# Find absolute position of objects when only distances are known.

I have a set of objects u1, u2,..., un and an algorithm which gives a R(1000) [1000 dimensional] vector for each object.

Table 1
t1  t2  ... t1000
u1 234 157 ... 92
u2 117 157 ... 39
...
un 234 157 ... 39


From this data, I'm calculating hamming distance between these objects:

Table 2
row col distance
u1  u2  200
u1  u3  500
u4  u2  200
u4  u3  900
...


I want to create a 2D plot in which each object is a point and distance between points comes from Table 2.

For that I need absolute positions of these objects in some 2D space. Is there a way to find or simulate absolute positions when only distances are known. If yes, how do I select what to put on x-axis and y-axis?

• If the points are in 1000 dimensions, there's no reason to think you can represent them in two. – Gerry Myerson Jun 28 at 9:30
• @Milloupe And satisfy $n\times n$ constraints for arbitrary $n$? – broncoAbierto Jun 28 at 9:47
• Unless the $n$ points are somehow coplanar. (dimensionality reduction?) – peterwhy Jun 28 at 9:51
• @peterwhy points are not coplanar. If I do dimensionality reduction, the distance between the points changes, both order and magnitude. – penguin Jun 28 at 11:07
• @Milloupe The distance in this case is hamming distance. How is it possible to plot all points in 1D? Take this example, 1 and 2 are 100 places apart, 2 and 3 are 500 places apart, 3 and 1 are 50 places apart. How do I place these points in 1 dimension? – penguin Jun 28 at 11:15

## 1 Answer

Figured out a way to this - Multidimensional Scaling https://en.wikipedia.org/wiki/Multidimensional_scaling

This may work accurately when "euclidean" distances are known. Not sure how much information does it compromise with hamming distances.