Estimate city positions from distance tables.

DISTANCE_TO_POSITION is a MATLAB program which estimates the positions of cities given a city-to-city distance table.

In 2D, the problem is singular. In particular, the position of one city is completely arbitrary, and one component of a second city is completely arbitrary (and a third city's position can be "flipped" about the line connecting cities one and two). To remove some of this singularity, the program assigns city #1 the position (0,0) and city #2 is given a zero Y coordinate.

In N-dimensional space, a similar set of constraints must be placed on the first N cities, or the least squares solver is likely to fail. (That is, we specify all N components of the first city to be 0, N-1 components of the second one, and so on, up to the N-th city which has a single zero component).

The computations carried out by this program assume that the cities lie on a plane. If the distance data is for international cities, the effect of spherical geometry may make the planar approximation very bad.

Once the nonlinear least squares problem is set up, MATLAB's LSQNONLIN function is called to seek a solution.


distance_to_position ( 'distance.txt' )
reads the distance information in 'distance.txt', estimates the positions of the cities, and writes out a position table in distance.coord.txt.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


DISTANCE_TO_POSITION is available in a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CHAIN_LETTER_TREE, a MATLAB program which analyzes a similarity matrix for 11 versions of a single chain letter and produces a dendrogram or tree diagram that suggests the degrees of relatedness and the evolutionary history of the letters.

CITIES, a dataset directory which contains sets of information about cities and the distances between them;

CITIES, a MATLAB library which handles various problems associated with a set of "cities" on a map.

DISTANCE_TO_POSITION_SPHERE, a MATLAB program which estimates the positions of cities on a sphere (such as the earth) based on a city-to-city distance table.

LAU_NP, a FORTRAN90 library which implements heuristic algorithms for various NP-hard combinatorial problems.

NMS, a FORTRAN90 library which includes a wide variety of numerical software.

PARTIAL_DIGEST, a FORTRAN90 library which solves the partial digest problem.

Source Code:

Examples and Tests:

CHAIN_LETTER is a problem involving perceived distances between pairs of 11 chain letters.

GRID04 is a problem involving distances between pairs of 4 points.

You can go up one level to the MATLAB source codes.

Last revised on 22 January 2014.