# ndindex¶

A Python library for manipulating indices of ndarrays.

ndindex is a library that allows representing and manipulating objects that can be valid indices to numpy arrays, i.e., slices, integers, ellipses, None, integer and boolean arrays, and tuples thereof. The goals of the library are

• Provide a uniform API to manipulate these objects. Unlike the standard index objects themselves like slice, int, and tuple, which do not share any methods in common related to being indices, ndindex classes can all be manipulated uniformly. For example, idx.args always gives the arguments used to construct idx.

• Give 100% correct semantics as defined by numpy’s ndarray. This means that ndindex will not make a transformation on an index object unless it is correct for all possible input array shapes. The only exception to this rule is that ndindex assumes that any given index will not raise IndexError (for instance, from an out of bounds integer index or from too few dimensions). For those operations where the array shape is known, there is a reduce() method to reduce an index to a simpler index that is equivalent for the given shape.

• Enable useful transformation and manipulation functions on index objects.

## Motivation¶

If you’ve ever worked with Python’s slice objects, you will quickly discover their limitations:

• Extracting the arguments of a slice is cumbersome. You have to write start, stop, step = s.start, s.stop, s.step. With ndindex you can write start, stop, step = s.args

• slice objects are not hashable. If you want to use them as dictionary keys, you have to use cumbersome translation back and forth to a hashable type such as tuple.

• slice makes no assumptions about what they are slicing. As a result, invalid slices like slice(0.5) or slice(0, 10, 0) are allowed. Also slices that would always be equivalent like slice(None, 10) and slice(0, 10) are unequal. To contrast, ndindex objects always assume they are indices to numpy arrays and type check their input. The reduce method can be used to put the arguments into canonical form.

• Once you generalizing slice objects to more general indices, it is difficult to work with them in a uniform way. For example, a[i] and a[(i,)] are always equivalent for numpy arrays, but tuple, slice, int, etc. are not related to one another. To contrast, all ndindex types have a uniform API, and all relevant operations on them produce ndindex objects.

• The above limitations can be annoying, but you might consider them worth living with. The real pain comes when you start trying to do slice arithmetic. Slices in Python behave fundamentally differently depending on whether the step is positive or negative and the start and stop are positive, negative, or None. Consider, for example, the meaning of the slice a[4:-2:-2], where a is a one-dimensional array. This slices every other element from the fifth element to the second from the last, but not including the second from last. The resulting array will have shape (0,) if the original shape is less than 1 or greater than 5, and shape (1,) otherwise. In ndindex, one can use len(Slice(4, -2, -2)) to compute the maximum length of this slice (1), or len(Slice(4, -2, -2).reduce(shape)) to compute the length for a specific array shape. See ndindex.Slice.__len__() and ndindex.Slice.reduce().

ndindex pre-codes common slice arithmetic into useful abstractions so you don’t have to try to figure out all the different cases yourself. And due to extensive testing (see below), you can be assured that ndindex is correct.

## Features¶

ndindex is still a work in progress. The following things are currently implemented:

• Classes do not canonicalize by default (the constructor only does basic type checking). Objects can be put into canonical form by calling reduce().

>>> from ndindex import Slice
>>> Slice(None, 12)
Slice(None, 12, None)
>>> Slice(None, 12).reduce()
Slice(0, 12, 1)


reduce() can also be called with a shape argument. idx.reduce(shape) reduces an index to an equivalent index over an array with the given shape.

>>> from numpy import arange
>>> Slice(2, -1).reduce((10,))
Slice(2, 9, 1)
>>> arange(10)[2:-1]
array([2, 3, 4, 5, 6, 7, 8])
>>> arange(10)[2:9:1]
array([2, 3, 4, 5, 6, 7, 8])

• Object arguments can be accessed with idx.args

>>> Slice(1, 3).args
(1, 3, None)

• All ndindex objects are hashable and can be used as dictionary keys.

• A real index object can be accessed with idx.raw. Use this to use an ndindex index to index an array.

>>> s = Slice(0, 2)
>>> arange(4)[s.raw]
array([0, 1])

• len() computes the maximum length of an index over a given axis.

>>> len(Slice(2, 10, 3))
3
>>> len(arange(10)[2:10:3])
3

• idx.isempty() returns True if an index always indexes to an empty array (an array with a 0 in its shape). isempty can also be called with a shape like idx.isempty(shape).

>>> Slice(0, 0).isempty()
True

• idx.expand(shape) expands an index so that it is as explicit as possible. An expanded index is always a Tuple where each indexed axis is indexed explicitly.

>>> from ndindex import Tuple
>>> Tuple(Slice(0, 10), ..., Slice(1, None)).expand((10, 11, 12))
Tuple(slice(0, 10, 1), slice(0, 11, 1), slice(1, 12, 1))

• idx.newshape(shape) returns the shape of a[idx], assuming a has shape shape.

>>> Tuple(0, ..., Slice(0, 5)).newshape((10, 10, 10))
(10, 5)

• i.as_subindex(j) produces an index k such that a[j][k] gives all the elements of a[j] that are also in a[i] (see the documentation for more information). This is useful for re-indexing an index onto chunks of an array.

>>> chunks = [Slice(0, 100), Slice(100, 200)]
>>> idx = Slice(50, 160)
>>> idx.as_subindex(chunks[0])
Slice(50, 100, 1)
>>> idx.as_subindex(chunks[1])
Slice(0, 60, 1)


The following things are not yet implemented, but are planned.

• Newaxis, IntegerArray, and BooleanArray types, so that all types of indexing are support.

• Composition: i1[i2] will create a new ndindex object i3 (when possible) so that a[i1][i2] == a[i3].

• Various set-like operations on indices, including “contains”, “union”, and “intersection”. For example, i1 + i2 will produce a single index so that a[i1 + i2] gives all the elements of a[i1] and a[i2].

• Support NEP 21 advanced indexing.

• Support for generating symbolic formulas for the above operations using SymPy.

And more. If there is something you would like to see this library be able to do, please open an issue. Pull requests are welcome as well.

## Testing and correctness¶

The most important priority for a library like this is correctness. Index manipulations, and especially slice manipulations, are complicated to code correctly, and the code for them typically involves dozens of different branches for different cases.

In order to assure correctness, all operations are tested extensively against numpy itself to ensure they give the same results. The basic idea is to take the pure Python index and the ndindex(index).raw, or in the case of a transformation, the before and after raw index, and index a numpy.arange with them (the input array itself doesn’t matter, so long as its values are distinct). If they do not give the same output array, or do not both produce the same error (like an IndexError), the code is not correct. For example, the reduce() method can be verified by checking that a[idx.raw] and a[idx.reduce(a.shape).raw] produce the same sub-arrays for all possible input arrays a and ndindex objects idx.

There are two primary types of tests that we employ to verify this:

• Exhaustive tests. These test every possible value in some range. For example, slice tests test all possible start, stop, and step values in the range [-10, 10], as well as None, on numpy.arange(n) for n in the range [0, 10]. This is the best type of test, because it checks every possible case. Unfortunately, it is often impossible to do full exhaustive testing due to combinatorial explosion.

• Hypothesis tests. Hypothesis is a library that can intelligently check a combinatorial search space of inputs. This requires writing hypothesis strategies that can generate all the relevant types of indices (see ndindex/tests/helpers.py). For more information on hypothesis, see https://hypothesis.readthedocs.io/en/latest/index.html. All tests have hypothesis tests, even if they are also tested exhaustively.

Why bother with hypothesis if the same thing is already tested exhaustively? The main reason is that hypothesis is much better at producing human-readable failure examples. When an exhaustive test fails, the failure will always be from the first set of inputs in the loop that produces a failure. Hypothesis on the other hand attempts to “shrink” the failure input to smallest input that still fails. For example, a failing exhaustive slice test might give Slice(-10, -9, -10) as a the failing example, but hypothesis would shrink it to Slice(-2, -1, -1). Another reason for the duplication is that hypothesis can sometimes test a slightly expanded test space without any additional consequences. For example, test_slice_reduce_hypothesis() in tests all types of array shapes, whereas test_slice_reduce_exhaustive() tests only 1-dimensional shapes. This doesn’t affect things because hypotheses will always shrink large shapes to a 1-dimensional shape in the case of a failure. Consequently every exhaustive test will also have a corresponding hypothesis test.