## Walking the Euler Path: PIN Cracking and DNA Sequencing

Continuing on to some cool applications of Eulerian paths.

The goal of this little graph experiment remains exploration of accelerating Eulerian path finding on the GPU. This is the final introductory post.

### Eulerian Path

Hierholzer algorithm works great. It’s linear in the number of edges, so as fast as we can possibly have. The idea is simple: pick a vertex, walk the graph, removing used edges from consideration and adding visited vertices to a stack, once we circle back to a vertex without edges – pop it from the stack and pre-pend it to the path. Once the stack is empty and all edges have been traversed – we have the path/cycle.

member this.FindEulerCycle (?start) = let mutable curVertex = defaultArg start 0 let stack = Stack<int>() let connections = Dictionary<int, int []>() let start = curVertex let mutable cycle = [] connections.Add(curVertex, this.GetConnectedVertices curVertex) let mutable first = true while stack.Count > 0 || first do first <- false let connected = connections.[curVertex] if connected.Length = 0 then cycle <- curVertex :: cycle curVertex <- stack.Pop() else stack.Push curVertex connections.[curVertex] <- connected.[1..] curVertex <- connected.[0] if not (connections.ContainsKey curVertex) then connections.Add(curVertex, this.GetConnectedVertices curVertex) let path = start::cycle if path.Length <> this.NumEdges + 1 then [] else start::cycle |> List.map (fun i -> verticesOrdinalToNames.[i])

Here we don’t check for pre-conditions on whether the graph has an Eulerian path/cycle, since this check is expensive enough that failure of the algorithm can serve as such a check.

Getting connected vertices (outgoing edges) is as fast as getting a sub-range. We only do it once for every vertex, then these are stored in a dictionary and mutated as we remove “used” edges, so the graph itself remains immutable. In our representation, getting outgoing edges is easy:

let getVertexConnections ordinal = let start = rowIndex.[ordinal] let end' = rowIndex.[ordinal + 1] - 1 colIndex.[start..end']

### De Bruijn Sequence

On a seemingly unrelated, but actually intimately related topic. Given an alphabet of m characters, create a cyclical sequence which:

- Contains all sub-sequences of length n, and
- Does not have any repeating sub-sequences of length n

The sequence is cyclical in a sense that we recover all its subsequences of length n by sliding a cyclical window over the sequence. So, for example, for the binary alphabet and n=3:

We can traverse the graph in order of the marked edges and record each edge label, thus getting the sequence: `01011100`

. This is a cyclical sequence, we just broke it in an arbitrary way. Sliding the n=3 length window we’ll get all the 3-digit binary numbers.

We get the sequence by first constructing the De Bruijn Graph from our sequence of numbers. The graph is constructed by taking all the sequences of length n – 1 and connecting them “prefix-to-suffix”, where for each sequence of length n, prefix (suffix) is the subsequence of the first (last) n – 1 characters of this sequence. So, for instance, in the above example, vertex ’00’ is a prefix of ‘001’, while ’01’ is its suffix. So while ’00’ and ’01’ are both vertices, they are linked with the edge that is labelled by the character necessary to create the entire number of length n (001) by moving from prefix to suffix: 00 -> 01, label: 1.

The resulting graph has a Eulerian cycle as it is easy enough to see by induction. We recover the sequence by traversing the cycle, and since we traverse all the edges only once, we’ll get exactly what we are looking for.

let prefix (s:string) = s.[..s.Length - 2] let suffix (s:string) = s.[1..] let prefSuf s = prefix s, suffix s // shorthand let numToBinary len n = let rec numToBinaryRec n len acc = if len = 0 then acc else numToBinaryRec (n >>> 1) (len - 1) (String.Format("{0}{1}", n &&& 0x1, acc)) numToBinaryRec n len "" let binaryDebruijnSeq n = if n <= 0 then failwith "n should be positive" let finish = pown 2 n let gr = [0..finish-1] |> List.map (numToBinary n >> prefSuf) |> List.groupBy fst |> List.map (fun (v, prefSuf) -> v + " -> " + (prefSuf |> List.map snd |> List.reduce (fun st e -> st + "," + e ))) |> DirectedGraph<string>.FromStrings let debruinSeq = gr.FindEulerPath() let debruinNum = debruinSeq |> List.windowed 2 |> List.mapi (fun i [p; s] -> "\"" + (i + 1).ToString() + ":" + s.[s.Length - 1].ToString() + "\"") gr.Visualize(euler = true, eulerLabels = debruinNum)

Here the function `binaryDeruijnSeq`

computes a prefix and a suffix of all n-digit binary numbers, then groups prefixes together and builds a collection of graph strings in my notation: , connecting a prefix to all its suffixes. After that, the collection is converted into an instance of a `DirectedGraph`

class, the Eulerian cycle is found and visualized in such a way, that starting from the green vertex, moving along the edges that mark the Eulerian cycle, we recover the De Bruijn sequnce by recording the edge labels.

### PIN Cracking

If we have a device protected by a 4-digit pin, such that punching in a few numbers in a sequence will unlock the device as long as there is a correct subsequence punched, we can use the De Bruijn approach above to generate a 10,000 long sequence that will necessarily yield the correct PIN in only 10,000 punches, as opposed to 40,000. See this article that describes it in some detail.

### DNA Sequencing

My favorite application, of course, is to DNA sequencing. DNA is sequenced from a bunch of reads. The reads are not very long – maybe around 300 nucleotides, maybe less. They are not always perfect either: some nucleotide or a few may not be produced correctly by the sequencer. Still, if we can gather enough of them together, align and error-correct, we could then build a De Bruijn graph much the same way as described above thus linking the reads together in a DNA sequence. This is of course a gross oversimplification, but it is the reason why I love Eulerian cycles and the source of my interest in speeding up algorithms of finding them.

In the future posts – more forays into the GPU-land in an attempt to speed up something already pretty fast and what came out of it.

## Walking the Euler Path: GPU for the Road

Continuation of the previous posts:

#### GPU Digression

I was going to talk about something else this week but figured I’d take advantage of the free-hand format and digress a bit.

Continuing the travel metaphor and remembering Julius Cesar’s “*alea* iacta”, we’ll talk about GPU algorithms, for which I invariably use my favorite *Aela*.CUDA library.

#### GPU Distinct

I have already talked about sorting & splitting non-negative integer arrays on the GPU. Another one in this small library is implementing distinct on the GPU. It is using the same ubiquitous scan algorithm as before:

let distinctGpu (dArr : DeviceMemory<int>) = use dSorted = sortGpu dArr use dGrouped = worker.Malloc<int>(dSorted.Length) let lp = LaunchParam(divup dSorted.Length blockSize, blockSize) worker.Launch <@ distinctSortedNums @> lp dSorted.Ptr dSorted.Length dGrouped.Ptr compactGpuWithKernel <@createDistinctMap @> dGrouped

- We first sort the array, so all non-distinct values get grouped together. (Using radix sort on the GPU), step complexity O(k), where k – maximum number of bits across all numbers in the array
- We then replace all values in the group except the first one with 0. One kernel invocation, so O(1) step complexity
- Compact: a variation on scan algorithm with O(log n) steps

So we have the O(log n) step and O(n) work complexity for this version of distinct. The regular linear distinct is O(n). So, is it worth it?

Here is how we test:

let mutable sample = Array.init N (fun i -> rnd.Next(0, 1000)) GpuDistinct.distinct sample

Here is the comparison:

Length: 2,097,152 CPU distinct: 00:00:00.0262776 GPU distinct: 00:00:02.9162098 Length: 26,214,400 CPU distinct: 00:00:00.5622276 GPU distinct: 00:00:03.2298218 Length: 262,144,000 CPU distinct: 00:00:03.8712437 GPU distinct: 00:00:05.7540822

Is all this complexity worth it? It’s hard to say, because as it is obvious from the above numbers, there is a lot of latency in the Alea.CUDA scan, which makes its application useful only once we have an array sufficiently large to hide this latency.

I could not do much in terms any further comparison – ran out of GPU memory before I ran out of .NET object size limitation.

The final comparison:

Length: 300,000,000 CPU distinct: 00:00:04.2019013 GPU distinct: 00:00:06.7728424

The CPU time increase ratio is 1.11, while the GPU increase was 1.18, while the increase in the size of our data is 1.14 – so not really informative: all we can see is that the work complexity is indeed O(n) in both cases, and that’s certainly nothing new. We could responsibly claim, however, that if it weren’t for the latency, our GPU implementation would be faster. Perhaps switching to C++ would confirm this statement.

### Computing Graph Properties

#### Motivation

Remember, for the visuals, we wanted to clearly identify vertices with certain numbers of incoming/outgoing edges. Another case: implementing the spanning tree algorithm, it is necessary to “convert” the directed graph to undirected. This is not a real conversion, we would just need to make sure that if (a -> b) exists in the graph, it means that (a b), i.e. – edges are connected no matter the direction. Our spanning tree should be using “weak” connectivity:

let euler = StrGraph.GenerateEulerGraph(8, 3, path=true) euler.Visualize(spanningTree=true, washNonSpanning=false)

Here red edges mark the “spanning” tree, this graph is “almost” strongly connected – it has an Euler path.

#### Graph as an Iterable

We need an ability to iterate over the vertices of our graph. So, we should be implementing `IEnumerable<DirectedGraph> `

to accomplish this, right? Wrong! What we want is the `AsEnumerable`

property. Makes things clean and easy. It uses `Seq.init`

method – which comes very handy any time we need to turn our data structure into an iterable quickly and cleanly.

member this.AsEnumerable = Seq.init nVertices (fun n -> nameFromOrdinal n, this.[nameFromOrdinal n])

Now we can also do ourselves a favor and decorate our class with the `StructuredFormatDisplay("{AsEnumerable}")`

to enable F# Interactive pretty printing of our graph:

[<StructuredFormatDisplay("{AsEnumerable}")>] type DirectedGraph<'a when 'a:comparison> (rowIndex : int seq, colIndex : int seq, verticesNameToOrdinal :

Now if we just type the name of an instantiated graph in the interactive, we’ll get something like:

val it : DirectedGraph = seq [("0", [|"2"|]); ("1", [|"2"|]); ("2", [|"3"; "4"; "5"|]); ("3", [|"5"; "6"; "7"|]); ...]

We can further improve on what we see by calling

gr.AsEnumerable |> Seq.toArray

to completely actualize the sequence and see the textual representation of the entire graph.

#### “Reverse” Graph

So, if we want all the above goodies (number of in/out edges per vertex, spanning tree), we need to extract the array of actual edges, as well as be able to compute the “reverse” graph. The “reverse” graph is defined as follows:

Given ,

That is for every edge of the original graph, the edges of the new one are created by reversing the original edges’ direction. In order to reverse the edges direction we must first obtain the edges themselves. If an edge is represented as a tuple , we can flip it, group by the first element, sort and thus obtain the two structures needed for the new, “reverse”, incidence matrix.

This can get time-consuming, that’s why we use F# lazy values to only invoke the computation once, when we actually need it:

let reverse = lazy ( let allExistingRows = [0..rowIndex.Length - 1] let subSeq = if hasCuda.Force() && rowIndex.Length >= gpuThresh then //use CUDA to reverse let start, end' = let dStart, dEnd = getEdgesGpu rowIndex colIndex sortStartEnd dStart dEnd Seq.zip end' start else asOrdinalsEnumerable () |> Seq.map (fun (i, verts) -> verts |> Seq.map (fun v -> (v, i))) |> Seq.collect id let grSeq = subSeq |> Seq.groupBy fst |> Seq.map (fun (key, sq) -> key, sq |> Seq.map snd |> Seq.toArray) let allRows : seq<int * int []> = allExistingRows.Except (grSeq |> Seq.map fst) |> Seq.map (fun e -> e, [||]) |> fun col -> col.Union grSeq |> Seq.sortBy fst let revRowIndex = allRows |> Seq.scan (fun st (key, v) -> st + v.Length) 0 |> Seq.take rowIndex.Length let revColIndex = allRows |> Seq.collect snd DirectedGraph(revRowIndex, revColIndex, verticesNameToOrdinal) ) member this.Reverse = reverse.Force()

On line 35, `.Force()`

will only call the computation once and cache the result. Each subsequent call to `.Force()`

will retrieve the cached value.

It’s worth mentioning what code on line 24 is doing. By now we have the array of all “terminal” vertices, which will become the new “outgoing” ones. However if the original graph had vertices with nothing going into them, they will have nothing going out of them in the current graph, and thus the new “reversed” `grSeq`

will be incomplete. We need to add another vertex with 0 outgoing edges:

let s = [|"a -> b, c, d, e"|]; let gr = StrGraph.FromStrings s gr.Visualize() gr.Reverse.Visualize()

#### Reversing on the GPU

The code above makes use of the GPU when it detects that the GPU is present and the graph is sufficiently large to warrant the GPU involvement. Right now, I am setting the threshold to .

I am only making this decision for generating the edges array, which is created on the GPU as two arrays: `start`

and `end'`

that hold the edge nodes. Further, this tuple of arrays in converted into the array of tuples – a data structure more suited for representing an edge.

It is possible to delegate more to the GPU if we know for sure we are not going to get into the situation handled on line 24 above. And we won’t, if we are dealing with Euler graphs. For now, let’s compare performance of just finding the edges part. The step complexity for the GPU implementation is O(1), this is a pleasantly parallel task, so things are easy.

[<Kernel;ReflectedDefinition>] let toEdgesKernel (rowIndex : deviceptr<int>) len (colIndex : deviceptr<int>) (start : deviceptr<int>) (end' : deviceptr<int>) = let idx = blockIdx.x * blockDim.x + threadIdx.x if idx < len - 1 then for vertex = rowIndex.[idx] to rowIndex.[idx + 1] - 1 do start.[vertex] <- idx end'.[vertex] <- colIndex.[vertex]

Here is the test:

let mutable N = 10 * 1024 * 1024 let k = 5 sw.Restart() let gr = StrGraph.GenerateEulerGraph(N, k) sw.Stop() printfn "Graph: %s vertices, %s edges generated in %A" (String.Format("{0:N0}", gr.NumVertices)) (String.Format("{0:N0}", gr.NumEdges)) sw.Elapsed sw.Restart() let starts, ends = getEdges gr.RowIndex gr.ColIndex sw.Stop() printfn "GPU edges: %A" sw.Elapsed sw.Restart() gr.OrdinalEdges sw.Stop() printfn "CPU edges: %A" sw.Elapsed

And the output:

Graph: 10,485,760 vertices, 31,458,372 edges generated in 00:00:18.9789697 GPU edges: 00:00:01.5234606 CPU edges: 00:00:16.5161326

Finally. I’m happy to take the win!

## Visualizing Graphs

#### Previously

### Generating and Visualizing Graphs

I can hardly overemphasize the importance of visusalizations. Many a bug had been immediately spotted just by looking at a visual of a complex data structure. I therefore decided to add visuals to the project as soon as the `DirectedGraph`

class was born.

#### Code & Prerequisits

Code is on GitHub.

- GraphViz: install and add the bin directory to the PATH
- EmguCV v3.1: install and add the bin directory to the PATH

#### DrawGraph

This is a small auxiliary component I wrote to make all future visualizations possible. And here is a sidebar. I didn’t want to write this component. I am not a fan of re-writing something that was written a hundred times before me, so the first thing I did was look for something similar I could use. Sure enough, I found a few things. How can I put it? Software engineering is great, but boy, do we tend to overengineer things! I know, I’m guilty of the same thing myself. All I wanted from the library was an ability to receive a text file written in GraphViz DSL, and get on the output a .png containing the picture of the graph. Just a very simple GraphViz driver, nothing more.

One library had me instantiate 3 (three!) classes, another developed a whole API of its own to build the GraphViz file… I ended up writing my own component, it has precisely 47 lines of code. the last 4 lines are aliasing a single function that does exactly what I wanted. It creates the png file and then immediately invokes the EmguCV image viewer to show it. After we’re done, it cleans up after itself, deleting the temporary png file. Here it is.

#### Taking it for a Ride

Just to see this work…

Another digression. *Love* the new feature that generates all the “#r” instructions for F# scripts and sticks them into one file! Yes, this one! Right-click on “References” in an F# project:

And the generated scripts auto-update as you recompile with new references! A+ for the feature, thank you so much.

Comes with a small gotcha, though: sometimes it doesn’t get the order of references quite right and then errors complaining of references not being loaded appear in the interactive. I spent quite a few painful hours wondering how is it that this reference was not loaded, when here it is! Then I realized: it was being loaded AFTER it was required by references coming after it).

#load "load-project-release.fsx" open DrawGraph createGraph "digraph{a->b; b->c; 2->1; d->b; b->b; a->d}" "dot.exe" None

Cool. Now I can take this and use my own function to generate a graph from a string adjacency list, visualize it, and even view some of its properties. Sort of make the graph “palpable”:

let sparse = ["a -> b, c, d"; "b -> a, c"; "d -> e, f"; "e -> f"; "1 -> 2, 3"; "3 -> 4, 5"; "x -> y, z"; "2 -> 5"] let grs = StrGraph.FromStrings sparse grs.Visualize(clusters = true)

`StrGraph.FromStrings`

does exactly what it says: it generates a graph from a sequence of strings, formatted like the `sparse`

list above.

My `Visualize`

function is a kitchen sink for all kinds of visuals, driven by its parameters. In the above example, it invokes graph partitioning to clearly mark connected components.

It is important to note, that this functionality was added to the visualizer not because I wanted to see connected components more clearly, but as a quick way to ensure that my partitioning implementation was indeed working correctly.

#### Generating Data and Looking at It

Now we have a class that builds graphs and even lets us look at them, so where do we get these graphs? The easiest thing (seemed at the time) was to create them.

Enter FsCheck. It’s not the easiest library to use, there is a learning curve and getting used to things takes time, but it’s very helpful. Their documentation is quite good too. The idea is to write a generator for your type and then use that generator to create as many samples as you like:

#load "load-project-release.fsx" open Graphs open FsCheck open System open DataGen let grGen = graphGen 3 50 let gr = grGen.Sample(15, 5).[2] gr.Visualize(into=3, out= 3)

This produces something like:

My function `graphGen len num`

generates a graph of text vertices where `len`

is the length of a vertex name and `num`

is the number of vertices. It returns an FsCheck generator that can then be sampled to get actual graphs. This was a one-off kind of experiment, so it’s in a completely separate module:

//DataGen.fs module DataGen open FsCheck open System open Graphs let nucl = Gen.choose(int 'A', int 'Z') |> Gen.map char let genVertex len = Gen.arrayOfLength len nucl |> Gen.map (fun c -> String(c)) let vertices len number = Gen.arrayOfLength number (genVertex len) |> Gen.map Array.distinct let graphGen len number = let verts = vertices len number let rnd = Random(int DateTime.UtcNow.Ticks) let pickFrom = verts |> Gen.map (fun lst -> lst.[rnd.Next(lst.Length)]) let pickTo = Gen.sized (fun n -> Gen.listOfLength (if n = 0 then 1 else n) pickFrom) Gen.sized <| (fun n -> Gen.map2 (fun from to' -> from, (to' |> Seq.reduce (fun acc v -> acc + ", " + v))) pickFrom pickTo |> Gen.arrayOfLength (if n = 0 then 1 else n) |> Gen.map (Array.distinctBy fst) |> Gen.map (fun arr -> arr |> Array.map (fun (a, b) -> a + " -> " + b)) ) |> Gen.map StrGraph.FromStrings

This whole module cascades different FsCheck generators to create a random graph.

The simplest of them `nucl`

, generates a random character. (Its name comes from the fact that originally I wanted to limit the alphabet to just four nucleotide characters A, C, G, T). Then this generator is used by `genVertex`

to generate a random string vertex, and finally `vertices`

creates an array of distinct random vertices.

`graphGen`

creates a sequence of strings that `FromStrings`

(above) understands. It first creates a string of “inbound” vertices and then adds an outbound vertex to each such string.

Sampling is a little tricky, for instance, the first parameter to the `Sample`

function, which, per documentation, controls sample size, in this case is responsible for complexity and connectivity of the resulting graphs.

#### On to Euler…

The script above also specifies a couple of optional parameters to the visualizer: `into`

will mark any vertex that has `into`

or more inbound connections in green. And `out`

will do the same for outbound connections and yellow. If the same vertex possesses both properties, it turns blue.

Inspired by all this success, I now want to write a function that would generate Eulerian graphs. The famous theorem states that being Eulerian (having an Euler cycle) for a directed graph is equivalent to being strongly connected and having in-degree of each vertex equal to its out-degree. Thus, the above properties of the visualizer are quite helpful in confirming that the brand new generator I have written for Eulerain graphs (`GenerateEulerGraph`

) is at the very least on track:

let gre = StrGraph.GenerateEulerGraph(10, 5) gre.Visualize(into=3, out=3)

Very encouraging! Whatever has at least 3 edges out, has at least 3 edges in. Not a definitive test, but the necessary condition of having only blue and transparent vertices in the case of an Eulerian graph is satisfied.

In the next post – more about Eulerian graphs, de Brujin sequences, building (and visualizing!) de Bruijn graphs, used for DNA sequence assembly.

## Walking the Euler Path: Intro

### Source Code

I’m thinking about a few posts in these series going very fast through the project. The source is on my GitHub, check out the tags since the master branch is still work in progress.

### Experimenting with Graph Algorithms with F# and GPU

Graphs play their role in bioinformatics which is my favorite area of computer science and software engineering lately. This relationship was the biggest motivator behind this project.

I have been experimenting with a few graph algorithms trying to parallelize them. This is interesting because these algorithms usually resist parallelization since they are fast in their serial version running in O(|E|) or O(|E| + |V|) time (E – the set of edges, V – the set of vertices of the graph). And of course I use any excuse to further explore the F# language.

### Representation

The object of this mini-study is a directed unweighted graph. The choice to represent it is simple: adjacency list or incidence matrix. Since I had CUDA in mind from the start, the latter was chosen, and since I had large graphs in mind, hundreds of millions, possibly billions of edges (limited only by the .NET object size: is it still a problem? I haven’t checked, and by the size of my GPU memory), sparse matrix data structure was picked.

#### Sparse Matrix Implementation

I first wrote a very bare-bones sparse matrix class, just to get my feet wet. Of all possible representations for a sparse matrix, I chose CSR (or CSC which is the transposition of CSR), the idea is intuitive and works great for a directed graph incidence matrix.

Briefly (taking CSR – Compressed Sparse Row as an example), we represent our matrix in 3 arrays: V, C, R. V – the array of non-zero values, written left-to-right, top-to-bottom. C – the array of column indices of the values in V. And C – the “boundary”, or “row index” array, built as follows: We start by recording the number of non-zero values per row in each element of R, starting with R[1]. R[0] = 0. Then we apply the scan operation (like the F# Seq.scan) to the row array, to produce the final result. The resulting array contains m + 1 (m – number of rows in the matrix) elements, its last entry equals the total number of non-zero values in the matrix). This array is used as a “slicer” or “indexer” into the column/value arrays: non-zero columns of row will be located in arrays V and C at the indices starting from R[i] and ending at R[i + 1] – 1. This is all pretty intuitive.

#### Overcoming F# Strong Typing

F# is a combination of strong typing and dynamic generic resolution, which makes it a challenge when you need to write a template for which it is natural to be resolved at compile time. Then sweet memories of C++ or Python invade… There exists a way to overcome all that and it is not pretty. To implement it I needed the old F# PowerPack with `INumeric`

included. Then I just coded the pattern explained in the blog post:

// SparseMatrix.fs /// <summary> /// Sparse matrix implementation with CSR and CSC storage /// </summary> [<StructuredFormatDisplay("{PrintMatrix}")>] type SparseMatrix<'a> (ops : INumeric<'a>, row : 'a seq, rowIndex : int seq, colIndex : int seq, rowSize, isCSR : bool) = .... static member CreateMatrix (row : 'a []) (isCSR : bool) = let ops = GlobalAssociations.GetNumericAssociation<'a>() let colIdx, vals = Array.zip [|0..row.Length - 1|] row |> Array.filter (fun (i, v) -> ops.Compare(v, ops.Zero) <> 0) |> Array.unzip SparseMatrix(ops, vals, [0; vals.Length], colIdx, row.Length, isCSR)

The idea is to use the `GlobalAssociations`

to smooth-talk the compiler into letting you do what you want. The pattern is to not directly use the constructor to create your object, but a static method instead, by means of which this “compiler-whispering” is hidden from the user.

My sparse matrix is built dynamically: it is first created with a single row through a call to `CreateMatrix`

and then rows can be appended to it by calling `AddValues row`

. The idea is to allow creation and storage of huge matrices dynamically. These matrices may be stored in large files for which representation in dense format in memory may not be feasible.

#### Representing the graph

So, at which point does it make sense to use a sparse matrix instead of a dense one in CSR/CSC? It’s easy to figure out:

If we have a matrix , then the answer is given by the equation: , here is the number of non-zero elements in the matrix.

For a graph the set V takes a place of rows, and E – that of columns. The above inequality becomes: , so our sparse structure becomes very economical for large, not to mention “really huge” graphs. (We don’t have the values array anymore, since all our values are just 0s and 1s).

And so the graph is born:

[<StructuredFormatDisplay("{AsEnumerable}")>] type DirectedGraph<'a when 'a:comparison> (rowIndex : int seq, colIndex : int seq, verticesNameToOrdinal : IDictionary<'a, int>) as this = let rowIndex = rowIndex.ToArray() let colIndex = colIndex.ToArray() let nEdges = colIndex.Length let verticesNameToOrdinal = verticesNameToOrdinal let nVertices = verticesNameToOrdinal.Count // vertices connected to the ordinal vertex let getVertexConnections ordinal = let start = rowIndex.[ordinal] let end' = rowIndex.[ordinal + 1] - 1 colIndex.[start..end']

This is not very useful, however, since it assumes that we already have `rowIndex`

for the CSR type “R” and `colIndex`

for the “C” arrays. It's like saying: "You want a graph? So, create a graph!". I would like to have a whole bunch of graph generators, and I do. I placed them all into the file `Generators.fs`

.

This is a good case for using type augmentations. When we need to implement something that “looks good” on the object, but doesn’t really belong to it.

In the next post I’ll talk about visualizing things, and vsiualization methods *really* have nothing to do with the graph itself. Nevertheless, it is natural to write:

myGraph.Visualize(euler=true)

instead of:

Visualize(myGraph, euler=true)

So we use type augmentations, for instance, going back to the generators:

//Generators.fs type Graphs.DirectedGraph<'a when 'a:comparison> with /// <summary> /// Create the graph from a file /// </summary> /// <param name="fileName"></param> static member FromFile (fileName : string) = if String.IsNullOrWhiteSpace fileName || not (File.Exists fileName) then failwith "Invalid file" let lines = File.ReadLines(fileName) DirectedGraph<string>.FromStrings(lines)

which creates a graph by reading a text file and calling another generator method at the end. This method actually calls the constructor to create an instance of the object. Keeps everything clean and separate.

This post was intended to briefly construct the skeleton. In the next we’ll put some meat on the bones and talk about visualizing stuff.

## GPU Split & Sort With Alea.CUDA

### Code

The complete source: here

### Intro

Starting to play with graph algorithms on the GPU (who wants to wait for nvGraph from NVIDIA, right?) As one of the precursor problems – sorting large arrays of non-negative integers came up. Radix sort is a simple effective algorithm quite suitable for GPU implementation.

### Least Significant Digit Radix Sort

The idea is simple. We take a set A of fixed size elements with lexicographical ordering defined. We represent each element in a numeric system of a given radix (e.g., 2). We then scan each element from right to left and group the elements based on the digit within the current scan window, preserving the original order of the elements. The algorithm is described here in more detail.

Here is a the pseudocode for radix = 2 (easy to extend for any radix):

k = sizeof(int) n = array.length for i = 0 to k all_0s = [] all_1s = [] for j = 0 to n if bit(i, array[j]) == 0 then all_0s.add(array[j]) else all_1s.add(array[j]) array = all_0s + all_1s

### GPU Implementation

This is a poster problem for the “split” GPU primitive. Split is what <code>GroupBy</code> is in LINQ (or <code>GROUP BY</code> in TSQL) where a sequence of entries are split (grouped) into a number of categories. The code above applies split k times to an array of length n, each time the categories are are “0” and “1”, and splitting/grouping are done based on the current digit of an entry.

The particular case of splitting into just two categories is easy to implement on the GPU. Chapter 39 of GPU Gems describes such implementation (search for “radix sort” on the page).

The algorithm described there shows how to implement the pseudocode above by computing the position of each member of the array in the new merged array (line 10 above) without an intermediary of accumulating lists. The new position of each member of the array is computed based on the exclusive scan where the value of scanned[i] = scanned[i – 1] + 1 when array[i-1] has 0 in the “current” position. (scanned[0] = 0). Thus, by the end of the scan, we know where in the new array the “1” category starts (it’s scanned[n – 1] + is0(array[n – 1]) – total length of the “0” category, and the new address of each member of the array is computed from the scan: for the “0” category – it is simply the value of scanned (each element of scanned is only increased when a 0 bit is encountered), and start_of_1 + (i – scanned[i]) for each member of the “1” category: its position in the original array minus the number of “0” category members up to this point, offset by the start of the “1” category.

The algorithm has two parts: sorting each block as described above and then merging the results. In our implementation we skip the second step, since Alea.CUDA `DeviceScanModule does the merge for us (inefficiently so at each iteration, but makes for simpler, more intuitive code).`

```
```### Coding with Alea.CUDA

The great thing about Alea.CUDA libray v2 is the Alea.CUDA.Unbound namespace that implements all kinds of scans (and reducers, since a reducer is just a scan that throws away almost all of its results).

let sort (arr : int []) =
let len = arr.Length
if len = 0 then [||]
else
let gridSize = divup arr.Length blockSize
let lp = LaunchParam(gridSize, blockSize)
// reducer to find the maximum number & get the number of iterations
// from it.
use reduceModule = new DeviceReduceModule<int>(target, <@ max @>)
use reducer = reduceModule.Create(len)
use scanModule = new DeviceScanModule<int>(target, <@ (+) @>)
use scanner = scanModule.Create(len)
use dArr = worker.Malloc(arr)
use dBits = worker.Malloc(len)
use numFalses = worker.Malloc(len)
use dArrTemp = worker.Malloc(len)
// Number of iterations = bit count of the maximum number
let numIter = reducer.Reduce(dArr.Ptr, len) |> getBitCount
let getArr i = if i &&& 1 = 0 then dArr else dArrTemp
let getOutArr i = getArr (i + 1)
for i = 0 to numIter - 1 do
// compute significant bits
worker.Launch <@ getNthSignificantReversedBit @> lp (getArr i).Ptr i len dBits.Ptr
// scan the bits to compute starting positions further down
scanner.ExclusiveScan(dBits.Ptr, numFalses.Ptr, 0, len)
// scatter
worker.Launch <@ scatter @> lp (getArr i).Ptr len numFalses.Ptr dBits.Ptr (getOutArr i).Ptr
(getOutArr (numIter - 1)).Gather()
let generateRandomData n =
if n <= 0 then failwith "n should be positive"
let seed = uint32 DateTime.Now.Second
// setup random number generator
use cudaRandom = (new XorShift7.CUDA.DefaultNormalRandomModuleF32(target)).Create(1, 1, seed) :> IRandom<float32>
use prngBuffer = cudaRandom.AllocCUDAStreamBuffer n
// create random numbers
cudaRandom.Fill(0, n, prngBuffer)
// transfer results from device to host
prngBuffer.Gather() |> Array.map (((*) (float32 n)) >> int >> (fun x -> if x = Int32.MinValue then Int32.MaxValue else abs x))

This is basically it. One small optimization - instead of looping 32 times (length of int in F#), we figure out the bit count the largest element and iterate fewer times.

Helper functions/kernels are straightforward enough:

let worker = Worker.Default
let target = GPUModuleTarget.Worker worker
let blockSize = 512
[<Kernel; ReflectedDefinition>]
let getNthSignificantReversedBit (arr : deviceptr<int>) (n : int) (len : int) (revBits : deviceptr<int>) =
let idx = blockIdx.x * blockDim.x + threadIdx.x
if idx < len then
revBits.[idx] <- ((arr.[idx] >>> n &&& 1) ^^^ 1)
[<Kernel; ReflectedDefinition>]
let scatter (arr : deviceptr<int>) (len: int) (falsesScan : deviceptr<int>) (revBits : deviceptr<int>) (out : deviceptr<int>) =
let idx = blockIdx.x * blockDim.x + threadIdx.x
if idx < len then
let totalFalses = falsesScan.[len - 1] + revBits.[len - 1]
// when the bit is equal to 1 - it will be offset by the scan value + totalFalses
// if it's equal to 0 - just the scan value contains the right address
let addr = if revBits.[idx] = 1 then falsesScan.[idx] else totalFalses + idx - falsesScan.[idx]
out.[addr] <- arr.[idx]
let getBitCount n =
let rec getNextPowerOfTwoRec n acc =
if n = 0 then acc
else getNextPowerOfTwoRec (n >>> 1) (acc + 1)
getNextPowerOfTwoRec n 0

### Generating Data and Testing with FsCheck

I find `FsCheck`

quite handy for quick testing and generating data with minimal setup in FSharp scripts:

let genNonNeg = Arb.generate<int> |> Gen.filter ((<=) 0)
type Marker =
static member arbNonNeg = genNonNeg |> Arb.fromGen
static member ``Sorting Correctly`` arr =
sort arr = Array.sort arr
Arb.registerByType(typeof<Marker>)
Check.QuickAll(typeof<Marker>)

Just a quick note: the first line may seem weird at first glance: looks like the filter condition is dropping non-positive numbers. But a more attentive second glance reveals that it is actually filtering out negative numbers (currying, prefix notation should help convince anyone in doubt).

### Large sets test.

I have performance tested this on large datasets (10, 000, 000 - 300, 000, 000) integers, any more and I'm out of memory on my 6 Gb Titan GPU. The chart below goes up to 100,000,000.

I have generated data for these tests on the GPU as well, using Alea.CUDA random generators, since FsCheck generator above is awfully slow when it comes to large enough arrays.

The code comes almost verbatim from this blog post:

let generateRandomData n =
if n <= 0 then failwith "n should be positive"
let seed = uint32 DateTime.Now.Second
// setup random number generator
use cudaRandom = (new XorShift7.CUDA.DefaultNormalRandomModuleF32(target)).Create(1, 1, seed) :> IRandom<float32>
use prngBuffer = cudaRandom.AllocCUDAStreamBuffer n
// create random numbers
cudaRandom.Fill(0, n, prngBuffer)
// transfer results from device to host
prngBuffer.Gather() |> Array.map (((*) (float32 n)) >> int >> (fun x -> if x = Int32.MinValue then Int32.MaxValue else abs x))

except I'm converting the generated array to integers in the [0; n] range. Since my radix sort work complexity is O(k * n) (step complexity O(k)) where `k = bitCount(max input)`

I figured this would give an adequate (kinda) comparison with the native F# `Array.orderBy `

.

### Optimization

It is clear from the above chart, that there is a lot of latency in our GPU implementation - the cost we incur for the coding nicety: by using `DeviceScanModule`

and `DeviceReduceModule`

we bracket the merges we would otherwise have to do by hand. Hence the first possible optimization: bite the bullet, do it right, with a single merge at the end of the process and perform a block-by-block sort with `BlockScanModule`

```
```

```
```## Capture Video in 2 Lines of Code

March 21, 2016
3 comments
Literally. Well almost. 2 meaningful lines + some boilerplate. This has got to be easier than even Python!

Using EmguCV, a wrapper around OpenCV and F# Interactive:

#I @"C:\<project directory>\packages\EmguCV.3.0.0\lib\net451"
#r "Emgu.CV.dll"
#r "Emgu.Util.dll"
#r "System.Windows.Forms"
open Emgu.CV
open System.Windows.Forms
let capture = new Capture()
Application.Idle.Add(fun _ -> CvInvoke.Imshow("Camera", capture.QueryFrame()))

## Look-and-say: [Alea.]CUDA

January 6, 2016
Leave a comment
Continuing the Advent of Code theme from the previous post. Figured since this year is going to be my year of CUDA, this would be a good opportunity to take it for a ride. A good April 1st post, but why wait?

So, how can we make this even faster than the already fast imperative solution?

### Naive Approach

The complete script for this is on GitHub.

The kernel computes one iteration of the look-and-say sequence. Every instance looks at a single group of repeating digits and outputs the result into an array. The new array is twice the size of the original one (representing a previous member of the sequence), every digit or the first digit of a group will produce two number (n, c) and every repeating digit will produce two “0”s on the output:

1 1 1 3 1 2 2 1 1 3 -> 3 1 0 0 0 0 1 3 1 1 2 2 0 0 2 1 0 0 1 3

[<Kernel; ReflectedDefinition>]
let lookAndSayKernel (arr : deviceptr<int8>) (len : int) (out : deviceptr<int8>) =
let ind = blockIdx.x * blockDim.x + threadIdx.x
if ind < len then
let c = arr.[ind]
let idxOut = 2 * ind
if ind = 0 || arr.[ind - 1] <> c then
let mutable i = 1
while ind + i < len && c = arr.[ind + i] do
i <- i + 1
out.[idxOut] <- int8 i
out.[idxOut + 1] <- c
else
out.[idxOut] <- 0y
out.[idxOut + 1] <- 0y

In the naive approach we bring the resulting sequence into the memory and repeat:

let lookAndSayGpu (s : string) n =
let arr = s |> Seq.map (fun c -> (string>>int8) c) |> Seq.toArray
let rec loop (arr : int8 []) n =
if n = 0 then arr.Length
else
let blockSize = 512
let gridSize = divup arr.Length blockSize
let lp = LaunchParam(gridSize, blockSize)
use dArr = worker.Malloc(arr)
use dOut = worker.Malloc(arr.Length * 2)
worker.Launch <@lookAndSayKernel@> lp dArr.Ptr arr.Length dOut.Ptr
let out = dOut.Gather()
let arr =
out
|> Array.filter (fun b -> b > 0y)
loop arr (n - 1)
loop arr n

This is almost not worth checking against the high performing imperative algorithm of the last post, but why not?

Not unexpectedly we are still falling short of the imperative solution. Not a problem. There are a few very low hanging fruit here that are just waiting to be picked.

### A Better Solution

The complete script for this is on GitHub.

There are two problems with our solution so far:

- Two much memory being moved between the GPU and RAM
- A significant part of the algorithm is executed on the CPU with memory allocated and destroyed in the process (that would be the array compacting part)

The following algorithm addresses both problems:

GenerateSequences(str, n) {

//Allocate the maximum amount of memory for all data structures used on the GPU

len = str.Length

initializeGpuMemory(str, len)

for i = 1 to n {

lookAndSayKernel(len)

compactResultStream(2 * len)

len = getNewLength()

}

}

Of the 3 steps in this iterative loop, there is only one where in the implementation we move data from the GPU. It would be easy enough to not do it if necessary, though.

At the core of this algorithm there is a stream compaction routine that is executed hopefully entirely on the GPU. Compaction is performed by copying only the values needed to the addresses in the output stream. These addresses are determined by running an inclusive scan on the array of integers that contains “1” in a meaningful position, and “0” in all the rest. The details of how scan is used for stream compaction and how scans are implemented on the GPU, are discussed in GPU Gems 3 (free from NVIDIA). I’m using a device scan that comes with Alea.CUDA, which, hopefully, minimizes memory copies between main memory and GPU.

Now that the stream has been compacted, we get the length of the new sequence as the last entry in the address map, obtained during compaction (and explained in the linked chapter of GPU Gems 3).

[<Kernel; ReflectedDefinition>]
let lookAndSayKernelSimple (arr : deviceptr<int>) (len : int) (out : deviceptr<int>) (flags : deviceptr<int>)=
let ind = blockIdx.x * blockDim.x + threadIdx.x
if ind < len then
let c = arr.[ind]
let idxOut = 2 * ind
let prevUnrepeated = if ind = 0 || arr.[ind - 1] <> c then 1 else 0
flags.[idxOut] <- prevUnrepeated
flags.[idxOut + 1] <- prevUnrepeated
if prevUnrepeated = 1 then
let mutable i = 1
while ind + i < len && c = arr.[ind + i] do
i <- i + 1
out.[idxOut] <- i
out.[idxOut + 1] <- c
else
out.[idxOut] <- 0
out.[idxOut + 1] <- 0
[<Kernel; ReflectedDefinition>]
let copyScanned (arr : deviceptr<int>) (out : deviceptr<int>) (len : int) (flags : deviceptr<int>) (addrmap : deviceptr<int>) =
let ind = blockIdx.x * blockDim.x + threadIdx.x
if ind < len && flags.[ind] > 0 then out.[addrmap.[ind] - 1] <- arr.[ind]

In the modified kernel, we also fill out the “flags” array which are scanned to obtain the address map for compacting the stream. The second kernel uses the result of the scan performed on these “flags” to produce the new compacted sequence. So for the above example:

original: 1 1 1 3 1 2 2 1 1 3

new: 3 1 0 0 0 0 1 3 1 1 2 2 0 0 2 1 0 0 1 3

flags: 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1

map: 1 2 2 2 2 2 3 4 5 6 7 8 8 8 9 10 10 10 11 12 (inclusive scan of the flags)

The last entry in the map is also the new length.

let lookAndSayGpuScan (s : string) n =
let maxLen = 20 * 1024 * 1024
let arr = s |> Seq.map (fun c -> (string>>int) c) |> Seq.toArray
use dOut = worker.Malloc(maxLen)
use dFlags = worker.Malloc(maxLen)
use dAddressMap = worker.Malloc(maxLen)
use dArr = worker.Malloc(maxLen)
dArr.Scatter(arr)
use scanModule = new DeviceScanModule<int>(GPUModuleTarget.Worker(worker), <@ (+) @>)
scanModule.Create(100) |> ignore
let sw = Stopwatch()
let rec loop n len =
if n = 0 then len
else
let blockSize = 512
let gridSize = divup len blockSize
let lp = LaunchParam(gridSize, blockSize)
use scan = scanModule.Create(2 * len)
worker.Launch <@lookAndSayKernelSimple@> lp dArr.Ptr len dOut.Ptr dFlags.Ptr
scan.InclusiveScan(dFlags.Ptr, dAddressMap.Ptr, 2 * len)
let gridSize = divup (2 * len) blockSize
let lp = LaunchParam(gridSize, blockSize)
worker.Launch <@copyScanned@> lp dOut.Ptr dArr.Ptr (2 * len) dFlags.Ptr dAddressMap.Ptr
let len = dAddressMap.GatherScalar(2 * len - 1)
loop (n - 1) len
sw.Start()
let res = loop n s.Length
sw.Stop()
res, sw.ElapsedMilliseconds

The complete code is above. Notice, that we are not cheating when we are measuring the speed of the internal loop and not including allocations. These can be factored out completely, so the only experiment that matters is what actually runs.

Now, we are finally fast! But it’s a mighty weird chart, probably because most of the time is spent in context switching and all other latency necessary to make this run…

So, takeaways: GPU programming makes us think differently, cool GPU algorithms exist, writing for the GPU is tough but rewarding, and we can really blow a small problem out of any proportion considered descent.

```
```