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labs/lab3-k-means/README.md

-# K-Means
-
-Use the following commands to make a fresh clone of your repository:
-
-```
-git clone -b kmeans git@gitlab.epfl.ch:lamp/student-repositories-s21/cs206-GASPAR.git cs206-kmeans
-```
-
-## Useful links
-
-  * [A guide to the Scala parallel collections](https://docs.scala-lang.org/overviews/parallel-collections/overview.html)
-  * [The API documentation of the Scala parallel collections](https://www.javadoc.io/doc/org.scala-lang.modules/scala-parallel-collections_2.13/latest/scala/collection/index.html)
-  * [The API documentation of the Scala standard library](https://www.scala-lang.org/files/archive/api/2.13.4)
-  * [The API documentation of the Java standard library](https://docs.oracle.com/en/java/javase/15/docs/api/index.html)
-
-**If you have issues with the IDE, try [reimporting the
-build](https://gitlab.epfl.ch/lamp/cs206/-/blob/master/labs/example-lab.md#ide-features-like-type-on-hover-or-go-to-definition-do-not-work),
-if you still have problems, use `compile` in sbt instead.**
-
-## Introduction
-
-In this assignment, you will implement the K-means algorithm for cluster
-detection, which is used to partition *n* vectors into *k* clusters.
-Here, vectors are separated into clusters based on their mutual similarity --
-vectors that are closer to each other in space are more likely to end up in the
-same cluster, and the distant vectors are likely to be in different clusters.
-K-means has many applications: it is used in data mining, image filtering and
-signal processing.
-
-Here is a simple example -- let's say that we have a set of vectors in 2D space,
-as shown in the following figure:
-
-![points](points.png)
-
-As a human, you can visually distinguish the three clusters of points in the
-image:
-
-![clusters](clusters.png)
-
-When the number of clusters, dimensions and vectors grows, it becomes difficult
-and even impossible to manually determine the clusters.
-K-means is a simple algorithm that takes a set of vectors (called *points*) and
-outputs as set of clusters as follows:
-
-1. Pick `k` points called *means*. This is called *initialization*.
-2. Associate each input point with the *mean* that is closest to it.
-   We obtain `k` *clusters* of points, and we refer to this process as
-   *classifying* the points.
-3. Update each mean to have the average value of the corresponding cluster.
-4. If the `k` means have significantly changed, go back to step 2.
-   If they did not, we say that the algorithm *converged*.
-5. The `k` means represent different clusters -- every point is in the cluster
-   corresponding to the closest mean.
-
-Above, two steps need additional discussion.
-First, how do we pick the initial `k` means?
-The initialization step can be done in many different ways -- we will just
-randomly pick some of the input vectors.
-Second, how do we know that the algorithm converged?
-We will check that, for each mean, the square distance between the old value of
-the mean and the new value of the mean is less than or equal to some value
-`eta`.
-
-For a better illustration, here are a few steps of the K-means algorithm.
-Initially, we pick a random set of means, shown with "X" in the figure:
-
-![step0](step0.png)
-
-Then, we classify the points according to the closest mean ("X").
-The means divide the space into regions, where each point is closer to the
-corresponding mean than any other mean -- in the figure, the dotted line depicts
-the borders of different regions:
-
-![step1](step1.png)
-
-All the points in the same region form one cluster. After having classified the
-points, we can update the mean values to the average of all the points in the
-cluster:
-
-![step2](step2.png)
-
-Each of the means was significantly updated.
-This is a good indication that the algorithm did not yet converge,
-so we repeat the steps again -- we first classify all the points:
-
-![step3](step3.png)
-
-And then we update the means again:
-
-![step4](step4.png)
-
-One of the means did not change at all in the last step.
-Still, other means have changed so we continue this process until the change
-in the position of each point drops below the `eta` value.
-
-At each iteration of K-means, we can associate multiple points to clusters,
-and compute the average of the `k` clusters, in parallel.
-Note that the association of a point to its cluster is independent of the
-other points in the input, and similarly, the computation of the average of a cluster
-is independent of the other clusters.
-Once all parallel tasks of the current iteration complete,
-the algorithm can proceed to the next iteration.
-
-K-means is an example of a *bulk synchronous parallel* algorithm (BSP).
-BSP algorithms are composed from a sequence of supersteps, each of which
-contains:
-
-- *parallel computation*, in which processes independently perform local
-  computations and produce some values
-- *communication*, in which processes exchange data
-- *barrier synchronisation*, during which processes wait until every process
-  finishes
-
-Data-parallel programming models are typically a good fit for BSP algorithms,
-as each bulk synchronous phase can correspond to some number of data-parallel
-operations.
-
-
-## Classifying the points
-
-In the first part of this assignment, you will classify the input points
-according to the square distance to the means.
-Input points are described with the following `Point` data-type:
-
-    class Point(val x: Double, val y: Double, val z: Double)
-
-You will start by implementing the `classify` method:
-
-    def classify(points: ParSeq[Point], means: ParSeq[Point]): ParMap[Point, ParSeq[Point]]
-
-This method take a sequence of points and a sequence of means, and return
-a map collection, which maps each mean to the sequence of points in the corresponding
-cluster.
-
-Hint: Use `groupBy` and the `findClosest` method, which is already defined for
-you. After that, make sure that all the means are in the resulting map, even if their
-sequences are empty.
-
-
-## Updating the means
-
-In the second part of this assignment, you will update the means corresponding
-to different clusters.
-
-Implement the method `update`, which takes the map of classified points produced
-in the previous step, and the sequence of previous means.
-The method returns the new sequence of means:
-
-    def update(classified: ParMap[Point, ParSeq[Point]], oldMeans: ParSeq[Point]): ParSeq[Point]
-
-Take care to preserve order in the resulting sequence -- the mean `i` in
-the resulting sequence must correspond to the mean `i` from `oldMeans`.
-
-Hint: Make sure you use the `findAverage` method that is predefined for you.
-
-
-## Detecting convergence
-
-Finally, you will implement convergence detection.
-The convergence detection method takes a sequence of old
-means and the sequence of updated means, and returns a boolean indicating if the
-algorithm converged or not.
-Given an `eta` parameter, `oldMeans` and `newMeans`, it returns `true` if the
-algorithm converged, and `false` otherwise:
-
-    def converged(eta: Double, oldMeans: ParSeq[Point], newMeans: ParSeq[Point])
-
-The algorithm converged iff the square distance between the old and the new mean is less
-than or equal to `eta`, for all means.
-
-Note: the means in the two lists are ordered -- the mean at `i` in `oldMeans`
-is the previous value of the mean at `i` in `newMeans`.
-
-Implement `converged`!
-
-
-## Running the algorithm
-
-We now have everything we need to run the K-means algorithm.
-We only need to combine the previously defined methods in the right way.
-
-The tail-recursive `kMeans` method takes a sequence of
-points `points`, previously computed sequence of means `means`, and the `eta`
-value:
-
-    @tailrec final def kMeans(points: ParSeq[Point],
-      means: ParSeq[Point], eta: Double): ParSeq[Point]
-
-The `kMeans` method should return the sequence of means,
-each corresponding to a specific cluster.
-
-Hint: `kMeans` implements the steps 2-4 from the K-means pseudocode.
-
-You can use the following command from within `sbt` to benchmark your code, the
-parallel time shoul be significantly faster than the sequential time if your
-computer has more than one core:
-
-    > runMain kmeans.KMeansRunner
-
-
-## Use cases
-
-And now for the fun part -- the K-means algorithm has a lot of use-cases!
-
-In image processing applications, it can be used to reduce the size of the
-color palette, thus compressing the image. This is done by turning a
-[true color image](http://en.wikipedia.org/wiki/Color_depth), where each pixel
-is encoded into 32 bits, into [indexed color](http://en.wikipedia.org/wiki/Indexed_color),
-where each pixel can be encoded with just a few bits. This is done by using k-means to
-"cluster" the important colors in the image, thus reducing its palette from
-24-bit (`2^24` colors) to just 32 indexed colors, chosen from the 24-bit palette.
-Here, pixels from the image are the input vectors,
-and their coordinates are the different color channels.
-
-This is the original true color (24-bit) image:
-
-![start](start.png)
-
-And this is the indexed color (32 colors) version of it:
-
-![uniform-random-30steps](uniform-random-30steps.png)
-
-So, thanks to your k-means implementation, ScalaShop can now compress images!
-You can start ScalaShop by invoking from `sbt`:
-
-    > runMain kmeans.fun.ScalaShop
-
-The k-means algorithm is very sensitive to the initial choice of means. There
-are three choice strategies implemented in ScalaShop:
-
-* `Uniform Choice` is the simplest strategy. It chooses `n` colors uniformly in
-the entire color space, regardless of the colors used in the image. If the image
-has a dominant color, the means created by this strategy will likely be very far
-away from the clusters formed by this dominant color. You can try setting the
-`Uniform Choice` strategy with 1, 10 and 30 steps. You will notice the initial
-choice is quite bad, but the quality improves as the k-means algorithm is applied
-in more steps.
-* `Random Sampling` is another simple strategy, but with better results. For the
-initial means, it randomly samples `n` colors from the image. This yields good
-results if the image has few dominant colors, but it cannot handle subtle nuances
-in the image. Again, if you try this strategy with 1, 10 and 30 k-means iteration
-steps, you will notice improvements as the k-means algorithm is ran more.
-* `Uniform Random` is the most complex strategy to pick means, but it also produces
-the best results. It works by uniformly splitting the color space in sub-spaces.
-It then counts the number of pixels that have colors belonging to that sub-space.
-Based on this number, it chooses a proportional number of means in the sub-space,
-by randomly sampling from the pixels in that sub-space. Therefore, if your image
-has dominant colors, this strategy will drop a proportional number of means for
-each dominant color, thus allowing the k-means algorithm to capture fine nuances.
-
-In the EPFL image now available in ScalaShop, the mountains are a good way to see
-how well each initial choice of means fares. You also have different strategies
-for deciding convergence:
-
-* `Steps` allows to run a fixed number of steps. After this, the k-means algorithm
-is stopped.
-* `Eta` corresponds to the means stability, as we showed earlier: if the
-means did not move much since the last iteration, the result is considered stable.
-* `Sound-to-noise` ratio is a more refined convergence strategy, which does not settle
-for stability but tries to minimize the difference between the true color image
-and the index color one. This strategy goes beyond `Eta`, but high Sound-to-noise
-ratios will prevent the k-means algorithm from finishing!
-
-With this in mind, enjoy ScalaShop, the ultimate image manipulation tool and
-the nice, warm, sunny photo of EPFL!