_idx, arg in enumerate(expr.args):
        arg_path = path_to_expr + (arg_idx,)
        for ordered_idx, (ordered_path, ordered_expr) in enumerate(
            if arg_path == ordered_path:

    return sorted(args_depth_from_end)

while len(unordered_nodes) > 0:
    next_in_order = min(
        filter(coffman_graham_filter_criteria, unordered_nodes.items()),
    path, _expr = next_in_order

    del unordered_nodes[tuple(path)]


In the second phase, nodes are processed in the reversed order from the first phase. Nodes are assigned to the lowest possible level that is at least one level above their downstream dependencies. The ordering of the first phase leads us to the smallest possible level stack (the minimal-length schedule). This is modified slightly to bump dependencies to the next level if the level is full.

levels = []

for idx, node in enumerate(reversed(ordered_nodes)):
    min_level = -1
    for inv_level_idx, level in enumerate(reversed(levels)):
        for maybe_dependency in level:
            if is_dependency_of(node, maybe_dependency):
                min_level = len(levels) - inv_level_idx - 1
                # need to go to a greater level
        if min_level > -1:

    for level_idx in range(min_level + 1, len(levels)):
        if len(levels[level_idx]) < width:
        # We didn't find an acceptable level, so put it at a new higher level

return levels

FilterVisitor Pattern

This solution leans on the visitor pattern and matching structure from the previous post to set up the problem, which I'll refer to as the FilterVisitor.

To start the topological sort, we want to create an ordered list of nodes with no dependencies (leaves, usually symbols and constants) and an unordered map. By visiting the tree and matching our target operations, we can push them into the set of unordered operations and push the leaves into the ordered list.

def visit_sympy_expr(expr, matcher, base=None):
    if base is None:
        base = tuple()

    if matcher(expr):
        yield base, expr

    for idx, arg in enumerate(expr.args):
        for result in visit_sympy_expr(arg, matcher, base + (idx,)):
            yield result

The matcher used for this example is:

def match_Add(expr):
    return expr.func == sympy.core.add.Add or len(expr.args) == 0


LLVM's auto-vectorization focuses on two use cases: 1. Loops 2. superword-level parallelism aka "SLP"

The loop vectorizer case is straightforward: process multiple iterations of a loop in parallel. The use case I'm covering is closer to the SLP case:

combine similar independent instructions into vector instructions


The SLP-vectorizer processes the code bottom-up, across basic blocks, in search of scalars to combine.

According to benchmarks by Michael Larabel from an old version of LLVM (circa 2013), the SLP vectorization isn't as impactful as the loop vectorizer.

Research into Superword Level Parallelism

This page references three papers for integrating superword-level parallelism into compilers (circa 2000). For example, this paper and a follow up project to implement it in LLVM suggest that performance improvements can be gained by being smart about additional optimization combinations applying SLP that goes beyond what the LLVM compiler achieves today.