computer science

Kinetic smallest enclosing disk

Kinetic smallest enclosing disk is a computer science 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 Kinetic smallest enclosing disk rather than just read about it. In short: A kinetic smallest enclosing disk data structure is a kinetic data structure that maintains the smallest enclosing disk of a set of moving points. 2D In 2 dimensions, the best known kinetic smallest enclosing disk data structure uses the farthest point delaunay triangulation of the point set to maintain the smallest enclosing disk. The farthest-point Delaunay triangulation is the dual of the farthest-point Voronoi d…

Key takeaways

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

Reference excerpt

A kinetic smallest enclosing disk data structure is a kinetic data structure that maintains the smallest enclosing disk of a set of moving points.

2D In 2 dimensions, the best known kinetic smallest enclosing disk data structure uses the farthest point delaunay triangulation of the point set to maintain the smallest enclosing disk. The farthest-point Delaunay triangulation is the dual of the farthest-point Voronoi diagram. It is known that if the farthest-point delaunay triangulation of a point set contains an acute triangle, the circumcircle of this triangle is the smallest enclosing disk. Otherwise, the smallest enclosing disk has the diameter of the point set as its diameter. Thus, by maintaining the kinetic diameter of the point set, the farthest-point delaunay triangulation, and whether or not the farthest-point delaunay triangulation has an acute triangle, the smallest enclosing disk can be maintained. This data structure is responsive and compact, but not local or efficient:

Responsiveness: This data structure requires O ( log 2 ⁡ n ) {\displaystyle O(\log ^{2}n)} time to process each certificate failure, and thus is responsive. Locality: A point can be involved in Θ ( n ) {\displaystyle \Theta (n)} certificates. Therefore, this data structure is not local. Compactness: This data structure requires O(n) certificates total, and thus is compact. Efficiency: This data structure has O ( n 3 + ϵ ) {\displaystyle O(n^{3+\epsilon })} events total.(for all ϵ > 0 {\displaystyle \epsilon >0} The best known lower bound on the number of changes to the smallest enclosing disk is Ω ( n 2 ) {\displaystyle \Omega (n^{2})} . Thus the efficiency of this data structure, the ratio of total events to external events, is O ( n 1 + ϵ ) {\displaystyle O(n^{1+\epsilon })} . The existence of kinetic data structure that has o ( n 3 + ϵ ) {\displaystyle o(n^{3+\epsilon })} events is an open problem.

Approximate 2D The smallest enclosing disk of a set of n moving points can be ε-approximated by a kinetic data structure that processes O ( 1 / ϵ 5 / 2 ) {\displaystyle O(1/\epsilon ^{5/2})} events and requires O ( ( n / ϵ ) log ⁡ n ) {\displaystyle O((n/{\sqrt {\epsilon }})\log n)} time total.

Higher dimensions In dimensions higher than 2, efficiently maintaining the smallest enclosing sphere of a set of moving points is an open problem.

References

Worked examples

Example 1 — a first encounter with Kinetic smallest enclosing disk

Start with the simplest possible case. Write down what Kinetic smallest enclosing disk claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, 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 Kinetic smallest enclosing disk 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 Kinetic smallest enclosing disk 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 Kinetic smallest enclosing disk

In research: Kinetic smallest enclosing disk appears in computer science 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 Kinetic smallest enclosing disk 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: Kinetic smallest enclosing disk is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational geometry, Kinetic data structures, so understanding it makes those chapters shorter.
In everyday life: Look for Kinetic smallest enclosing disk 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 Kinetic smallest enclosing disk in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kinetic smallest enclosing disk 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 Kinetic smallest enclosing disk out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kinetic smallest enclosing disk in simple terms?

A kinetic smallest enclosing disk data structure is a kinetic data structure that maintains the smallest enclosing disk of a set of moving points. 2D In 2 dimensions, the best known kinetic smallest enclosing disk data structure uses the farthest point delaunay triangulation of the point set to mai…

Why does Kinetic smallest enclosing disk matter?

Because it connects several computer science 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 Kinetic smallest enclosing disk?

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 Kinetic smallest enclosing disk.