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Sort sequences with respect to the similarity of consecutive items.

Project description

Sort sequences with respect to the similarity of consecutive items.

Definition

Given a sequence of items \((x_i, y_i)\), where each item is represented by two values \(x, y\), the goal is to sort the sequence such that the following loss is minimal:

\begin{equation*} L = \sum_{i=1}^{N-1} \mu(y_i, x_{i+1}) \end{equation*}

where \(\mu\) denotes a suitable metric for the items’ values.

Example

Given the items

>>> items = [
...     (0.4, 0.6),
...     (0.0, 0.2),
...     (0.8, 1.0),
...     (0.6, 0.8),
...     (0.2, 0.4),
... ]

together with the L1 distance \(\mu: (x, y) \rightarrow |x-y|\), the current loss is

>>> abs(0.6 - 0.0) + abs(0.2 - 0.8) + abs(1.0 - 0.6) + abs(0.8 - 0.2)
2.2

Clearly the optimal sort order which minimizes the loss is

>>> optimal = [
   ...     (0.0, 0.2),
   ...     (0.2, 0.4),
   ...     (0.4, 1.6),
   ...     (0.6, 0.8),
   ...     (0.8, 1.0),
   ... ]

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