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-# 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-s22/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:
-
-
-
-As a human, you can visually distinguish the three clusters of points in the
-image:
-
-
-
-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:
-
-
-
-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:
-
-
-
-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:
-
-
-
-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:
-
-
-
-And then we update the means again:
-
-
-
-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:
-
-
-
-And this is the indexed color (32 colors) version of it:
-
-
-
-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!
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-# Barnes-Hut Simulation
-
-Use the following commands to make a fresh clone of your repository:
-
-```sh
-git clone -b barneshut git@gitlab.epfl.ch:lamp/student-repositories-s22/cs206-GASPAR.git cs206-barneshut
-```
-
-## Useful links
-
- * [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 parallel Barnes-Hut algorithm for *N-body simulation*.
-N-body simulation is a simulation of a system of *N* particles that interact with physical
-forces, such as gravity or electrostatic force.
-Given initial positions and velocities of all the *particles* (or *bodies*), the N-body simulation
-computes the new positions and velocities of the particles as the time progresses.
-It does so by dividing time into discrete short intervals, and computing the
-positions of the particles after each interval.
-
-<!--
-Let us recall some basics of classical mechanics.
-What makes a particle, such as an atom, a comet or a star, move through space?
-First, the particle moves through space as a consequence of having some initial velocity
-compared to other particles.
-Second, the particle changes its velocity as a result of external forces.
-These forces are in direct correlation with the proximity of other particles.
-For example, Earth moves through space with some initial velocity,
-which is constantly being changed due to the gravitational force between the Earth and the Sun.
--->
-
-Before we study the Barnes-Hut algorithm for the N-body simulation problem,
-we will focus on a simpler algorithm -- the *direct sum N-body algorithm*.
-The direct sum algorithm consists of multiple iterations, each of which performs
-the following steps for each particle:
-
-1. The particle position is updated according to its current velocity (`delta` is a short time period).
-
- x' = x + v_x * delta
- y' = y + v_y * delta
-
-2. The net force on the particle is computed by adding the individual forces from all the other particles.
-
- F_x = F_1x + F_2x + F_3x + ... + F_Nx
- F_y = F_1y + F_2y + F_3y + ... + F_Ny
-
-3. The particle velocity is updated according to the net force on that particle.
-
- v_x' = v_x + F_x / mass * delta
- v_y' = v_y + F_y / mass * delta
-
-In this exercise, we will assume that the force between particles is the *gravitational force*
-from classical mechanics. Let's recall the formula for the gravitational force between two stellar bodies:
-
- F = G * (m1 * m2) / distance^2
-
-Above, `F` is the absolute value of the gravitational force, `m1` and `m2` are the masses of the two bodies,
-and `r` is the distance between them. `G` is the gravitational constant.
-
-<!--
-The gravitational force vector always points towards the other body.
-The `F_x` and `F_y` components of the gravitational force on the body `m1`
-can be computed by observing the following triangle similarity:
-
-```scala
-distance = math.sqrt((x2 - x1) ^ 2 + (y2 - y1) ^ 2)
-F_x / F = (x2 - x1) / distance
-F_y / F = (y2 - y1) / distance
-```
-
-This is shown in the following figure:
-
-
--->
-
-For each particle, the net force is computed by summing the components of individual forces from all other particles,
-as shown in the following figure:
-
-
-
-The direct sum N-body algorithm is very simple, but also inefficient.
-Since we need to update `N` particles, and compute `N - 1` force contributions for each of those particles,
-the overall complexity of an iteration step of this algorithm is `O(N^2)`.
-As the number of particles grows larger, the direct sum N-body algorithm becomes prohibitively expensive.
-
-The Barnes-Hut algorithm is an optimization of the direct sum N-body algorithm,
-and is based on the following observation:
-
-> If a cluster of bodies is sufficiently distant from a body *A*, the net force on *A*
-> from that cluster can be approximated with one big body with the mass of all the
-> bodies in the cluster, positioned at the center of mass of the cluster.
-
-This is illustrated in the following figure:
-
-
-
-To take advantage of this observation, the Barnes-Hut algorithm relies on
-a *quadtree* -- a data structure that divides the space into cells, and answers queries
-such as 'What is the total mass and the center of mass of all the particles in this cell?'.
-The following figure shows an example of a quadtree for 6 bodies:
-
-
-
-Above, the total force from the bodies *B*, *C*, *D* and *E* on the body *A* can be approximated
-by a single body with *mass* equal to the sum of masses *B*, *C*, *D* and *E*,
-positioned at the center of mass of bodies *B*, *C*, *D* and *E*.
-The center of mass `(massX, massY)` is computed as follows:
-
-```scala
-mass = m_B + m_C + m_D + m_E
-massX = (m_B * x_B + m_C * x_C + m_D * x_D + m_E * x_E) / mass
-massY = (m_B * y_B + m_C * y_C + m_D * y_D + m_E * y_E) / mass
-```
-An iteration of the Barnes-Hut algorithm is composed of the following steps:
-
-1. Construct the quadtree for the current arrangement of the bodies.
- 1. Determine the *boundaries*, i.e. the square into which all bodies fit.
- 2. Construct a quadtree that covers the boundaries and contains all the bodies.
-2. Update the bodies -- for each body:
- 1. Update the body position according to its current velocity.
- 2. Using the quadtree, compute the net force on the body by adding the individual forces from all the other bodies.
- 3. Update the velocity according to the net force on that body.
-
-It turns out that, for most spatial distribution of bodies,
-the expected number of cells that contribute to the net force on a body is `log n`,
-so the overall complexity of the Barnes-Hut algorithm is `O(n log n)`.
-
-Now that we covered all the necessary theory, let's finally dig into the implementation!
-You will implement:
-
-- a quadtree and its combiner data structure
-- an operation that computes the total force on a body using the quadtree
-- a simulation step of the Barnes-Hut algorithm
-
-Since this assignment consists of multiple components,
-we will follow the principles of *test-driven development* and test each component separately,
-before moving on to the next component.
-That way, if anything goes wrong, we will more precisely know where the error is.
-It is always better to detect errors sooner, rather than later.
-
-
-## Data Structures
-
-We will start by implementing the necessary data structures: the quadtree,
-the body data-type and the sector matrix.
-You will find the stubs in the `package.scala` file of the `barneshut` package.
-
-
-### Quadtree Data Structure
-
-In this part of the assignment, we implement the quadtree data structure,
-denoted with the abstract data-type `Quad`.
-Every `Quad` represents a square cell of space, and can be one of the following node types:
-
-- an `Empty` node, which represents an empty quadtree
-- a `Leaf` node, which represents one or more bodies
-- a `Fork` node, which divides a spatial cell into four quadrants
-
-The definition of `Quad` is as follows:
-
-```scala
-sealed abstract class Quad {
- def massX: Float
- def massY: Float
- def mass: Float
- def centerX: Float
- def centerY: Float
- def size: Float
- def total: Int
- def insert(b: Body): Quad
-}
-```
-
-Here, `massX` and `massY` represent the center of mass of the bodies in the respective cell,
-`mass` is the total mass of bodies in that cell, `centerX` and `centerY` are the coordinates
-of the center of the cell, `size` is the length of the side of the cell,
-and `total` is the total number of bodies in the cell.
-
-Note that we consider the top left corner to be at coordinate (0, 0). We also consider the x axis to grow to the right and the y axis to the bottom.
-
-
-
-The method `insert` creates a new quadtree which additionally contains the body `b`,
-and covers the same area in space as the original quadtree.
-Quadtree is an *immutable* data structure -- `insert` does not modify the existing `Quad` object.
-Note that `Body` has the following signature:
-
-```scala
-class Body(val mass: Float, val x: Float, val y: Float, val xspeed: Float, val yspeed: Float)
-```
-
-In this part of the exercise, you only need to know about body's position `x` and `y`.
-
-Let's start by implementing the simplest `Quad` type -- the empty quadtree:
-
-```scala
-case class Empty(centerX: Float, centerY: Float, size: Float) extends Quad
-```
-
-The center and the size of the `Empty` quadtree are specified in its constructor.
-The `Empty` tree does not contain any bodies, so we specify that its center of mass is equal to its center.
-
-Next, let's implement the `Fork` quadtree:
-
-```scala
-case class Fork(nw: Quad, ne: Quad, sw: Quad, se: Quad) extends Quad
-```
-
-This node is specified by four child quadtrees `nw`, `ne`, `sw` and `se`,
-in the northwest, northeast, southwest and southeast quadrant, respectively.
-
-The northwest is located on the top left, northeast on the top right, southwest on the bottom left and southeast on the bottom right.
-
-The constructor assumes that the children nodes that represent four adjacent cells of the
-same size and adjacent to each other, as in the earlier figure.
-The center of the `Fork` quadtree is then specified by, say, the
-lower right corner of the quadtree `nw`. If the `Fork` quadtree is empty, the
-center of mass coincides with the center.
-
-Inserting into a `Fork` is recursive -- it updates the respective child and creates a new `Fork`.
-
-Finally, the `Leaf` quadtree represents one or more bodies:
-
-```scala
-case class Leaf(centerX: Float, centerY: Float, size: Float, bodies: Seq[Body])
-extends Quad
-```
-
-If the `size` of a `Leaf` is greater than a predefined constant `minimumSize`,
-inserting an additonal body into that `Leaf` quadtree creates a `Fork` quadtree
-with empty children, and adds all the bodies into that `Fork` (including the new body).
-Otherwise, inserting creates another `Leaf` with all the existing bodies and the new one.
-
-
-### The Body Data-Type
-
-Next, we can implement the `Body` data-type:
-
-```scala
-class Body(val mass: Float, val x: Float, val y: Float, val xspeed: Float, val yspeed: Float) {
- def updated(quad: Quad): Body = ???
-}
-```
-
-Here, `xspeed` and `yspeed` represent the velocity of the body, `mass` is its mass,
-and `x` and `y` are the coordinates of the body.
-
-The most interesting method on the `Body` is `updated` -- it takes a quadtree and
-returns the updated version of the `Body`:
-
-```scala
-def updated(quad: Quad): Body
-```
-
-This method is already half-completed for you -- you only need to implement
-its nested method `traverse`, which goes through the quadtree and proceeds casewise:
-
-- empty quadtree does not affect the net force
-- each body in a leaf quadtree adds some net force
-- a fork quadtree that is sufficiently far away acts as a single point of mass
-- a fork quadtree that is not sufficiently far away must be recursively traversed
-
-When are we allowed to approximate a cluster of bodies with a single point?
-The heuristic that is used is that the size of the cell divided by the distance
-`dist` between the center of mass and the particle is less than some constant `theta`:
-
-```scala
-quad.size / dist < theta
-```
-
-Hint: make sure you use the `distance` to compute distance between points,
-the `theta` value for the condition, and `addForce` to add force contributions!
-
-Before proceeding, make sure to run tests against your `Quad` and `Body` implementations.
-
-
-### The Sector Matrix
-
-The last data structure that we will implement is the *sector matrix*.
-In this data structure, we will use the auxiliary class `Boundaries`, which
-contains the `minX`, `maxX`, `minY` and `maxY` fields for the boundaries of the scene:
-
-```scala
-class Boundaries {
- var minX: Float
- var minY: Float
- var maxX: Float
- var maxY: Float
- def size = math.max(maxX - minX, maxY - minY)
-}
-```
-
-We will also rely on the `ConcBuffer` data structure, mentioned in the lecture:
-
-```scala
-class ConcBuffer[T]
-```
-
-The `ConcBuffer` class comes with efficient `+=`, `combine` and `foreach` operations,
-which add elements into the buffer, combine two buffers and traverse the buffer, respectively.
-The sector matrix additionally has the `toQuad` method, which returns a quadtree that contains all
-the elements previously added with the `+=` method.
-Recall from the lectures that this combination of methods make the `ConcBuffer` a *combiner*.
-
-The `SectorMatrix` is just a square matrix that covers a square region of space
-specified by the boundaries:
-
-```scala
-class SectorMatrix(val boundaries: Boundaries, val sectorPrecision: Int) {
- val sectorSize = boundaries.size / sectorPrecision
- val matrix = new Array[ConcBuffer[Body]](sectorPrecision * sectorPrecision)
- for (i <- 0 until matrix.length) matrix(i) = new ConcBuffer
- def apply(x: Int, y: Int) = matrix(y * sectorPrecision + x)
-}
-```
-
-The `sectorPrecision` argument denotes the width and height of the matrix, and
-each entry contains a `ConcBuffer[Body]` object. Effectively, the `SectorMatrix` is a *combiner* --
-it partitions the square region of space into `sectorPrecision` times `sectorPrecision` buckets, called *sectors*.
-
-
-
-Combiners such as the `SectorMatrix` are used in parallel programming to partition the results into
-some intermediate form that is more amenable to parallelization.
-Recall from the lecture that one of the ways to implement a *combiner* is by using
-a bucket data structure -- we will do exactly that in this part of the exercise!
-We will add three methods on the `SectorMatrix` that will make it a combiner.
-We start with the `+=` method:
-
-```scala
-def +=(b: Body): SectorMatrix = {
- ???
- this
-}
-```
-
-This method should use the body position, `boundaries` and `sectorPrecision`
-to determine the sector into which the body should go into,
-and add the body into the corresponding `ConcBuffer` object.
-
-Importantly, if the `Body` lies outside of the `Boundaries`, it should be considered
-to be located at the closest point within the `Boundaries` for the purpose of finding which `ConcBuffer` should hold the body.
-
-Next, we implement the `combine` method, which takes another `SectorMatrix`,
-and creates a `SectorMatrix` which contains the elements of both input
-`SectorMatrix` data structures:
-
-```scala
-def combine(that: SectorMatrix): SectorMatrix
-```
-
-This method calls `combine` on the pair of `ConcBuffer`s in `this` and `that`
-matrices to produce the `ConcBuffer` for the resulting matrix.
-You can safely assume that combine will only be called on matrices of same dimensions, boundaries and sector precision.
-
-
-
-
-The nice thing about the sector matrix is that a quadtree can be constructed
-in parallel for each sector. Those little quadtrees can then be linked together.
-The `toQuad` method on the `SectorMatrix` does this:
-
-```scala
-def toQuad(parallelism: Int): Quad
-```
-
-This method is already implemented -- you can examine
-it if you would like to know how it works.
-
-<!--
-Finally, we will implement the `toQuad` method, which uses the `SectorMatrix`
-to create a `Quad` tree in parallel:
-
-```scala
-def toQuad(parallelism: Int): Quad = {
- def BALANCING_FACTOR = 4
- def quad(x: Int, y: Int, span: Int, achievedParallelism: Int): Quad = {
- if (span == 1) {
- ???
- } else {
- ???
- }
- }
- quad(0, 0, sectorPrecision, 1)
-}
-```
-
-The `toQuad` method takes the desired degree of `parallelism` as an argument,
-and then calls the recursive `quad` method.
-The `quad` method operates on a subsquare of the matrix starting at `x`, `y` coordinates,
-with the length `span`.
-The fourth parameter, `achievedParallelism`, describes how many tasks have been
-created at a given call depth.
-
-Implement the `quad` method so that:
-
-1. If `span` is 1, the method sequentially converts the bodies in that sector to a quadtree.
- Use `foldLeft` and the `insert` method on the bodies in each sector to get the quadtree.
-2. Otherwise, if `parallelism > 1 && achievedParallelism < parallelism * BALANCING_FACTOR`,
- the method recursively, and in parallel, creates quadtrees for the `4` subsquares
- and links them together into a bigger quadtree.
-3. Otherwise, if the `parallelism == 1 || achievedParallelism >= parallelism * BALANCING_FACTOR`,
- the method does the same as in case 2, but without parallelism.
-
-Don't forget to multiply `achievedParallelism` by `4` at each step.
--->
-
-Congratulations, you just implemented your first combiner!
-Before proceeding, make sure to run those unit tests.
-
-
-## Implementing Barnes-Hut
-
-Now that we have all the right data structures ready and polished,
-implementing Barnes-Hut becomes a piece of cake.
-
-Take a look at the file `Simulator.scala`, which contains the implementation
-of the Barnes-Hut simulator, and in particular the `step` method.
-The `step` method represents one step in the simulation:
-
-```scala
-def step(bodies: Seq[Body]): (Seq[Body], Quad) = {
- // 1. compute boundaries
- val boundaries = computeBoundaries(bodies)
-
- // 2. compute sector matrix
- val sectorMatrix = computeSectorMatrix(bodies, boundaries)
-
- // 3. compute quadtree
- val quad = computeQuad(sectorMatrix)
-
- // 4. eliminate outliers
- val filteredBodies = eliminateOutliers(bodies, sectorMatrix, quad)
-
- // 5. update body velocities and positions
- val newBodies = updateBodies(filteredBodies, quad)
-
- (newBodies, quad)
-}
-```
-
-The pre-existing code in `step` nicely summarizes what this method does.
-
-
-### Computing the Scene Boundaries
-
-First, we must compute the boundaries of all the bodies in the scene.
-Since bodies move and the boundaries dynamically change,
-we must do this in every iteration of the algorithm.
-The `computeBoundaries` method is already implemented -- it uses the `aggregate`
-combinator on the sequence of bodies to compute the boundaries:
-
-```scala
-def computeBoundaries(bodies: Seq[Body]): Boundaries = {
- val parBodies = bodies.par
- parBodies.tasksupport = taskSupport
- parBodies.aggregate(new Boundaries)(updateBoundaries, mergeBoundaries)
-}
-```
-
-How does this work? The `aggregate` method divides the input sequence into a
-number of chunks. For each of the chunks, it uses the `new Boundaries` expression
-to create the accumulation value, and then folds the values in that chunk
-calling `updateBoundaries` on each body, in the same way a `foldLeft` operation would.
-Finally, `aggregate` combines the results of different chunks using a reduction tree and `mergeBoundaries`.
-
-So, we need the `updateBoundaries` method:
-
-```scala
-def updateBoundaries(boundaries: Boundaries, body: Body): Boundaries
-```
-
-Given an existing `boundaries` object and a body, the `updateBoundaries` updates
-the `minX`, `minY`, `maxX` and `maxY` values so that the boundaries include the body.
-
-Next, the `mergeBoundaries` method creates a new `Boundaries` object, which represents
-the smallest rectangle that contains both the input boundaries:
-
-```scala
-def mergeBoundaries(a: Boundaries, b: Boundaries): Boundaries
-```
-
-Question: Is `mergeBoundaries` associative? Is it commutative? Does it need to be commutative?
-
-Implement these two methods, and test that they work correctly!
-
-
-### Building the Quadtree
-
-Next, we need to build a `Quad` tree from the sequence of bodies.
-We will first implement the `computeSectorMatrix` method to get the `SectorMatrix`:
-
-```scala
-def computeSectorMatrix(bodies: Seq[Body], boundaries: Boundaries): SectorMatrix
-```
-
-Hint: aggregate the `SectorMatrix` from the sequence of bodies, the same way it was used for boundaries.
-Use the SECTOR_PRECISION constant when creating a new `SectorMatrix`.
-
-<!--
-Next, implement `computeQuad`, which converts `SectorMatrix` to a `Quad`:
-
- def computeQuad(sectorMatrix: SectorMatrix): Quad
-
-Hint: use `taskSupport.parallelismLevel` to determine the desired parallelism.
--->
-
-Test that these methods work correctly before proceeding!
-
-
-### Eliminating Outliers
-
-During the execution of the Barnes-Hut algorithm, some of the bodies tend to
-move far away from most of the other bodies. There are many ways to deal with such *outliers*,
-but to keep things simple, we will eliminate bodies that move too fast and too far away.
-
-We will not go into details of how this works, but if you'd like to know more,
-you can try to understand how the `eliminateOutliers` method works.
-
-
-### Updating Bodies
-
-The `updateBodies` method uses the quadtree to map each body from the
-previous iteration of the algorithm to a new iteration:
-
-```scala
-def updateBodies(bodies: Seq[Body], quad: Quad): Seq[Body]
-```
-
-Recall that we already implemented the `updated` method which updates a single body.
-
-
-## Running Barnes-Hut
-
-At last, the parallel Barnes-Hut algorithm is implemented.
-Note that, despite all the work, we kept our Barnes-Hut algorithm implementation simple
-and avoided the details that a more realistic implementation must address. In particular:
-
-- we represented each body as a single point in space
-- we restricted the simulation to two-dimensional space
-- we ignored close encounter effects, such as body collision or tearing
-- we ignored any relativistic effects, and assume classical mechanics
-- we ignored errors induced by floating point computations
-
-You can now run it as follows:
-
- > runMain barneshut.BarnesHut
-
-To visualize the quadtree, press the *Show quad* button, and then hit the *Start/Pause* button.
-
-Play with the parallelism level and the number of bodies, and observe the average speedups in the lower right corner.
-Then sit back, and enjoy the show!
-
diff --git a/labs/lab4-barnes-hut-simulation/cell.png b/labs/lab4-barnes-hut-simulation/cell.png
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