Graph reduction

Key takeaways

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

Reference excerpt

In computer science, graph reduction implements an efficient version of non-strict evaluation, an evaluation strategy where the arguments to a function are not immediately evaluated. This form of non-strict evaluation is also known as lazy evaluation and used in functional programming languages. The technique was first developed by Chris Wadsworth in 1971.

Motivation A simple example of evaluating an arithmetic expression follows:

( ( 2 + 2 ) + ( 2 + 2 ) ) + ( 3 + 3 )

= ( ( 2 + 2 ) + ( 2 + 2 ) ) + 6

= ( ( 2 + 2 ) + 4 ) + 6

= ( 4 + 4 ) + 6

= 8 + 6

= 14 {\displaystyle {\begin{aligned}&{}&&((2+2)+(2+2))+(3+3)\\&{}&=&((2+2)+(2+2))+6\\&{}&=&((2+2)+4)+6\\&{}&=&(4+4)+6\\&{}&=&8+6\\&{}&=&14\end{aligned}}}

The above reduction sequence employs a strategy known as outermost tree reduction. The same expression can be evaluated using innermost tree reduction, yielding the reduction sequence:

( ( 2 + 2 ) + ( 2 + 2 ) ) + ( 3 + 3 )

= ( ( 2 + 2 ) + 4 ) + ( 3 + 3 )

= ( 4 + 4 ) + ( 3 + 3 )

= ( 4 + 4 ) + 6

= 8 + 6

= 14 {\displaystyle {\begin{aligned}&{}&&((2+2)+(2+2))+(3+3)\\&{}&=&((2+2)+4)+(3+3)\\&{}&=&(4+4)+(3+3)\\&{}&=&(4+4)+6\\&{}&=&8+6\\&{}&=&14\end{aligned}}}

Notice that the reduction order is made explicit by the addition of parentheses. This expression could also have been simply evaluated right to left, because addition is an associative operation. Represented as a tree, the expression above looks like this:

This is where the term tree reduction comes from. When represented as a tree, we can think of innermost reduction as working from the bottom up, while outermost works from the top down. The expression can also be represented as a directed acyclic graph, allowing sub-expressions to be shared:

As for trees, outermost and innermost reduction also applies to graphs. Hence we have graph reduction. Now evaluation with outermost graph reduction can proceed as follows:

Notice that evaluation now only requires four steps. Outermost graph reduction is referred to as lazy evaluation and innermost graph reduction is referred to as eager evaluation.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Graph reduction

Start with the simplest possible case. Write down what Graph reduction 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 Graph reduction 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 Graph reduction 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 Graph reduction

In research: Graph reduction 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 Graph reduction 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: Graph reduction is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph algorithms, Graph rewriting, Implementation of functional programming languages, so understanding it makes those chapters shorter.
In everyday life: Look for Graph reduction 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 Graph reduction in 20 minutes

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

Frequently asked questions

What is Graph reduction in simple terms?

In computer science, graph reduction implements an efficient version of non-strict evaluation, an evaluation strategy where the arguments to a function are not immediately evaluated. This form of non-strict evaluation is also known as lazy evaluation and used in functional programming languages.

Why does Graph reduction 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 Graph reduction?

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 Graph reduction.