computer science

Minimum routing cost spanning tree

Minimum routing cost spanning tree 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 Minimum routing cost spanning tree rather than just read about it. In short: In computer science, the minimum routing cost spanning tree of a weighted graph is a spanning tree minimizing the sum of pairwise distances between vertices in the tree. It is also called the optimum distance spanning tree, shortest total path length spanning tree, minimum total distance spanning tree, or minimum average distance spanning tree.

Key takeaways

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

Reference excerpt

In computer science, the minimum routing cost spanning tree of a weighted graph is a spanning tree minimizing the sum of pairwise distances between vertices in the tree. It is also called the optimum distance spanning tree, shortest total path length spanning tree, minimum total distance spanning tree, or minimum average distance spanning tree. In an unweighted graph, this is the spanning tree of minimum Wiener index. Hu (1974) writes that the problem of constructing these trees was proposed by Francesco Maffioli. It is NP-hard to construct it, even for unweighted graphs. However, it has a polynomial-time approximation scheme. The approximation works by choosing a number k {\displaystyle k} that depends on the approximation ratio but not on the number of vertices of the input graph, and by searching among all trees with k {\displaystyle k} internal nodes. The minimum routing cost spanning tree of an unweighted interval graph can be constructed in linear time. A polynomial time algorithm is also known for distance-hereditary graphs, weighted so that the weighted distances are hereditary.

Fairness considerations Several works assume that different people may have different preferences on edges in the graph, and the goal is to find a spanning tree that is "socially" best.

Darmann, Klamler and Pferschy present a greedy algorithm that finds such a spanning tree. Escoffier, Gourvès and Monnot study the problem under the egalitarian rule - maximizing the smallest utility of an agent. Galand, Perny and Spanjaard study the problem under the criterion of minimizing the Choquet integral.

See also Optimal network design - the problem of finding a spanning set (not necessarily a tree) that minimizes the sum of shortest path lengths.

References

Worked examples

Example 1 — a first encounter with Minimum routing cost spanning tree

Start with the simplest possible case. Write down what Minimum routing cost 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 routing cost 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 routing cost 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 routing cost spanning tree

In research: Minimum routing cost 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 routing cost 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 routing cost spanning tree is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algorithms and data structures stubs, NP-complete problems, Spanning tree, so understanding it makes those chapters shorter.
In everyday life: Look for Minimum routing cost 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.

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How to study Minimum routing cost spanning tree in 20 minutes

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

Frequently asked questions

What is Minimum routing cost spanning tree in simple terms?

Why does Minimum routing cost 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 routing cost 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 routing cost spanning tree.