E-graph
An E-graph (Equality Graph) is a data structure used in various areas of computer science, particularly in program optimization, automated reasoning, and symbolic computation. It efficiently represents a set of expressions and their equivalences.
At its core, an E-graph is a directed acyclic graph (DAG) where nodes, called E-nodes, represent expressions or sub-expressions. Importantly, E-nodes are not just isolated nodes; they are grouped into equivalence classes called E-classes. An E-class represents the set of expressions that are known to be equivalent. Therefore, each E-node belongs to exactly one E-class, and an E-class may contain multiple E-nodes.
The key advantage of E-graphs lies in their ability to compactly represent a large number of equivalent expressions. This compaction is achieved through two main mechanisms:
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Hashing: Similar expressions are automatically merged into the same E-node, preventing redundant representation. This is usually done structurally (e.g., expressions with the same operator and operands are considered identical).
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Equivalence Classes: When an equivalence between two expressions is discovered (e.g., through algebraic simplification rules), their respective E-nodes are merged into the same E-class. This efficiently propagates the equivalence throughout the E-graph.
Maintaining an E-graph involves several operations:
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Adding an expression: This involves checking if an equivalent expression already exists in the graph. If so, the new expression is added as an E-node in the existing E-class. Otherwise, a new E-node and E-class are created.
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Finding an expression: Determining if a particular expression is already represented in the graph.
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Merging E-classes: Combining two E-classes when it is determined that the expressions they represent are equivalent. This operation is central to propagating equivalence information.
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Rebuilding: After significant modifications, the E-graph may need to be rebuilt to ensure that equivalences are fully exploited and the graph remains compact. This process typically involves traversing the graph and applying known simplification rules to merge equivalent expressions.
E-graphs provide a powerful way to represent and reason about expressions, enabling efficient optimization and simplification techniques in various applications. They are particularly effective in scenarios where a large number of equivalent expressions need to be considered.