[CINN][Add Backend Pass Comment No.1] Add comment for EliminateCommonFactorOfLocalIndex

paddle/cinn/optim/eliminate_common_factor_of_local_index.h

@@ -20,9 +20,78 @@ namespace cinn {
 namespace optim {
 
 /**
- * Given Expr AST, analyze the Greatest Common Divisor (GCD) of local variable
- * indexes. Then each local index divides it's GCD value. This optimization
- * could help analysising the space allocated for local variables.
+ * Simplifies index expressions for local memory access in GPU kernels.
+ *
+ * This pass optimizes scenarios where GPU kernels use complex index expressions
+ * to access local memory. It's particularly effective for compute-intensive
+ * algorithms that heavily utilize shared memory.
+ *
+ * The pass analyzes and simplifies index expressions for local GPU memory
+ * accesses. It extracts common factors (GCD, offset, or symbolic) and
+ * transforms indices into a more efficient iterator-based form. This
+ * optimization affects Load and Store operations targeting GPU local memory.
+ *
+ * Execution flow:
+ * 1. Collect all index expressions for local GPU memory accesses.
+ * 2. Apply three types of common factor elimination:
+ *    a) GCD (Greatest Common Divisor) elimination.
+ *       e.g., ([2i, 4i], [4i, 3i]) -> ([i, 4i], [2i, 3i])
+ *    b) Offset elimination.
+ *       e.g., ([i+2, i+3], [i+4, i+6]) -> ([i, i], [i+2, i+3])
+ *    c) Symbolic common factor elimination.
+ *       e.g., ([C, 2], [3C, 4]) -> ([1, 2], [3, 4])
+ * 3. Transform simplified indices into iterator-based forms.
+ *       e.g., [i, 0, 0] -> [0, 0, i]
+ * 4. Update the IR, replacing original indices with simplified versions.
+ *
+ * Key benefits:
+ * 1. Reduces computational overhead in index calculations.
+ * 2. Decreases the required size of local memory buffers.
+ *
+ * Safety and correctness:
+ * This optimization is safe because local tensors are initialized within the
+ * function, and the IR doesn't depend on their values before initialization.
+ * The pass ensures that equivalent indices before simplification remain
+ * equivalent after, and distinct indices remain distinct.
+ *
+ * Performance impact:
+ * - Simplifies complex index calculations, reducing arithmetic operations.
+ * - Can reduce the size of local memory buffers, e.g., transforming indices
+ *   [2i] and [4i] to [i] and [2i], potentially halving the required buffer
+ * size.
+ * - Works well with subsequent optimizations like ResizeBufferToMaxVarRange for
+ *   further memory savings.
+ *
+ * Example 1: GCD elimination
+ * Input:
+ *   for (int i = 0; i < 100; i++) {
+ *     local_tensor[4*i, 0, 0] = global_tensor[i, 0, 0];
+ *     local_tensor[2*i, 0, 0] = local_tensor[4*i, 0, 0];
+ *   }
+ * Output:
+ *   for (int i = 0; i < 100; i++) {
+ *     local_tensor[0, 0, 2*i] = global_tensor[i, 0, 0];
+ *     local_tensor[0, 0, i] = local_tensor[0, 0, 2*i];
+ *   }
+ *
+ * Example 2: Offset elimination
+ * Input:
+ *   for (int i = 0; i < 100; i++) {
+ *     local_tensor[i+2, 0, 0] = global_tensor[i, 0, 0];
+ *     local_tensor[i+4, 0, 0] = local_tensor[i+2, 0, 0];
+ *   }
+ * Output:
+ *   for (int i = 0; i < 100; i++) {
+ *     local_tensor[0, 0, i] = global_tensor[i, 0, 0];
+ *     local_tensor[0, 0, i+2] = local_tensor[0, 0, i];
+ *   }
+ *
+ * Example 3: Symbolic common factor elimination
+ * Input:
+ *     local_tensor[C, 0, 0] = global_tensor[i, 0, 0];
+ *     // (C is a symbolic constant)
+ * Output:
+ *     local_tensor[0, 0, 0] = global_tensor[i, 0, 0];
  */
 void EliminateCommonFactorOfLocalIndex(ir::Expr* expr);