Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 DiscreteHausdorffDistance

Source Notebook

Find the Hausdorff distance between two sets of elements

Contributed by: Sjoerd Smit

ResourceFunction["DiscreteHausdorffDistance"][set1, set2]

returns the Hausdorff distance between two sets, which is the maximum distance you need to travel from a point in one set to reach a point in the other.

Details and Options

Given a distance function d that measures the minimum distance from an element to a set, the Hausdorff distance is given by:
ResourceFunction["DiscreteHausdorffDistance"] takes the same options as Nearest.

Examples

Basic Examples (2) 

Measure the distance between two sets of numbers:

In[1]:=
ResourceFunction["DiscreteHausdorffDistance"][{0, 1/2, 2}, {1, 3, -2}]
Out[1]=

This is the maximum of the following distances:

In[2]:=
dists1 = Nearest[{0, 1/2, 2} -> "Distance", {1, 3, -2}]
Out[2]=
In[3]:=
dists2 = Nearest[{1, 3, -2} -> "Distance", {0, 1/2, 2}]
Out[3]=
In[4]:=
Max[dists1, dists2]
Out[4]=

Scope (1) 

Use a custom DistanceFunction for sets of strings:

In[5]:=
ResourceFunction[ "DiscreteHausdorffDistance"][{"aa", "bb", "cc"}, {"ab", "de"}, DistanceFunction -> HammingDistance]
Out[5]=

Applications (2) 

Compute the distance between sets of points in 2D space:

In[6]:=
SeedRandom[1234]; set1 = RandomPoint[Circle[], 5]; set2 = RandomPoint[Circle[{0, 0}, 2], 10]; ResourceFunction["DiscreteHausdorffDistance"][set1, set2]
Out[9]=

Show the maximum distances from one set to the other as arrows. DiscreteHausdorffDistance is equal to the longest of these two arrows:

In[10]:=
arrows = MapApply[ Replace[ First@TakeLargestBy[ Nearest[#1 -> All, #2, 1][[All, 1]] -> All, Key["Distance"], 1 ], assoc_Association :> Arrow[{ #2[[assoc["Index"]]], assoc["Element", "Element"] }] ] &, { {set1, set2}, {set2, set1} } ]; Show[ListPlot[{set1, set2}, AspectRatio -> 1], Graphics[arrows]]
Out[11]=

Publisher

Sjoerd Smit

Version History

  • 1.0.0 – 26 April 2023

Related Resources

License Information