From f7b40a6d15f91fec31385aedced0a86e9222a7e5 Mon Sep 17 00:00:00 2001
From: Olivier Blanvillain <olivier.blanvillain@gmail.com>
Date: Mon, 1 Mar 2021 11:28:01 +0100
Subject: [PATCH] Add labs/lab2-reductions-and-prefix-sums
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diff --git a/labs/lab2-reductions-and-prefix-sums/README.md b/labs/lab2-reductions-and-prefix-sums/README.md
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+# Reductions and Prefix Sums
+
+Use the following commands to make a fresh clone of your repository:
+
+```
+git clone -b reductions git@gitlab.epfl.ch:lamp/student-repositories-s21/cs206-GASPAR.git cs206-reductions
+```
+
+## 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 several variants of reduction and prefix
+sum algorithms. Each of the three parts of the assignment will exercise a
+different aspect of parallel programming:
+
+- choosing the right parallelization threshold
+- identifying the correct reduction operator
+- identifying the correct prefix sum operator
+
+We will use the `parallel` construct, defined in the file `package.scala`, as in the lecture to start parallel
+computations. Every `parallel` construct invocation takes two tasks as input and
+outputs the corresponding results as a tuple of two elements. You are not allowed
+to use the `task` construct in this assignment.
+
+## Parallel Counting Change
+
+If you took the course Functional Programming in Scala, you surely recall the
+assignment in which you had to count the number of ways in which you can make
+the change for a given amount of money.
+The text of that assignment was as follows:
+
+> Write a recursive function that counts how many different ways you can
+> make change for an amount, given a list of coin denominations. For
+> example, there are 3 ways to give change for 4 if you have coins with
+> denomination 1 and 2: 1+1+1+1, 1+1+2, 2+2.
+
+In this assignment, you will repeat the same task, but this time, your
+implementation will be parallel.
+Start with the sequential version of this problem once more -- the `countChange`
+function takes the amount of money and the list of different coin denominations.
+It returns the total number of different ways you can give change:
+
+```scala
+def countChange(money: Int, coins: List[Int]): Int
+```
+
+Note that the solution to this problem is recursive.
+In every recursive call, we either decide to continue subtracting the next coin
+in the `coins` list from the `money` amount, or we decide to drop the coin from the list of coins.
+For example, if we have 4 CHF, and coin denominations of 1 and 2, the call
+graph, in which every node depicts one invocation of the `countChange` method,
+is as follows:
+
+ 4,[1, 2]
+ 3,[1, 2] + 4,[2]
+ 2,[1, 2] + 3,[2] 2,[2] + 4,[]
+ 1,[1, 2] + 2,[2] 1,[2] + 3,[] 0,[2] + 2,[] 0
+ 0,[1, 2] + 1,[2] 1 0 0 1 0
+ 1 0
+
+We can take advantage of this recursive structure by evaluating different
+subtrees in parallel.
+This is the next part of the assignment -- implement the method `parCountChange`
+that counts the amount of change in parallel:
+
+```scala
+def parCountChange(money: Int, coins: List[Int], threshold: Threshold): Int
+```
+
+As we learned in the lectures, the `parCountChange` should not spawn parallel
+computations after reaching the leaf in the call graph -- the synchronization
+costs of doing this are way too high.
+Instead, we need to *agglomerate* parts of the computation.
+We do this by calling the sequential `countChange` method when we decide that
+the amount of work is lower than a certain value, called the *threshold*.
+To separate the concern of deciding on the threshold value from the
+implementation of our parallel algorithm, we implement the threshold
+functionality in a separate function, described by the `Threshold` type alias:
+
+```scala
+type Threshold = (Int, List[Int]) => Boolean
+```
+
+When a `threshold` function returns `true` for a given amount of money and the
+given list of coins, the sequential `countChange` implementation must be called.
+
+Implement `parCountChange`!
+
+Now that we have the `parCountChange` method, we ask ourselves what is the right
+implementation of the `threshold` function?
+Recall the examples from the lectures, such as summing the array values and
+computing the norm, where this was easy -- we exactly knew the amount of work
+required to traverse a subrange of the array, so `threshold` could return `true`
+when the length of the subrange was smaller than a certain value.
+
+Sadly, the total amount of work for a given `parCountChange` invocation is hard
+to evaluate from the remaining amount of money and a list of coins.
+In fact, the amount of work directly corresponds to the count that
+`parCountChange` returns, which is the value that we are trying to compute.
+Counting change is a canonical example of a task-parallel problem in which the
+partitioning the workload across processors is *solution-driven* -- to know how
+to optimally partition the work, we would first need to solve the problem
+itself.
+
+For this reason, many parallel algorithms in practice rely on heuristics to
+assess the amount of work in a subtask.
+We will implement several such heuristics in this exercise, and assess the
+effect on performance.
+First, implement the `moneyThreshold` method, which creates a threshold function
+that returns `true` when the amount of money is less than or equal to `2 / 3` of
+the starting amount:
+
+```scala
+def moneyThreshold(startingMoney: Int): Threshold
+```
+
+Remember that `a / b` will return the integer division of `a` and `b` when both
+operands are `Int`s. To avoid this problem, be sure to always do the multiplication of
+`startingMoney` by `2` before doing the division by `3`.
+
+Now run the `ParallelCountChange` application in sbt and observe the speedup:
+
+```
+> runMain reductions.ParallelCountChangeRunner
+```
+
+The previous heuristic did not take into account how many coins were left on the
+coins list, so try two other heuristics.
+Implement the method `totalCoinsThreshold`, which returns a threshold function
+that returns `true` when the number of coins is less than or equal to the `2 / 3` of the
+initial number of coins:
+
+```scala
+def totalCoinsThreshold(totalCoins: Int): Threshold
+```
+
+Again, be careful about the order of operations.
+
+Then, implement the method `combinedThreshold`, which returns a threshold
+function that returns `true` when the amount of money multiplied with the number
+of remaining coins is less than or equal to the starting money multiplied with
+the initial number of coins divided by `2`:
+
+```scala
+def combinedThreshold(startingMoney: Int, allCoins: List[Int]): Threshold
+```
+
+Which of the three threshold heuristics gives the best speedup?
+Can you think of a heuristic that improves performance even more?
+
+
+## Parallel Parentheses Balancing
+
+In this part of the assignment, we recall the Parenthesis Balancing assignment
+that might be familiar to you from the Functional Programming in Scala course.
+Here, the task is to, given an array of characters, decide if the parentheses in
+the array are balanced.
+
+Let us recall a few examples of strings in which parentheses are correctly
+balanced:
+
+```
+(if (zero? x) max (/ 1 x))
+I told him (that it's not (yet) done). (But he wasn't listening)
+```
+
+Similarly, the parentheses in the following strings are not balanced:
+
+```
+(o_()
+:-)
+())(
+```
+
+Implement a sequential function `balance`, which returns `true` iff the
+parentheses in the array are balanced:
+
+```scala
+def balance(chars: Array[Char]): Boolean
+```
+
+Next, you will implement a parallel version of this method.
+By now, you're already an expert at implementing the structure of a reduction
+algorithm, so you should have no problem there.
+The tricky part in parallel parentheses balancing is choosing the reduction
+operator -- you probably implemented `balance` by keeping an integer
+accumulator, incrementing it for left parentheses and decrementing it for the
+right ones, taking care that this accumulator does not drop below zero.
+Parallel parentheses balancing will require a bit more ingenuity on your part,
+so we will give you a hint -- you will need two integer values for the
+accumulator.
+
+Implement the `parBalance` method, which checks if the parentheses in the input
+array are balanced using two helper methods `reduce` and `traverse`.
+These methods implement the parallel reduction and the sequential traversal
+part, respectively:
+
+```scala
+def parBalance(chars: Array[Char], threshold: Int): Boolean = {
+ def traverse(idx: Int, until: Int, _???_: Int, _???_: Int): ???
+
+ def reduce(from: Int, until: Int): ??? = ???
+
+ reduce(0, chars.length) == ???
+}
+```
+
+In this case, we again use the fixed threshold parameter, as we did in the
+lectures. Sections with size smaller or equal to the threshold should be processed sequentially.
+For maximum performance, use a `while` loop in the `traverse` method, or make
+`traverse` tail-recursive -- do not use a `Range`.
+
+Now, run the `ParallelParenthesesBalancing` application from sbt:
+
+```
+> runMain reductions.ParallelParenthesesBalancingRunner
+```
+
+How large was your speedup?
+
+If you are looking for additional challenges, prove that your reduction operator
+is associative!
+
+
+## Line of Sight
+
+In the last part of the exercise, you will be implementing an entirely new
+parallel algorithm -- you will apply the prefix sum algorithm to computing the
+line-of-sight in two-dimensional terrain.
+
+Imagine that you are standing at the zero of a coordinate system.
+The curve to your right describes the terrain that you are facing.
+This is shown in the following figure:
+
+
+
+The task of the line-of-sight algorithm is to compute the visibility of each
+point of the terrain, as shown in the following figure, where the visible area
+is above of the full line, and the obscured terrain is shown with a dotted
+line.
+
+
+
+What is the necessary and sufficient condition for a point on the terrain to be
+visibile from the zero of the coordinate system, where you are standing?
+Imagine that the terrain heights are represented with an array of numbers.
+We can compute (the tangent of) the viewing angle of each point on the terrain by dividing the
+height of the terrain `xs(i)` with the distance from the viewing point `i`,
+as shown in the following figure:
+
+
+
+It turns out that if the viewing angle of some point B is **lower** than the
+viewing angle of an earlier point A, then the point B is not visible, as shown
+in the following figure:
+
+
+
+This simple realization allows us to easily compute the line-of-sight on the
+terrain -- if you were a sequential programmer, you would traverse the array of
+height values from the beginning to the end, and write the maximum angle seen so
+far into the output array.
+
+Implement the sequential `lineOfSight` method, which, for each height entry in
+the `input` array (except for input(0) which is the location of the observer and
+is always zero), writes the maximum angle until that point into the `output`
+array (output(0) should be 0):
+
+```scala
+def lineOfSight(input: Array[Float], output: Array[Float]): Unit
+```
+
+We keep things simple -- instead of outputting an array of booleans denoting the
+visibilities, we only output the angles.
+
+Note that what we call an angle in this assignment is actually the tangent
+of the angle. Indeed, `xs(i)` is the opposing side of the angle and `i` the adjacent side.
+The ratio `xs(i) / i` that you compute is in fact the tangent of the angle!
+Since the tangent of an angle is strictly increasing between
+0° and 90°, it is a perfectly good replacement for the actual angle in our use case.
+Keep this in mind and make sure that you do not apply any trigonometic functions on the tangent!
+
+When we see a sequential algorithm that produces a sequence of values by
+traversing the input from left to right, this is an indication that the
+algorithm might have a parallel prefix sum variant.
+So let's try to implement one!
+
+Recall what you learned in the lectures -- the first phase of the parallel
+prefix sum algorithm is the *upsweep* phase.
+Here, the algorithm constructs the reduction tree by traversing parts of the
+input array in parallel.
+Implement the method `upsweepSequential`, which returns the maximum angle in a
+given part of the array, and the method `upsweep`, which returns the reduction
+tree over parts of the input array. If the length of the given part of the input
+array is less than or equal to `threshold`, then `upsweep` calls `upsweepSequential`.
+Note that the part of the input array that needs to traversed is represented
+using indices 'from' (inclusive) and 'until' (or 'end') (exclusive).
+
+```scala
+def upsweepSequential(input: Array[Float], from: Int, until: Int): Float
+```
+
+```scala
+def upsweep(input: Array[Float], from: Int, end: Int, threshold: Int): Tree
+```
+
+The `Tree` data type is either a `Leaf` or an inner `Node`, and it contains the
+maximum angle in the corresponding part of the array.
+Note that when the number of elements in a part of the input array,
+which is `(end - from)`, is smaller or equal to the threshold, the sequential `upsweepSequential`
+has to be invoked, and you should return a `Leaf`.
+Otherwise, you should process the part of the input array in parallel, and return
+a `Node`. Make sure that the work is evenly distributed between the parallel computations.
+
+The second phase is called *downsweep* -- here, the algorithm uses the tree to
+push the maximum angle in the corresponding *prefix* of the array to the leaves
+of the tree, and outputs the values.
+Implement the methods `downsweep` which processes parts of the tree in parallel,
+and the method `downsweepSequential`, which traverses the parts of the array
+corresponding to leaves of the tree and writes the final angles into the
+`output` array:
+
+```scala
+def downsweep(input: Array[Float], output: Array[Float],
+ startingAngle: Float, tree: Tree): Unit
+
+def downsweepSequential(input: Array[Float], output: Array[Float],
+ startingAngle: Float, from: Int, until: Int): Unit
+```
+
+Finally, implement `parLineOfSight` using the `upsweep` and `downsweep` methods:
+
+```scala
+def parLineOfSight(input: Array[Float], output: Array[Float],
+ threshold: Int): Unit
+```
+
+Now, run the `LineOfSight` application in sbt and observe the relative speedups:
+
+```
+> runMain reductions.LineOfSightRunner
+```
+
+How large is the speedup compared to the number of cores in your processor?
+Can you explain your results?
diff --git a/labs/lab2-reductions-and-prefix-sums/angle-example-2.svg b/labs/lab2-reductions-and-prefix-sums/angle-example-2.svg
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new file mode 100755
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