Representation of arrays in memory

C-style transformation of non-contiguous to contiguous layout

Let us consider a 2-D array, x = numpy.random.rand(4,3). The shape of the array is therefore 4 × 3 array ( numpy.shape(x) = (4,3)). Also any element in the array is associated with its respective, index (n₀, n₁).

Then, for a C-style non-contiguous layout with shape, (d₀, d₁, …, d_N−1), the value associated with (n₀, n₁, …, n_N−1) is located in the contiguous layout with index = n^C.

If,

n_i ≜ value of i^th index in the non-contiguous layout. eg. for (2, 1) entry in x is (n₀ = 2, n₁ = 1)

For C-style non-contiguous layout with shape (d₀, d₁, …, d_N−1) the value in (n₀, n₁, …, n_N−1) is located in the contiguous layout with index n^C

$expression for index in contiguous layout mapped from index in C-style non-contiguous layout$

such that, for d_k, d_{k + 1}, …, d_m−1, d_m if k > m

$products of d sub j is 1 if start index is greater than final index$

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Examples illustrating transformation to contiguous layout in C-style

Eg.1: Location of element in x with index (2, 1)

For a C-style non-contiguous layout x with shape, (d₀ = 4, d₁ = 3) i.e. (d₀, d₂₋₁) ∴ N = 2, the value in (n₀ = 2, n₁ = 1) of the non-contiguous layout is located in the contiguous layout with index n^C given by,

$index in contiguous layout for (2, 1) index in C-style non-contiguous layout is equal to 7$

index in C-style non-contiguous layout into respective mapped index in contiguous layout

Eg.2: Location of element in x with index (3, 1)

For a C-style non-contiguous layout x with shape, (d₀ = 4, d₁ = 3) the value in (n₀ = 3, n₁ = 1) of the non-contiguous layout is located in the contiguous layout with index n^Cgiven by,

$index in contiguous layout for (3, 1) index in C-style non-contiguous layout is equal to 10$

Notice that to move from the element with index (2, 1) in the array x to the element at (3, 1), three elements within the array must be jumped.

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Stride in an array

The number of elements that must be jumped to get to the next element (eg. next arrow) is given by the stride.

Stride in N-dimensional C-style array

$expression of index in contiguous layout in terms of stride$

where

Stride is
$stride is equal to product of d sub i's$

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