physics

Relative neighborhood graph

Relative neighborhood graph is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Relative neighborhood graph rather than just read about it. In short: In computational geometry, the relative neighborhood graph (RNG) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points p {\displaystyle p} and q {\displaystyle q} by an edge whenever there does not exist a third point r {\displaystyle r} that is closer to both p {\displaystyle p} and q {\displaystyle q} than they are to each other. This graph was proposed by Godfried Touss…

Key takeaways

Relative neighborhood graph belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Relative neighborhood graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Relative neighborhood graph from memory before moving on to harder problems.

Reference excerpt

In computational geometry, the relative neighborhood graph (RNG) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points p {\displaystyle p} and q {\displaystyle q} by an edge whenever there does not exist a third point r {\displaystyle r} that is closer to both p {\displaystyle p} and q {\displaystyle q} than they are to each other. This graph was proposed by Godfried Toussaint in 1980 as a way of defining a structure from a set of points that would match human perceptions of the shape of the set.

Algorithms Supowit (1983) showed how to construct the relative neighborhood graph of n {\displaystyle n} points in the plane efficiently in O ( n log ⁡ n ) {\displaystyle O(n\log n)} time. It can be computed in O ( n ) {\displaystyle O(n)} expected time, for random set of points distributed uniformly in the unit square. The relative neighborhood graph can be computed in linear time from the Delaunay triangulation of the point set.

Generalizations Because it is defined only in terms of the distances between points, the relative neighborhood graph can be defined for point sets in any dimension, and for non-Euclidean metrics. Computing the relative neighborhood graph, for higher-dimensional point sets, can be done in time O ( n 2 ) {\displaystyle O(n^{2})} .

Related graphs The relative neighborhood graph is an example of a lens-based beta skeleton. It is a subgraph of the Delaunay triangulation. In turn, the Euclidean minimum spanning tree is a subgraph of it, from which it follows that it is a connected graph. The Urquhart graph, the graph formed by removing the longest edge from every triangle in the Delaunay triangulation, was originally proposed as a fast method to compute the relative neighborhood graph. Although the Urquhart graph sometimes differs from the relative neighborhood graph it can be used as an approximation to the relative neighborhood graph.

References

Illustrations

Worked examples

Example 1 — a first encounter with Relative neighborhood graph

Start with the simplest possible case. Write down what Relative neighborhood graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Relative neighborhood graph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Relative neighborhood graph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Relative neighborhood graph

In research: Relative neighborhood graph appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Relative neighborhood graph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Relative neighborhood graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometric graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Relative neighborhood graph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Relative neighborhood graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Relative neighborhood graph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Relative neighborhood graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Relative neighborhood graph in simple terms?

Why does Relative neighborhood graph matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Relative neighborhood graph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Relative neighborhood graph.