50 lines
3.9 KiB
Markdown
50 lines
3.9 KiB
Markdown
# Exercises
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Here let be listed exercises for the readers of this wiki. You can allow yourself to as many helpers and resources as you find challenging: with each problem you should either find out you know the solution or learn something new while solving it.
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1. What's the difference between [free software](free_software.md) and [open source](open_source.md)?
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2. Say we have an algorithm that finds all pairs of equal numbers in an array of numbers of length *N* and adds all of these (unordered) pairs to a set *S*. The algorithm is (pseudocode): `for i := 0 to N: for j := 0 to N: if numbers[i] == numbers[j]: add(S,pair(i,j))`. How can we optimize the algorithm in terms of its execution speed (i.e. make it perform fewer operations while keeping its results the same)? How did the asymptotic time complexity ("big O") class change?
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3. In computer graphics, what is the difference between ray casting, ray tracing and path tracing?
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4. Why are manhole lids round and not square?
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5. Give one real-life example of each of the following binary [relation](relation.md) types: [transitive](transitivity.md) but not equivalence, non-transitive, [antisymmetric](antisymmetric.md) and symmetric at the same time, [asymmetric](asymmetry.md), non-trivial [equivalence](equivalence.md).
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## Solutions
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A solution to each problem should be listed here -- keep in mind there may exist other solutions that those listed here.
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**solution 1**:
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Both movements share very similar rules of licensing and technically free software and open-source are largely the same. However, free software is fundamentally aiming for the creation of ethical software -- that which respects its user's freedom -- while open source is a later movement that tries to adapt free software for the business and abandons the pursuit of ethics, i.e. it becomes corrupted by capitalism and no longer minds e.g. proprietary dependencies.
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**solution 2**:
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In the given algorithm we compare all numbers twice. This can be avoided by not comparing a number to previous numbers in the array (because these have already been compared). Additionally we don't have to compare the same number to itself, a number will always be equal to itself:
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```
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for i := 0 to N:
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add(S,i,i) // no need to compare
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for i := 0 to N:
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for j := i + 1 to N:
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if numbers[i] == numbers[j]:
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add(S,pair(i,j))
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```
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While the first algorithm performs N^2 comparisons, the new one only needs N - 1 + N - 2 + N - 3 + ... ~= N * N / 2 = N^2 / 2 comparisons. Even though the new version is always twice as fast, its time complexity class remains the same, that is O(N^(2)).
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**solution 3**:
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They are all image-order methods of 3D [rendering](rendering.md). [Ray casting](ray_casting.md) casts a single ray for each screen pixel and determines the pixel color from a single hit of the ray. [Ray tracing](ray_tracing.md) is a [recursive](recursion.md) form of ray casting -- it recursively spawns secondary rays from the first hit to more accurately determine the pixel color, allowing for effects such as shadows, reflections or refractions. Path tracing is a method also based on casting rays, but except for the primary rays the rays are cast at random (i.e. it is a [Monte Carlo](monte_carlo.md) method) to approximately solve the rendering equation, progressively computing a more accurate version of the image (i.e. the image contains significant noise at the beginning which lowers with more iterations performed) -- this allows computing [global illumination](global_illumination.md), i.e. a very realistic lighting that the two previous methods can't achieve.
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**solution 4**:
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Round lid can't fall into the hole.
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**solution 5**:
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- transitive, not equivalence: *A* is a descendant from *B*.
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- non-transitive: *A* is a friend of *B*.
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- symmetric: *A* had sex with *B*.
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- antisymmetric and symmetric: *A* is *B*.
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- asymmetric: *A* is a mother of *B*.
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- non-trivial equivalence: *A* is of same age as *B*. |