问题
Suppose we are given a small number of objects and "distances" between them -- what algorithm exists for fitting these objects to points in two-dimensional space in a way that approximates these distances?
The difficulty here is that the "distance" is not distance in Euclidean space -- this is why we can only fit/approximate.
(for those interested in what the notion of distance is precisely, it is the symmetric distance metric on the power set of a (finite) set).
回答1:
Given that the number of objects is small, you can create an undirected weighted graph, where these objects would be nodes and the edge between any two nodes has the weight that corresponds to the distance between these two objects. You end up with n*(n-1)/2 edges.
Once the graph is created, there are a lot of visualization software and algorithms that correspond to graphs.
回答2:
Try a triangulation method, something like this;
Start by taking three objects with known distances between them, and create a triangle in an arbitrary grid based on the side lengths.
For each additional object that has not been placed, find at least three other objects that have been placed that you have known distances to, and use those distances to place the object using distance / distance intersection (i.e. the intersection point of the three circles centred around the fixed points with radii of the distances)
Repeat until all objects have been placed, or no more objects can be placed.
For unplaced objects, you could start another similar exercise, and then use any available distances to relate the separate clusters. Look up triangulation and trilateration networks for more info.
Edit: As per the comment below, where the distances are approximate and include an element of error, the above approach may be used to establish provisional coordinates for each object, and those coordinates may then be adjusted using a least squares method such as variation of coordinates This would also cater for weighting distances based on their magnitude as required. For a more detailed description, check Ghilani & Wolf's book on the subject. This depends very much on the nature of the differences between your distances and how you would like your objects represented in Euclidean space based on those distances. The relationship needs to be modelled and applied as part of any solution.
回答3:
This is an example of Multidimensional Scaling, or more generally, Nonlinear dimensionality reduction. There are a fair amount of tools/libraries for doing this available (see the second link for a list).
来源：https://stackoverflow.com/questions/9180161/algorithm-for-fitting-abstract-distances-in-2d