dominosort 0.1

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:

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),
   ... ]

Related topics

Note that for the special case where \(x_i = y_i\) and the \(x_i\) represent 2d-coordinates this corresponds to the Travelling Salesman Problem without return to the origin.