New content: Add definition for shape broadcasting

This change introduces a new section for Algorithms, following APIs,
to collect algorithms referenced throughout the specification.

A section for Broadcasting is introduced, which defines broadcasting
shapes and gives an explicit algorithm matching WebNN implementations
of NumPy's General Broadcasting Rules. Definitions for "broadcastable"
and "unidirectionally broadcastable" are introduced. The previous
definition of "broadcast-shapes" is removed in favor of these new
algorithms.

For webmachinelearning#324, webmachinelearning#462, and potentially webmachinelearning#523.

Author: inexorabletash

index.bs

@@ -2439,8 +2439,8 @@ partial interface MLGraphBuilder {
     1. If |a|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}} is not equal to |b|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}}, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
     1. Let |descriptor| be a new {{MLOperandDescriptor}}.
     1. Set |descriptor|.{{MLOperandDescriptor/dataType}} to |a|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}}.
-    1. Set |descriptor|.{{MLOperandDescriptor/dimensions}} to the result of running the [=MLGraphBuilder/broadcast-shapes=] steps given |a|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}} and |b|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}}.
-        1. If that [=exception/throws=] an error, re-[=exception/throw=] the error.
+    1. Set |descriptor|.{{MLOperandDescriptor/dimensions}} to the result of [=broadcasting the shapes=] |a|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}} and |b|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}}.
+        1. If that returns failure, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
     1. If any of the following sub-steps fail, [=exception/throw=] an "{{OperationError}}" {{DOMException}}.
         1. Let |output| be the result of [=creating an MLOperand=] given [=this=] and |descriptor|.
         1. Make a request to the underlying platform to:
@@ -2454,21 +2454,6 @@ partial interface MLGraphBuilder {
   </div>
 </details>
 
-<details open algorithm>
-  <summary>
-    To <dfn for="MLGraphBuilder">broadcast-shapes</dfn> given |shape1| and |shape2|, run the following steps:
-  </summary>
-  <div class=algorithm-steps>
-    1. [=Assert=]: The type of |shape1| and |shape2| is `sequence of unsigned long`.
-    1. Let |output| be the result of invoking the [=implementation-defined=] shape broadcast on |shape1| and |shape2|.
-            1. If that fails, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
-    1. Return |output|.
-        <div class = "note">
-        The most common implementation is that two shapes are compatible, when each of their corresponding dimensions are equal, or one of them is 1. The output shape consists of the maximum of the corresponding dimensions.
-        </div>
-  </div>
-</details>
-
 <details open>
   <summary>
     The element-wise binary operation algorithms invoke the [=MLGraphBuilder/element-wise-binary-op | create element-wise binary operation=] steps as follows.
@@ -2576,8 +2561,8 @@ Although operations *greaterOrEqual* and *lesserOrEqual* can each be implemented
         1. If |a|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}} is not equal to |b|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}}, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
     1. Let |descriptor| be a new {{MLOperandDescriptor}}.
     1. Set |descriptor|.{{MLOperandDescriptor/dataType}} to {{MLOperandDataType/"uint8"}}.
-    1. Set |descriptor|.{{MLOperandDescriptor/dimensions}} to the result of running the [=MLGraphBuilder/broadcast-shapes=] steps given |a|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}} and |b|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}}.
-        1. If that [=exception/throws=] an error, re-[=exception/throw=] the error.
+    1. Set |descriptor|.{{MLOperandDescriptor/dimensions}} to the result of [=broadcasting the shapes=] |a|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}} and |b|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}}.
+        1. If that returns failure, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
     1. If any of the following sub-steps fail, [=exception/throw=] an "{{OperationError}}" {{DOMException}}.
         1. Let |output| be the result of [=creating an MLOperand=] given [=this=] and |descriptor|.
         1. Make a request to the underlying platform to:
@@ -3066,7 +3051,7 @@ partial interface MLGraphBuilder {
 </div>
 
 ### gemm ### {#api-mlgraphbuilder-gemm}
-Calculate the [general matrix multiplication of the Basic Linear Algebra Subprograms](https://en.wikipedia.org/wiki/Basic_Linear_Algebra_Subprograms#Level_3). The calculation follows the expression `alpha * A * B + beta * C`, where `A` is a 2-D tensor with shape [M, K] or [K, M], `B` is a 2-D tensor with shape [K, N] or [N, K], and `C` is broadcastable to the shape [M, N]. `A` and `B` may optionally be transposed prior to the calculation.
+Calculate the [general matrix multiplication of the Basic Linear Algebra Subprograms](https://en.wikipedia.org/wiki/Basic_Linear_Algebra_Subprograms#Level_3). The calculation follows the expression `alpha * A * B + beta * C`, where `A` is a 2-D tensor with shape [M, K] or [K, M], `B` is a 2-D tensor with shape [K, N] or [N, K], and `C` is [=broadcastable=] to the shape [M, N]. `A` and `B` may optionally be transposed prior to the calculation.
 
 <script type=idl>
 dictionary MLGemmOptions {
@@ -3086,7 +3071,7 @@ partial interface MLGraphBuilder {
 <dl dfn-type=dict-member dfn-for=MLGemmOptions>
     : <dfn>c</dfn>
     ::
-        An {{MLOperand}}. Specifies the third input tensor. It is either a scalar, or of the shape that is unidirectionally broadcastable to the shape [M, N] according to [[!numpy-broadcasting-rule]]. When it is not specified, the computation is done as if *c* is a scalar 0.0.
+        An {{MLOperand}}. Specifies the third input tensor. It is either a scalar, or of the shape that is [=unidirectionally broadcastable=] to the shape [M, N]. When it is not specified, the computation is done as if *c* is a scalar 0.0.
 
     : <dfn>alpha</dfn>
     ::
@@ -3127,7 +3112,7 @@ partial interface MLGraphBuilder {
     1. If |options|.{{MLGemmOptions/aTranspose}} is true, then let |shapeA| be the reverse array of |shapeA|.
     1. If |options|.{{MLGemmOptions/bTranspose}} is true, then let |shapeB| be the reverse array of |shapeB|.
     1. If |shapeA|[1] is not equal to |shapeB|[0], then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
-    1. If |options|.{{MLGemmOptions/c}} [=map/exists=] and is not unidirectionally broadcastable to the shape [|shapeA|[0], |shapeB|[1]] according to the [[!numpy-broadcasting-rule]], then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
+    1. If |options|.{{MLGemmOptions/c}} [=map/exists=] and is not [=unidirectionally broadcastable=] to the shape [|shapeA|[0], |shapeB|[1]], then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
         <div class="note">
             Type compatibility between |a|, |b| and |options|.{{MLGemmOptions/c}} can be also checked.
         </div>
@@ -4971,7 +4956,7 @@ partial interface MLGraphBuilder {
 <div>
     **Arguments:**
         - *input*: an {{MLOperand}}. The input tensor.
-        - *slope*: an {{MLOperand}}. The slope tensor. Its shape is either the same as, or unidirectionally broadcastable to the shape of input tensor *input* according to [[!numpy-broadcasting-rule]].
+        - *slope*: an {{MLOperand}}. The slope tensor. Its shape is either the same as, or [=unidirectionally broadcastable=] to the shape of input tensor *input*.
 
     **Returns:**
         - an {{MLOperand}}. The output tensor of the same shape as *input*.
@@ -4986,8 +4971,8 @@ partial interface MLGraphBuilder {
     1. [=Assert=]: the type of |input| and |slope| is {{MLOperand}}.
     1. Let |descriptor| be a new {{MLOperandDescriptor}}.
     1. Set |descriptor|.{{MLOperandDescriptor/dataType}} to |input|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}}.
-    1. Set |descriptor|.{{MLOperandDescriptor/dimensions}} to the result of running the [=MLGraphBuilder/broadcast-shapes=] steps given |input|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}} and |slope|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}}.
-        1. If that [=exception/throws=] an error, re-[=exception/throw=] the error.
+    1. Set |descriptor|.{{MLOperandDescriptor/dimensions}} to the result of [=broadcasting the shapes=] |input|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}} and |slope|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}}.
+        1. If that returns failure, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
     1. If any of the following sub-steps fail, [=exception/throw=] an "{{OperationError}}" {{DOMException}}.
         1. Let |output| be the result of [=creating an MLOperand=] given [=this=] and |descriptor|.
         1. Make a request to the underlying platform to:
@@ -6127,9 +6112,9 @@ partial interface MLGraphBuilder {
     1. If |input|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}} is not equal to |other|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}}, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
     1. Let |descriptor| be a new {{MLOperandDescriptor}}.
     1. Set |descriptor|.{{MLOperandDescriptor/dataType}} to |input|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dataType}}.
-    1. Set |descriptor|.{{MLOperandDescriptor/dimensions}} to the result of running the [=MLGraphBuilder/broadcast-shapes=] steps given |input|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}} and |other|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}}.
-        1. If that [=exception/throws=] an error, re-[=exception/throw=] the error.
-    1. If |condition| is not unidirectionally broadcastable to |descriptor|.{{MLOperandDescriptor/dimensions}} according to the [[!numpy-broadcasting-rule]], then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
+    1. Set |descriptor|.{{MLOperandDescriptor/dimensions}} to the result of [=broadcasting the shapes=] |input|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}} and |other|.{{MLOperand/[[descriptor]]}}.{{MLOperandDescriptor/dimensions}}.
+        1. If that returns failure, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
+    1. If |condition| is not [=unidirectionally broadcastable=] to |descriptor|.{{MLOperandDescriptor/dimensions}}, then [=exception/throw=] a "{{DataError}}" {{DOMException}}.
     1. If any of the following sub-steps fail, [=exception/throw=] an "{{OperationError}}" {{DOMException}}.
         1. Let |output| be the result of [=creating an MLOperand=] given [=this=] and |descriptor|.
         1. Make a request to the underlying platform to:
@@ -6346,6 +6331,36 @@ dictionary MLOperandDescriptor {
   </div>
 </details>
 
+Algorithms {#algorithms}
+=====================
+
+## Broadcasting ## {#algorithms-broadcasting}
+
+Broadcasting refers to how operations treat tensors with different shapes, and follow the precedent set by [[!numpy-broadcasting-rule]].
+
+<div algorithm>
+To <dfn>broadcast the shapes</dfn> |A| and |B|, perform the following steps. |A| and |B| are [=/lists=] of positive integers, representing the dimensions of tensors, and the steps return a new [=/list=] of positive integers, or failure.
+
+1. Let |sizeA| be the [=list/size=] of |A|.
+1. Let |sizeB| be the [=list/size=] of |B|.
+1. Let |output| be a new [=/list=].
+1. [=list/For each=] |index| in [=the range=] 0 to |sizeA|, exclusive:
+    1. Let |dimA| be |A|[|sizeA| - |index| - 1] if |index| < |sizeA|, or 1 otherwise.
+    1. Let |dimB| be |B|[|sizeB| - |index| - 1] if |index| < |sizeB|, or 1 otherwise.
+    1. If |dimA| is not equal to |dimB| and |dimA| is not equal to 1, then return failure.
+    1. [=list/Prepend=] |dimA| to |output|.
+1. Return |output|.
+
+</div>
+
+<div algorithm>
+|A| is <dfn>broadcastable</dfn> to |B| if [=broadcasting the shapes=] |A| and |B| does not result in failure.
+</div>
+
+<div algorithm>
+|A| is <dfn>unidirectionally broadcastable</dfn> to |B| if |A| is [=broadcastable=] to |B|, and |B| is [=broadcastable=] to |A|.
+</div>
+
 Examples {#examples}
 =====================