mathematics

Nested dissection

Nested dissection is a mathematics 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 Nested dissection rather than just read about it. In short: In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff.

Key takeaways

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

Reference excerpt

In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff. Nested dissection consists of the following steps:

Form an undirected graph in which the vertices represent rows and columns of the system of linear equations, and an edge represents a nonzero entry in the sparse matrix representing the system. Recursively partition the graph into subgraphs using separators, small subsets of vertices the removal of which allows the graph to be partitioned into subgraphs with at most a constant fraction of the number of vertices. Perform Cholesky decomposition (a variant of Gaussian elimination for symmetric matrices), ordering the elimination of the variables by the recursive structure of the partition: each of the two subgraphs formed by removing the separator is eliminated first, and then the separator vertices are eliminated. As a consequence of this algorithm, the fill-in (the set of nonzero matrix entries created in the Cholesky decomposition that are not part of the input matrix structure) is limited to at most the square of the separator size at each level of the recursive partition. In particular, for planar graphs (frequently arising in the solution of sparse linear systems derived from two-dimensional finite element method meshes) the resulting matrix has O(n log n) nonzeros, due to the planar separator theorem guaranteeing separators of size O(√n). For arbitrary graphs there is a nested dissection that guarantees fill-in within a O ( min { d log 4 ⁡ n , m 1 / 4 log 3.5 ⁡ n } ) {\displaystyle O(\min\{{\sqrt {d}}\log ^{4}n,m^{1/4}\log ^{3.5}n\})} factor of optimal, where d is the maximum degree and m is the number of non-zeros.

See also Cycle rank of a graph, or a symmetric Boolean matrix, measures the minimum parallel time needed to perform Cholesky decomposition Vertex separator

Notes

References George, J. Alan (1973), "Nested dissection of a regular finite element mesh", SIAM Journal on Numerical Analysis, 10 (2): 345–363, Bibcode:1973SJNA...10..345G, doi:10.1137/0710032, JSTOR 2156361. Gilbert, John R. (1988), "Some nested dissection order is nearly optimal", Information Processing Letters, 26 (6): 325–328, doi:10.1016/0020-0190(88)90191-3, hdl:1813/6607. Gilbert, John R.; Tarjan, Robert E. (1986), "The analysis of a nested dissection algorithm", Numerische Mathematik, 50 (4): 377–404, doi:10.1007/BF01396660. Lipton, Richard J.; Rose, Donald J.; Tarjan, Robert E. (1979), "Generalized nested dissection", SIAM Journal on Numerical Analysis, 16 (2): 346–358, Bibcode:1979SJNA...16..346L, doi:10.1137/0716027, JSTOR 2156840. Agrawal, Ajit; Klein, Philip; Ravi, R. (1993), "Cutting down on Fill Using Nested Dissection: Provably Good Elimination Orderings", Graph Theory and Sparse Matrix Computation, The IMA Volumes in Mathematics and its Applications, vol. 56, Springer New York, pp. 31–55, doi:10.1007/978-1-4613-8369-7_2, ISBN 978-1-4613-8371-0.

Worked examples

Example 1 — a first encounter with Nested dissection

Start with the simplest possible case. Write down what Nested dissection claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Nested dissection 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 Nested dissection 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 Nested dissection

In research: Nested dissection appears in mathematics 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 Nested dissection 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: Nested dissection is common in secondary-school and first-year university syllabi. It links to neighbouring topics Numerical linear algebra, Sparse matrices, so understanding it makes those chapters shorter.
In everyday life: Look for Nested dissection 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 Nested dissection in 20 minutes

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

Frequently asked questions

What is Nested dissection in simple terms?

Why does Nested dissection matter?

Because it connects several mathematics 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 Nested dissection?

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 Nested dissection.