Minimum-diameter spanning tree

Key takeaways

Minimum-diameter spanning tree 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 Minimum-diameter spanning tree to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Minimum-diameter spanning tree from memory before moving on to harder problems.

Reference excerpt

In metric geometry and computational geometry, a minimum-diameter spanning tree of a finite set of points in a metric space is a spanning tree in which the diameter (the longest path length in the tree between two of its points) is as small as possible.

In general metric spaces It is always possible to find a minimum-diameter spanning tree with one or two vertices that are not leaves. This can be proven by transforming any other tree into a tree of this special form, without increasing its diameter. To do so, consider the longest path in any given tree (its diameter path), and the vertex or edge at the midpoint of this path. If there is a vertex at the midpoint, it is the non-leaf vertex of a star, whose diameter is at most that of the given tree. If the midpoint is interior to an edge of the given tree, then there exists a tree that includes this edge, and in which every remaining vertex is a leaf connected to the endpoint of this edge that is nearest in the given tree, with diameter at most that of the given tree. Because of this special form, it is possible to construct a minimum-diameter spanning tree of n {\displaystyle n} points in time O ( n 3 ) {\displaystyle O(n^{3})} , assuming that the distance between two points can be computed in constant time. To do so, test all candidates for the single point or pair of points that are not leaves. Each single point can be tested in O ( n ) {\displaystyle O(n)} time. Each pair of points can also be tested in O ( n ) {\displaystyle O(n)} , after a precomputation step in which, for each point, the other points are sorted by their distance from it. To test a pair of points, start with a tree in which all remaining points are attached to one point of the pair, and then in decreasing order by distance from that point, reattach these points one at a time to the other point of the pair, keeping track of the diameter of the tree at each step. There are O ( n 2 ) {\displaystyle O(n^{2})} candidate pairs of non-leaf points, each of which can be evaluated in time O ( n ) {\displaystyle O(n)} , giving a total time bound of O ( n 3 ) {\displaystyle O(n^{3})} . The problem of constructing a minimum-diameter spanning tree is different from computing the diameter of the given points, the maximum pairwise distance. For some sets of points, the diameter of the points and the diameter of their minimum-diameter spanning tree are equal; for others (for instance, three equidistant points) these two distances can differ from each other by a factor of two.

In graphs

For the metric space of shortest-path distances on a weighted undirected graph, a minimum-diameter spanning tree can also be a spanning tree of the graph, a tree whose edges all belong to the graph. However, this may require it to have more than two non-leaf vertices. In this case, the problem is equivalent to finding an absolute 1-center of the graph. This is a point in a metric space obtained from the given graph by replacing each edge by a continuous interval of the same length. That is, it can either be a vertex or it can lie partway along any edge of the given graph. Among such points, the absolute 1-center is a point minimizing the maximum distance to all vertices. The shortest-path tree from this point to all vertices in the graph is a minimum-diameter spanning tree of the graph. The absolute 1-center problem was introduced long before the first study of the minimum-diameter spanning tree problem, and in a graph with n {\displaystyle n} vertices and m {\displaystyle m} edges it can be solved in time O ( m n + n 2 log ⁡ n ) {\displaystyle O(mn+n^{2}\log n)} .

In the Euclidean plane The exact solution of the minimum-diameter spanning tree problem, in the Euclidean plane, can be sped up from O ( n 3 ) {\displaystyle O(n^{3})} to n 17 / 6 + o ( 1 ) {\displaystyle n^{17/6+o(1)}} , at the expense of using complicated range search data structures. The same method extends to higher dimensions, with smaller reductions in the exponent compared to the cubic algorithm. In d {\displaystyle d} dimensions, the time bound for this method is

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Minimum-diameter spanning tree

Start with the simplest possible case. Write down what Minimum-diameter spanning tree 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 Minimum-diameter spanning tree 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 Minimum-diameter spanning tree 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 Minimum-diameter spanning tree

In research: Minimum-diameter spanning tree 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 Minimum-diameter spanning tree 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: Minimum-diameter spanning tree is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational problems in graph theory, Geometric algorithms, Spanning tree, so understanding it makes those chapters shorter.
In everyday life: Look for Minimum-diameter spanning tree 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.

How to study Minimum-diameter spanning tree in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Minimum-diameter spanning tree 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 Minimum-diameter spanning tree out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Minimum-diameter spanning tree in simple terms?

In metric geometry and computational geometry, a minimum-diameter spanning tree of a finite set of points in a metric space is a spanning tree in which the diameter (the longest path length in the tree between two of its points) is as small as possible. In general metric spaces It is always possibl…

Why does Minimum-diameter spanning tree 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 Minimum-diameter spanning tree?

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 Minimum-diameter spanning tree.