computer science

Monge array

Monge array 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 Monge array rather than just read about it. In short: In mathematics applied to computer science, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge. An m-by-n matrix is said to be a Monge array if, for all i , j , k , ℓ {\displaystyle i,j,k,\ell } such that 1 ≤ i < k ≤ m and 1 ≤ j < ℓ ≤ n , {\displaystyle 1\leq i<k\leq m{\text{ and }}1\leq j<\ell \leq n,} one obtains A [ i , j ] + A [ k , ℓ ] ≤…

Key takeaways

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

Reference excerpt

In mathematics applied to computer science, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge. An m-by-n matrix is said to be a Monge array if, for all i , j , k , ℓ {\displaystyle i,j,k,\ell } such that

1 ≤ i < k ≤ m and 1 ≤ j < ℓ ≤ n , {\displaystyle 1\leq i<k\leq m{\text{ and }}1\leq j<\ell \leq n,}

one obtains

A [ i , j ] + A [ k , ℓ ] ≤ A [ i , ℓ ] + A [ k , j ] . {\displaystyle A[i,j]+A[k,\ell ]\leq A[i,\ell ]+A[k,j].}

In other words, for any two rows and two columns of a Monge array, the four elements at the intersection points (a 2 × 2 sub-matrix) have the property that the sum of the upper-left and lower-right elements (on the main diagonal) is less than or equal to the sum of the lower-left and upper-right elements (on the antidiagonal). This matrix is a Monge array:

[ 10 17 13 28 23 17 22 16 29 23 24 28 22 34 24 11 13 6 17 7 45 44 32 37 23 36 33 19 21 6 75 66 51 53 34 ] {\displaystyle {\begin{bmatrix}10&17&13&28&23\\17&22&16&29&23\\24&28&22&34&24\\11&13&6&17&7\\45&44&32&37&23\\36&33&19&21&6\\75&66&51&53&34\end{bmatrix}}}

For example, take the intersection of rows 2 and 4 with columns 1 and 5. The four elements are

[ 17 23 11 7 ] . {\displaystyle {\begin{bmatrix}17&23\\11&7\end{bmatrix}}.}

The sum of the upper-left and lower-right elements (17 + 7 = 24) is not larger than the sum of the lower-left and upper-right elements (23 + 11 = 34).

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Monge array

Start with the simplest possible case. Write down what Monge array 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 Monge array 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 Monge array 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 Monge array

In research: Monge array 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 Monge array 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: Monge array is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theoretical computer science, so understanding it makes those chapters shorter.
In everyday life: Look for Monge array 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 Monge array in 20 minutes

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

Frequently asked questions

What is Monge array in simple terms?

Why does Monge array 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 Monge array?

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 Monge array.