Coding Day

I would like to mention a question that posed on stackoverflow a year back, with a slight change.

Given a function which produces a random integer in the range 0 to 4, write a function which produces a random integer in the range 0 to 6.

Simply, we have let ran5() = new Random().Next(5), we would like to write ran7 using ran5, and without any other random number generator
The answers are rather interesting for this question, as it is important to have a uniformly distributed random number generator.
The first attempt, is a naive way to generate the random number by defining the sample set small.

//random number generator with 6 numbers
let ran = new Random()
let random5() = ran.Next(5)
 
//naive
let rantbl = [| 0 .. 6 |]
 
let rec random7() = let i,j = random5(), random5()                   
                    let sum = i + j
                    if  sum < rantbl.Length then rantbl.[sum]
                    else random7()

Now, if you run the sample, it will generate the random numbers to 7 just fine, but it won’t be uniformly distributed, and will converge to a number as we run infinite times.
However, if we define our initial set uniformly to match out number generator, in this case we can generate 5 numbers, so a set of 25, we can hit each element by the multiplication of those two generators

//based on
let randomTable = [|for i in 0 .. 24 do if i / 7<3 then yield i % 7
                                        else yield -1 |] 
let rec random7_2() = let i,j = random5(), random5()
                      let index = i  * 5 + j
                      if  randomTable.[index] <> -1 then randomTable.[index]
                      else random7_2()
 
let sampler sample = seq { for i in 0 .. 100000 do yield sample() }

We can estimate of the value of a random variable and predict the error, which is proportional to the iterations. Since we have 2 sets, we can analyse and compare their statistical properties.

let variance sample N =  let mean = sample |> Seq.average_by (fun i-> float i)
                         (sample |> Seq.sum_by (fun i->(float i - mean) * (float i - mean) ))
                                                     / N
let standardDeviation varian= sqrt (varian)
 
let standardError sample =  let N = float (Seq.length sample)
                            let vari = variance sample N
                            let stDev = standardDeviation vari
                            stDev / (sqrt N)
 
let sample = sampler random7
standardError(sampler random7 )
standardError (sampler random7_2)
 
// uniform distribution
let errorSet mySet = let N = Seq.length mySet |> float
                     mySet |> Seq.count_by (fun e-> e)  |> Seq.map (fun (i,j) -> i, (float j)/N * 100.0) |> Map.of_seq
 
errorSet (sampler random7)
errorSet (sampler random7_2)

Random Number Generation

Downloads: 80 File Name: random.fs

Pi Calculation

Today is the day, 03/14 2010. Happy day to all!

The Monte Carlo method generates multiple trials to determine the expected value of a random variable.

Calculating Pi is generally the hello world of Monte Carlo Method in Stochastic Calculus. So for today, I will try to give a sample calculation of pi as monte carlo in F#. There are very good articles on the Monte carlo and how the pi calculation is formulated.

Generally these are the steps to follow, before running the simulation

1. Define the input set
2. Generate randomly the input set
3. Filter the random set using some computations
4. Fold the result to the final result

Pi calculation formula :

open System
let calcPi() =
    let ran = new Random()
    let distances = seq { for i in 0 .. 100000 do
                            let x,y  = ran.NextDouble(), ran.NextDouble()
                            yield sqrt (x * x + y * y)
         }
 
    let count = distances |> Seq.filter (fun distance -> distance <= 1.0) 
                              |> Seq.length     
  4.0 * (float count) / (float (Seq.length distances))
 
let errorRate() = (1.0 - (calcPi() / Math.PI)) * 100.0
 
let print5 value = for i in 0 .. 5 do printf "%f\n" (value())
print5 (errorRate)

Pi Calculation

Downloads: 125 File Name: pi.fs

Real-World functional programming has been written by Tomas Petricek and Jon Skeet.
Tomasz’s and Jon’s book is built around how to think functionally when programming and how to make best use of functional paradigm for real world scenerios. The book is written  mainly for the C# programmers who would like to switch/learn more of the functional world. It also features snippets with some technical background of the ideas involved.
As the title suggests the book is not focussed on one language, instead a mix of C# and F#, and it does a great job in bringing these worlds together,which is to me best of both worlds. Although, samples are all in functional style, when the problem needs the functional beauty to express, F# takes place. This gives any .NET developer to see when to use the necessary language for a particular problem.

The book starts slowly with the functional structures, and goes to monads, and to reactive libraries.  I found the language of the book clear to understand, and easy to follow. The samples in the beginning of a chapter starts with some really simple constructs, but at the end of a chapter they become more complicated, and a nice programs to reflect the idea of the chapter.

Applied functional programming is probably the most sophisticated part of the book. Some ideas mainly inspired from research papers (cited at the end) blended with modern languages and libraries and applied somewhat differently. Especially composable functional libraries and reactive functional programs were insightful and open the mind with new possibilities.

Finally, I would recommend this book whoever wants to switch to functional programming and also learn new techniques in general. Each sample is crafted well, and represent real value with its tutorial writing style.

Although it has been a while since Brian posted the poker problem in his blog, I haven’t got the chance to look at it until I came across in ProjectEuler Problem 54. It is not the elegant or best solution at all, but just wanted to join the crew and can confirm it works with the problem’s 1000 games’ dataset.

Hope this helps.

#light
(* Problem 54 *)
type suit = |Spades |Hearts |Clubs |Diamonds 
type card=  |Ace = 14 |Two = 2  |Three =3  |Four = 4 |Five = 5 |Six = 6 
            |Seven =7  |Eight = 8 |Nine = 9 |Ten = 10 |Jack = 11 |Queen =12 |King  =13
type acard = (card * suit)  
 
let carder x:card= enum x
 
let card_value = function
    | 'A' -> card.Ace
    | 'K' -> card.King
    | 'Q' -> card.Queen
    | 'J' -> card.Jack  
    | 'T' -> card.Ten
    | c ->  carder(System.Int32.Parse(c.ToString()))
 
let suit_value = function
    | 'S' -> Spades
    | 'H' -> Hearts    
    | 'C' -> Clubs
    | 'D' -> Diamonds    
    | a -> invalid_arg (a.ToString())
 
let Create (str: string) :acard = (card_value str.[0],suit_value str.[1])
 
let isstraigh (mycards:acard list)  = 
    let mycards = List.sort_by (fun (a,b) -> a,b) mycards
    let rec isstr previouscard mycards   (straightlist : acard list) =
        if straightlist.Length >= 5 then straightlist
        else match mycards with  
                | cur :: rest -> if int (fst cur) = int ((fst previouscard)) + 1 then isstr cur rest  (cur::straightlist)
                                 else isstr cur rest  []                           
                | _ -> []
    let head = List.hd mycards
    isstr head (List.tl mycards) [head]
 
let pairl (mycards : acard list) groupfunction minelementCount =
                 mycards |> Seq.group_by groupfunction
                         |> Seq.filter (fun a -> Seq.length (snd a) >= minelementCount)
                         |> Seq.to_list
                         |> List.unzip
 
 
type Ranks =
    | Highest of card
    | Pair of  card
    | TwoPair of card*card
    | Three of card
    | Straight of card
    | Flush of card 
    | FullHouse of  card*card
    | Four of  card
    | StraightFlush of card
 
type Player = |One |Two  |Noone
 
let rank (mycards :  acard list) =     
    let traverseL  (l :'a list)  = if not l.IsEmpty then l  |> List.hd |> Seq.to_list
                                   else []                            
    let isflush mycards = snd (pairl mycards snd 5)  |> traverseL
 
    let FofF l = traverseL l |> List.hd |> fst
 
    let flush = isflush mycards
    let straight = isstraigh mycards
 
    let ispair count= pairl mycards fst count                                                             
 
    let four,fours = let f,s = ispair 4 
                     f,s|> traverseL
    let three,threes = ispair 3 
    let two,twos = ispair 2
 
    let maxcard c = fst (List.max c)
 
    if not flush.IsEmpty && not straight.IsEmpty then StraightFlush(maxcard flush),flush
    elif  not fours.IsEmpty then Four(four.Head),fours
    elif not threes.IsEmpty && not twos.IsEmpty && FofF twos <> FofF threes then FullHouse(two.Head, three.Head), List.append (threes |> traverseL) (traverseL twos)
    elif not flush.IsEmpty then Flush(maxcard flush),flush
    elif not straight.IsEmpty then Straight(maxcard straight),straight
    elif not three.IsEmpty then Three( three.Head), threes |> traverseL
    elif List.length two = 2 then TwoPair(two.Head,two.Tail.Head),Seq.append (twos.Head) (twos.Tail.Head) |> Seq.to_list
    elif not (twos |> traverseL).IsEmpty then Pair(two.Head), twos |> traverseL
    else Highest(maxcard mycards), [List.max mycards]
 
let play input =
    let convert (line: string) = let l = line.Split([|' '|])                                  
                                 [|[for j in 0 .. 4 do yield Create( l.[j])]; [for j in 5 .. 9 do yield Create( l.[j])]|]                                                                  
    let playercrds = convert input              
    let rec iswinner pcards=               
                let ranks,rankcards = pcards |> Array.map (rank)  |> Array.unzip
 
                let removecards (mainlist) (toberemoved)  =                         
                        mainlist |> Array.map2 (fun rem main-> main |> List.filter (fun c->
                                List.fold_left(fun ac x-> if x = c then ac && false else ac && true) true rem)) toberemoved
 
                if ranks.[0]>ranks.[1] then One
                elif ranks.[0]<ranks.[1] then Two
                else iswinner  (removecards  pcards rankcards) 
 
    iswinner playercrds
 
 
play  "5H 5C 6S 7S KD 2C 3S 8S 8D TD" 
play  "5D 8C 9S JS AC 2C 5C 7D 8S QH"
play  "2D 9C AS AH AC 3D 6D 7D TD QD"
play  "4D 6S 9H QH QC 3D 6D 7H QD QS"  // prob pair queens look at the highes
play  "2H 2D 4C 4D 4S 3C 3D 3S 9S 9D" 
 
play  "2H 2D 4C 4D 4S 2H 2D 4C 4D 4S" 
 
play  "TH 8D 6C 4D 3S TH 8D 6C 4D 4S" 
 
 
 
let rdinput =   use file = System.IO.File.OpenText("poker.txt")
                let p1count = ref 0  
 
                while not file.EndOfStream do
                 if play (file.ReadLine()) = One then p1count := !p1count+1
 
                file.Close()
                !p1count

Expert F# is written by Don Syme, Adam Granicz, and Antonio Cisternino. That book is a definite source to learn and to expertise F#. Although I have read most of the draft chapters before, the final book has a lot more and valuable information for every type of developer. It really helped me to delve into some advanced features of the language that is uncommon in other languages.

Review

“Expert F#” is a book that focuses on more experienced developers. Although the first chapters look like introduction chapters, actually there are more than introduction chapters that explains the core F# libraries with examples expecting the reader to know the basics.

This book is built primarily on the language and the effective usage of its libraries. The first 10 chapters are more based on the language, libraries and techniques that will support to develop fluently. However; the real fun starts after chapter 11, that makes to write real world programs with more advanced techniques and patterns for common tasks.The authors paid a lot of attention to cover different scenarios and problem domains, that is why symbolic representations, lexing and interoperating chapters exist.

“Expert F#” gives different approaches to solve the problems using self explanatory and terse examples. It is nice to apply some sophisticated techniques immediately in the interactive window.

My favourite chapter of the book was reactive, asynchronous and concurrent programming. Nowadays, concurrency is such a hot topic and this chapter shows how to write reactive concurrent programs using a very neat syntax that programmers could benefit.

Finally, I would strongly recommend this book whoever wants to learn functional programming and F#. Considering the productization of F#, the language will be improved more and more and this book will be the best guide to follow up with that wave.

Introduction

For many years, parallel computing is an important area for research in high performance computing. Super computers dominated the industry all the time. However with the cost of obtaining a fast computer and a fast network, cluster computing considered as a good alternative. High Performance Computing market grew rapidly, mainly because of the clusters intensified. According to a research, clusters represent 50% of the High Performance Computing system revenue at the end of 2005.

The idea of cluster computing is to have many machines on a high-speed network, clusters of computers running the same program. Recently, with the invention and adoption of multi-core CPU systems for desktops, it has become even more important. MPI makes even easier for people to build supercomputers by the usage of powerful computers, high speed networks and powerful libraries.

Message Passing Interface (MPI) is the standard of message passing in a distributed computing environment. Its benefit for researchers is invaluable.

MPICH is an open source, portable implementation of Message Passing Interface (MPI) for developing distributed memory application .

The goal of MPI Tools is to make easy to write programs that runs on a cluster of machines. Also make the transition and the portability easy for existing programs in cluster. Using MPITools, it is possible to create distributed functional applications with F#. Although it is primarily developed for .NET framework, it can run on any CLI implementation.

Implementation

The first step involved to make MPICH available to use for F# platform. A wrapping library is implemented for MPICH. Mainly used MPICH functions made available to F#. Those MPI functions are implemented with the effective usage of types. MPI_Init,, MPI_Comm_size, MPI_Comm_rank, MPI_Finalize, MPI_Send, MPI_Recv, MPI_Abort,  MPI_Barrier, MPI_Bcast , MPI_Gather, MPI_Scatter, MPI_Reduce. In reality, you could write distributed programs with just the first six of those function as I will show on the samples.  When using the library you don’t have to worry about the types and data sizes as you usually do in C programming. The only thing important is the order of communication the same as socket programming.

Because MPICH is unmanaged library, it important to make the data types compatible using the interoperability libraries in .NET framework.  All of the exposed data types and functions are defined in “mpi.h” file in MPICH distribution. If you want to use a different MPI implementation then it is needed to change those functions definitions appropriately based on the documentation.

[]
extern int MPI_Send( void *buf, int count, int MPI_Datatype,
int dest, int tag, int MPI_Comm)

Once the value data types such as int, char, byte, double and float types implemented which are pretty same with C implementation. Next step was to make the reference types of the virtual machine available. Unfortunately not all reference types are possible to send out to wire because of the state or impureness of the type. The types have to be serializable in order to send or receive. To make that possible binary serialization is used and passed as a byte array to the MPI. Implementing the reference types made also possible to pass the functions and lambda functions in to the channel.

For the MPI development, the key factor is the data types. The parties have to agree with the file types. Also the size of the file types should be fixed in order to communicate. However the types are properly handled by the library using the sophisticated type system capabilities. In the programs the order becomes really important. In order to get to the internals of MPI Tools, here is an implementation for standard MPICH type definitions and a type converter for it (shortened for simplicity).

type MPI_Datatype =
| MPI_CHAR           =  0x4c000101
| MPI_SIGNED_CHAR    =  0x4c000118

let private TypeConvert (t) =
let res =
match (box t) with
| : ? byte -> MPI_Datatype.MPI_BYTE
| : ? char -> MPI_Datatype.MPI_CHAR
| _ -> failwith “not implemented data type
Enum.to_int res

The complicated, many parameter function calls in the unmanaged MPICH library becomes powerful function with a few arguments in the .NET library. For instance previously defined 6 argument MPI_Send function becomes a three argument polymorphic function. To make it easy, actually send function becomes in different flavours. Actually most of the communication functions come in different versions for different types. The version below is used for singular types. There are two more flavours one for arrays and other for matrix types.

let send(data,destination, tag)
'a * int * int -> unit

let sendArray(data : 'a array,destination,tag)
a array * int * int -> unit

let sendMatrix (data : matrix,destination,tag)
matrix * int * int -> unit

Similarly, the same pattern goes for the receive function. However, this time it is needed to specify the return type as a generic argument of the function.

let receive<'a>(source,tag)
int*int -> 'a
 
let receiveArray<'a> (source, tag)
int * int -> 'a array
 
let receiveMatrix(source, tag)
int * int -> matrix

The other functions of MPI are implemented in a similar manner. You could also check the project as a tutorial as well. The library uses effectively active patterns, discriminated unions, interoperability and other functional structures

Usage

First of all MPICH needs to be installed prior to usage. The library is used just like another .NET library in your programs. However the execution is relatively different than usual. The programs have to be executed using the MPI daemon called "mpiexec". You could look at more on how to configure a cluster in the MPICH documentation. To run the process in n processor or processes “-n” switch needs to be given as a command line argument followed by the name of the compiled program.

mpiexec -n 2 test.exe

Here is a very simple ping pong application using MPI Tools. You can find more samples on the MPI Tools Source code.

#light
#I @"..\MPITools.Bindings\"
#r @"MPITools.Bindings.dll"
open MPITools
MPI.initialize()
 
let procSize = MPI.size()
let curProcess =  MPI.rank()
 
let pingpong() =
if curProcess = 0 then
let i =  0
MPI.send(i,1,0)
let b = MPI.receive(1,0)
()
elif curProcess = 1 then
let b = MPI.receive(0,0)
MPI.send(b+1,0,0)
pingpong()
MPI.finalize()

Conclusion

You can download MPI Tools from codeplex. Using MPI Tools, the distributed programs will be short, expressive and well typed with the help of the glorified type system of F#.

MPI Tools is built with F# 1.9.3.7 Compiler for the .NET Framework 2.0. However it would possibly work with any CLI implementation. In the future, some more MPI functions will be implemented, including some helper functions that hides the imperative style programming. and the side effects.

I hope it will help to solve your high computation problems effectively. Please feel free to ask questions or to contribute to the project.

Have fun!