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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-s21/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!
+
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