The CLRS Algorithmic Reasoning Benchmark

Learning representations of algorithms is an emerging area of machine learning, seeking to bridge concepts from neural networks with classical algorithms. The CLRS Algorithmic Reasoning Benchmark (CLRS) consolidates and extends previous work torward evaluation algorithmic reasoning by providing a suite of implementations of classical algorithms. These algorithms have been selected from the third edition of the standard Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein.

Getting started

The CLRS Algorithmic Reasoning Benchmark can be installed with pip, either from PyPI:

pip install dm-clrs

or directly from GitHub (updated more frequently):

pip install git+git://github.com/deepmind/clrs.git

You may prefer to install it in a virtual environment if any requirements clash with your Python installation:

python3 -m venv clrs_env
source clrs_env/bin/activate
pip install git+git://github.com/deepmind/clrs.git

Once installed you can run our example baseline model:

python3 -m clrs.examples.run

If this is the first run of the example, the dataset will be downloaded and stored in --dataset_path (default '/tmp/CLRS30'). Alternatively, you can also download and extract https://storage.googleapis.com/dm-clrs/CLRS30_v1.0.0.tar.gz

Algorithms as graphs

CLRS implements the selected algorithms in an idiomatic way, which aligns as closely as possible to the original CLRS 3ed pseudocode. By controlling the input data distribution to conform to the preconditions we are able to automatically generate input/output pairs. We additionally provide trajectories of "hints" that expose the internal state of each algorithm, to both optionally simplify the learning challenge and to distinguish between different algorithms that solve the same overall task (e.g. sorting).

In the most generic sense, algorithms can be seen as manipulating sets of objects, along with any relations between them (which can themselves be decomposed into binary relations). Accordingly, we study all of the algorithms in this benchmark using a graph representation. In the event that objects obey a more strict ordered structure (e.g. arrays or rooted trees), we impose this ordering through inclusion of predecessor links.

How it works

For each algorithm, we provide a canonical set of train, eval and test trajectories for benchmarking out-of-distribution generalization.

Here, "problem size" refers to e.g. the length of an array or number of nodes in a graph, depending on the algorithm. "multiplier" is an algorithm-specific factor that increases the number of available eval and test trajectories to compensate for paucity of evaluation signals. "multiplier" is 1 for all algorithms except:

• Maximum subarray (Kadane), for which "multiplier" is 32.
• Quick select, minimum, binary search, string matchers (both naive and KMP), and segment intersection, for which "multiplier" is 64.

The trajectories can be used like so:

train_ds, num_samples, spec = clrs.create_dataset(
      folder='/tmp/CLRS30', algorithm='bfs',
      split='train', batch_size=32)

for i, feedback in enumerate(train_ds.as_numpy_iterator()):
  if i == 0:
    model.init(feedback.features, initial_seed)
  loss = model.feedback(rng_key, feedback.features)

Here, feedback is a namedtuple with the following structure:

Feedback = collections.namedtuple('Feedback', ['features', 'outputs'])
Features = collections.namedtuple('Features', ['inputs', 'hints', 'lengths'])

where the content of Features can be used for training and outputs is reserved for evaluation. Each field of the tuple is an ndarray with a leading batch dimension. Because hints are provided for the full algorithm trajectory, these contain an additional time dimension padded up to the maximum length max(T) of any trajectory within the dataset. The lengths field specifies the true length t <= max(T) for each trajectory, which can be used e.g. for loss masking.

Please see the examples directory for full working Graph Neural Network (GNN) examples using JAX and the DeepMind JAX Ecosystem of libraries.

What we provide

Algorithms

Our initial CLRS-30 benchmark includes the following 30 algorithms. We aim to support more algorithms in the future.

• Sorting
  • Insertion sort
  • Bubble sort
  • Heapsort (Williams, 1964)
  • Quicksort (Hoare, 1962)
• Searching
  • Minimum
  • Binary search
  • Quickselect (Hoare, 1961)
• Divide and conquer
  • Maximum subarray (Kadane's variant) (Bentley, 1984)
• Greedy
  • Activity selection (Gavril, 1972)
  • Task scheduling (Lawler, 1985)
• Dynamic programming
  • Matrix chain multiplication
  • Longest common subsequence
  • Optimal binary search tree (Aho et al., 1974)
• Graphs
  • Depth-first search (Moore, 1959)
  • Breadth-first search (Moore, 1959)
  • Topological sorting (Knuth, 1973)
  • Articulation points
  • Bridges
  • Kosaraju's strongly connected components algorithm (Aho et al., 1974)
  • Kruskal's minimum spanning tree algorithm (Kruskal, 1956)
  • Prim's minimum spanning tree algorithm (Prim, 1957)
  • Bellman-Ford algorithm for single-source shortest paths (Bellman, 1958)
  • Dijkstra's algorithm for single-source shortest paths (Dijkstra et al., 1959)
  • Directed acyclic graph single-source shortest paths
  • Floyd-Warshall algorithm for all-pairs shortest-paths (Floyd, 1962)
• Strings
  • Naïve string matching
  • Knuth-Morris-Pratt (KMP) string matcher (Knuth et al., 1977)
• Geometry
  • Segment intersection
  • Graham scan convex hull algorithm (Graham, 1972)
  • Jarvis' march convex hull algorithm (Jarvis, 1973)

Baselines

Models consist of a processor and a number of encoders and decoders. We provide JAX implementations of the following GNN baseline processors:

• Deep Sets (Zaheer et al., NIPS 2017)
• End-to-End Memory Networks (Sukhbaatar et al., NIPS 2015)
• Graph Attention Networks (Veličković et al., ICLR 2018)
• Graph Attention Networks v2 (Brody et al., ICLR 2022)
• Message-Passing Neural Networks (Gilmer et al., ICML 2017)
• Pointer Graph Networks (Veličković et al., NeurIPS 2020)

If you want to implement a new processor, the easiest way is to add it in the processors.py file and make it available through the get_processor_factory method there. A processor should have a __call__ method like this:

__call__(self,
         node_fts, edge_fts, graph_fts,
         adj_mat, hidden,
         nb_nodes, batch_size)

where node_fts, edge_fts and graph_fts will be float arrays of shape batch_size x nb_nodes x H, batch_size x nb_nodes x nb_nodes x H, and batch_size x H with encoded features for nodes, edges and graph respectively, adj_mat a batch_size x nb_nodes x nb_nodes boolean array of connectivity built from hints and inputs, and hidden a batch_size x nb_nodes x H float array with the previous-step outputs of the processor. The method should return a batch_size x nb_nodes x H float array.

For more fundamentally different baselines, it is necessary to create a new class that extends the Model API (as found within clrs/_src/model.py). clrs/_src/baselines.py provides one example of how this can be done.

Creating your own dataset

We provide a tensorflow_dataset generator class in dataset.py. This file can be modified to generate different versions of the available algorithms, and it can be built by using tfds build after following the installation instructions at https://www.tensorflow.org/datasets.

Alternatively, you can generate samples without going through tfds by instantiating samplers with the build_sampler method in clrs/_src/samplers.py, like so:

sampler, spec = clrs.build_sampler(
    name='bfs',
    seed=42,
    num_samples=1000,
    length=16)

def _iterate_sampler(batch_size):
  while True:
    yield sampler.next(batch_size)

for feedback in _iterate_sampler(batch_size=32):
  ...

Adding new algorithms

Adding a new algorithm to the task suite requires the following steps:

1. Determine the input/hint/output specification of your algorithm, and include it within the SPECS dictionary of clrs/_src/specs.py.
2. Implement the desired algorithm in an abstractified form. Examples of this can be found throughout the clrs/_src/algorithms/ folder.

• Choose appropriate moments within the algorithm’s execution to create probes that capture the inputs, outputs and all intermediate state (using the probing.push function).
• Once generated, probes must be formatted using the probing.finalize method, and should be returned together with the algorithm output.

3. Implement an appropriate input data sampler for your algorithm, and include it in the SAMPLERS dictionary within clrs/_src/samplers.py.

Once the algorithm has been added in this way, it can be accessed with the build_sampler method, and will also be incorporated to the dataset if regenerated with the generator class in dataset.py, as described above.

Citation

To cite the CLRS Algorithmic Reasoning Benchmark:

@article{deepmind2022clrs,
  title={The CLRS Algorithmic Reasoning Benchmark},
  author={Petar Veli\v{c}kovi\'{c} and Adri\`{a} Puigdom\`{e}nech Badia and
    David Budden and Razvan Pascanu and Andrea Banino and Misha Dashevskiy and
    Raia Hadsell and Charles Blundell},
  journal={arXiv preprint arXiv:2205.15659},
  year={2022}
}