## Tuesday, 9. December 2008

Dec 09

Die Dynamics NAV 2009 Entwickler-Hilfe ist ab heute auf MSDN verfÃƒÂ¼gbar. Damit ist es in KÃƒÂ¼rze auch mÃƒÂ¶glich ÃƒÂ¼ber Suchmaschinen in der Hilfe zu suchen. Laut dem Microsoft Dynamics NAV Team Blog wird die Hilfe dort auch stÃƒÂ¤ndig aktualisiert, so dass NAV 2009-Entwickler und -Administratoren ab heute wohl die Onlinevariante bevorzugen sollten.

Unter anderem werden auch folgende Themen beschrieben:

Tags:

Dynamics NAV 2009,

msdn
## Monday, 8. December 2008

Dec 08

Last time I showed how one can solve the Euler Project – problem 53 with the help of Dynamic Programming and a two-dimensional array. This time I will use another technique called (automatic) memoization.

First of all I need a memoization-function. There are two different approaches, one with a System.Collections.Generic.Dictionary and one with an immutable Map. You can read Don Syme’s blog posting to get more information about the differences.

let memoizeWithMap f =
let cache = ref Map.empty
fun x ->
match (!cache).TryFind(x) with
| Some res -> res
| None ->
let res = f x
cache := (!cache).Add(x,res)
res

open System.Collections.Generic
let memoizeWithDictionary f =
let cache = Dictionary<_, _>()
fun x ->
let (found,result) = cache.TryGetValue(x)
if found then
result
else
let res = f x
cache.[x] <- res
res

With the help of one of this generic memoization-functions I can create a recursive function for the binomial coefficients, which memoizes intermediate data:

let rec binomial =
let f (n,k) =
if k = 0I || n = k then
1I
else
binomial(n - 1I,k) + binomial(n - 1I,k - 1I)
f |> memoizeWithDictionary
let answer =
seq { for n in 2I .. 100I do
for k in 2I .. n -> n,k }
|> Seq.filter (fun (a,b) -> binomial (a,b) > 1000000I )
|> Seq.length
answer |> printf "Answer: %A"

It turns out that memoizeWithDictionary is much faster for this problem. But with 160ms it is still slower than the comparable version with a two-dimensional array (45ms).

Tags:

caching,

euler project,

F#,

memoization
Dec 08

Claudio Cherubino (from fsharp.it/) posted his solution to Euler Project – problem 53. As I dealed a lot with Dynamic Programming in the recent time, I tried to solve the problem with a dynamic program in F#.

Project Euler – Problem 53:

How many values of C(*n*,*k*), for 1 Ã¢â€°Â¤ *n* Ã¢â€°Â¤ 100, exceed one-million?

Remark: C(n,k) are the binomial coefficients.

As it turned out, this is not that complicated if one knows the recursive function for the binomial coefficients (see Wikipedia):

with

This is easily transformed into a F# program:

let binomials = Array2.create 101 101 1I
let answer = ref 0
for n in [1..100] do
for k in [1..n - 1] do
binomials.[n, k] <- binomials.[n - 1,k] + binomials.[n - 1,k - 1]
if binomials.[n, k] > 1000000I then
answer := !answer + 1
!answer |> printf "Answer: %A"

Claudio’s program took 1315ms on my computer. The dynamic program needs only 63ms. But we can still do better if we use the symmetry of Pascal’s triangle.

This leads to an algorithm, which calculates only half of the binomial coefficients.

let binomials = Array2.create 101 101 1I
let answer = ref 0
for n in [1..100] do
for k in [1..n/2] do
let b = binomials.[n - 1,k] + binomials.[n - 1,k - 1]
binomials.[n, k] <- b
binomials.[n, n - k] <- b
if b > 1000000I then
if k = n-k then
answer := !answer + 1
else
answer := !answer + 2

!answer |> printf "Answer: %A"

This version needs only 45ms – but we are not ready. I mentioned Pascal’s triangle and its symmetry. But we can use another property. We don’t have to calculate the complete row, if we exceed 100000. All values behind this threshold have to be greater.

let binomials = Array2.create 101 101 1I
let answer = ref 0
for n in [1..100] do
let threshold_reached = ref false
let c = ref 0
for k in [1..n/2] do
if not !threshold_reached then
let b = binomials.[n - 1,k] + binomials.[n - 1,k - 1]
binomials.[n, k] <- b
binomials.[n, n - k] <- b
if b > 1000000I then
threshold_reached := true
else
c := !c + 1
if !threshold_reached then
answer := !answer + (n - 1) - (2 * !c)

!answer |> printf "Answer: %A"

This final version took only 29 ms.

In the next posting I will show a version using memoization.

Tags:

binomial coefficient,

dynamic program,

F#,

problem 53,

project euler
## Sunday, 7. December 2008

Dec 07

The “Subset Sum”-problem is given as the following:

SUBSET SUM

Input: Numbers a_{1}, a_{2}, . . . , a_{n}, S Ã¢Ë†Ë† N.

Question: Is there a subset I Ã¢Å â€ {1,…,n} with Ã¢Ë†â€˜ a_{i} = S?

Finding a solution for this decision problem is a very easy task in F#.

let hasSubsetSum_Naive (numbers: int list) S =
let rec hasSubsetSum (a: int list) lastSum =
match a with
| [] -> false
| x::rest ->
if lastSum + x = S then
true
elif lastSum + x > S then
false
else
hasSubsetSum rest lastSum || hasSubsetSum rest (lastSum+x)
hasSubsetSum numbers 0
let numbers = [ 5;4;3;6;7;12 ]
let searchedSum = 33
hasSubsetSum_Naive numbers searchedSum

Of course this small program can be easily adjusted to give the subset I back. Unfortunately this naÃƒÂ¯ve approach leads to a running time of O(2^n).

But if S is small we can build a dynamic program and decide the problem in O(n*S) time.

This can be easily transformed into F# code.

let hasSubsetSum (numbers: int array) S =
if numbers |> Array.exists (fun x -> x = S) then
true
else
let a = numbers |> Array.filter (fun x -> x < S)
let n = a.Length
if n = 0 then
false
else
let v = Array2.create n (S+1) 0
let u = Array2.create n (S+1) false
let t = Array2.create n (S+1) 0
for j in [1..S] do
for i in [0..n-1] do
if j - a.[i] >= 0 && not u.[i,j - a.[i]] then
v.[i,j] <- t.[i,j - a.[i]] + a.[i]
if ((i = 0) || (i > 0 && t.[i-1,j] <> j)) && v.[i,j] = j then
u.[i,j] <- true
if v.[i,j] = j then
t.[i,j] <- j
else
if i > 0 then
t.[i,j] <- max t.[i-1,j] t.[i,j-1]
else
t.[i,j] <- t.[0,j-1]
**t.[n-1,S] = S**

Tags:

decision problem,

dynamic programming,

F#,

SubsetSum
## Friday, 5. December 2008

Dec 05

If we want to calculate the distance between two objects, we need some sort of distance function. I recently showed how we can compute the “edit distance” between two strings (see Damerau-Levenshtein-Distance in F# – part I, part II and part III). This time I will consider the distance between two points in the plane and on the surface of the earth.

First of all we need some sort of location type, which stores information about the coordinates (I used this type before). I will use the “Units of Measure”-feature in F# to distinct between <degree> vs. <radian> and <m> vs. <km>. If you are not familiar with this concept see Andrew Kennedy’s excellent blog series about Units of Measures in F#.

open Microsoft.FSharp.Math.SI
[<Measure>]
type degree
[<Measure>]
type radian =
static member per_degree = Math.PI/180.0<degree/radian>
[<Measure>]
type km =
static member per_meter = 1000.<m/km>

type Location =
{Latitude:float<degree>;
Longitude:float<degree> }
member x.YPos = x.Longitude
member x.XPos = x.Latitude
member x.Longitude_radian = x.Longitude * radian.per_degree
member x.Latitude_radian = x.Latitude * radian.per_degree
member x.Coordinates = (x.Longitude,x.Latitude)
static member CreateLocation(latitude, longitude) =
{new Location with
Latitude = latitude and
Longitude = longitude}

Now we can easily compute the Euclidean distance for two points in the plane.

let sq x = x * x
let CalcEuclideanDistance (s:Location) (t:Location) =
let d1 = (t.Longitude - s.Longitude) / 1.<degree>
let d2 = (t.Latitude - s.Latitude) / 1.<degree>
((sq d1 + sq d2) |> Math.Sqrt) * 1.<m>

If we want to approximate the distance of two cities on the surface of earth, we can use the “Great circle distance“.

“The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere (as opposed to going through the sphere’s interior)” – Wikipedia

let kmFactor = 6372.795<km>
let sin_rad x = Math.Sin(x / 1.<radian>)
let cos_rad x = Math.Cos(x / 1.<radian>)
/// Great Circle Distance
let CalcGreatCircleDistance (s:Location) (t:Location)=
if s = t then
0.<m>
else
let factor = kmFactor * km.per_meter
let b =
sin_rad s.Latitude_radian * sin_rad t.Latitude_radian +
cos_rad s.Latitude_radian * cos_rad t.Latitude_radian *
cos_rad (t.Longitude_radian - s.Longitude_radian)
Math.Acos(b) * factor

Tags:

distance on the surface of the earth,

Euclidean distance,

F#,

Great circle distance
## Tuesday, 2. December 2008

Dec 02

In one of my next postings I will show a algorithm, that calculates line segment intersections. As a prerequisite we need an algorithm, which calculates the intersection of two lines. In this post I will consider the following intersection of two lines in the euclidean plan:

We can write this as an equation system with two variables and two equations**:**

- Ax + r (Bx – Ax) = Cx + s (Dx – Cx)
- Ay + r (By – Ay) = Cy + s (Dy – Cy)

After solving this system we can calculate the intersection:

- Px = Ax + r (Bx – Ax)
- Py = Ay + r (By – Ay)

Putting this all together we can derive the following F#-code:

let calcIntersection (A:Location, B:Location, C:Location, D:Location) =
let (Ax,Ay,Bx,By,Cx,Cy,Dx,Dy) =
(A.XPos, A.YPos, B.XPos, B.YPos, C.XPos, C.YPos, D.XPos, D.YPos)
let d = (Bx-Ax)*(Dy-Cy)-(By-Ay)*(Dx-Cx)
if d = 0 then
// parallel lines ==> no intersection in euclidean plane
None
else

let q = (Ay-Cy)*(Dx-Cx)-(Ax-Cx)*(Dy-Cy)
let r = q / d
let p = (Ay-Cy)*(Bx-Ax)-(Ax-Cx)*(By-Ay)
let s = p / d
if r < 0. or r > 1. or s < 0. or s > 1. then
None // intersection is not within the line segments
else
Some(
(Ax+r*(Bx-Ax)), // Px
(Ay+r*(By-Ay))) // Py

Remarks:

- If one given line degenerates into a point, then the algorithm gives no intersection.
- If the given lines are parallel, then the algorithm gives no intersection, even if the lines have an overlap.

Tags:

Euclidean space,

F#,

line intersection,

line segment intersection
## Monday, 17. November 2008

Nov 17

Heute ist es endlich soweit – Dynamics NAV 2009 steht offiziell zum Download (1.2GB) auf PartnerSource bereit. Der Download ist zwar bereits seit Samstag verfügbar, aber aus unbekannten strategischen Gründen war diese Information mal wieder "Partner Confidential" und es wurde darum gebeten nicht vor dem 19.11.2009 darüber zu bloggen. Da die Katze jetzt aber offiziell aus dem Sack ist: "Happy Downloading!" ðŸ˜‰

Weitere Information sind im Launch Portal zu finden.

Tags:

Dynamics NAV 2009,

Navision
## Monday, 10. November 2008

Nov 10

Last time I showed how the immutable set implementation in F# can be used to get a immutable sorted list. As a result of using sets, the shown version doesn’t support repeated items. This lack can be wiped out by using an additional dictionary (immutable "Map" in F#) which stores the count of each item.

At first I define two basic helper functions for the dictionary:

module MapHelper =
let addToMap map idx =
let value = Map.tryfind idx map
match value with
| Some(x) -> Map.add idx (x+1) map
| None -> Map.add idx 1 map
let removeFromMap map idx =
let value = Map.tryfind idx map
match value with
| Some(x) ->
if x > 1 then
Map.add idx (x-1) map
else
Map.remove idx map
| None -> map
open MapHelper

Now I can adjust my sorted list implementation:

// a immutable sorted list - based on F# set
type 'a SortedFList =
{items: Tagged.Set<'a,Collections.Generic.IComparer<'a>>;
numbers: Map<'a,int>;
count: int}
member x.Min = x.items.MinimumElement
member x.Items =
**seq {
for item in x.items ****do
for number in [1..x.GetCount item] ****do
yield item}**
member x.Length = x.count
member x.IsEmpty = x.items.IsEmpty
member x.GetCount item =
match Map.tryfind item x.numbers with
| None -> 0
| Some(y) -> y
static member FromList(list, sortFunction) =
let comparer = FComparer<'a>.Create(sortFunction)
let m = list |> List.fold_left addToMap Map.empty
{new 'a SortedFList with
items = Tagged.Set<'a>.Create(comparer,list) and
numbers = m and
count = list.Length}
static member FromListWithDefaultComparer(list) =
SortedFList<'a>.FromList(list,compare)
static member Empty(sortFunction) =
SortedFList<'a>.FromList([],sortFunction)
static member EmptyWithDefaultComparer() =
SortedFList<'a>.Empty(compare)
member x.Add(y) =
{x with
items = x.items.Add(y);
numbers = addToMap x.numbers y;
count = x.count + 1}
member x.Remove(y) =
if x.GetCount y > 0 then
{x with
items = x.items.Remove(y);
numbers = removeFromMap x.numbers y;
count = x.count - 1}
else
x

Tags:

F#,

immutable map vs. dictionary,

immutable set,

immutable sorted list,

red-black trees
Nov 10

F# supports a powerful implementation of immutable lists (see Dustin’s introduction). But for my current work I needed a sorted list and of course I wanted it the F#-way, which means immutable. I didn’t want to reinvent the wheel so I asked my question in the hubFS forum.

It turned out (thanks to Robert) that F# uses red-black trees to give a fast implementation for the immutable set. This means that a set in F# stores all elements in a *balanced binary tree* and searching for an element costs only *O(log n)*. As a side effect all elements can be visited in the underlying order, which means the set implementation gives a possibility for implementing a sorted list. The only limitation is that a set usually don’t store objects more than once.

The next problem I got, was that my sorted list needs a specific ordering function. The standard F# set implementation uses structural comparison to give an order, which is not the order I need. But Laurent mentioned the *Set.Make* function in the F# PowerPack. This function creates a new set type for the given ordering function. The only problem with Set.Make is that it was only designed for OCaml compatibility. But with this hint I was able to put all information together and got a nice immutable sorted list implementation for F#:

// wrap orderingFunction as Collections.Generic.IComparer
type 'a FComparer =
{compareF: 'a -> 'a -> int}
static member Create(compareF) =
{new 'a FComparer with compareF = compareF}
interface Collections.Generic.IComparer<'a> with
member x.Compare(a, b) = x.compareF a b

// a immutable sorted list - based on F# set
type 'a SortedFList =
{items: Tagged.Set<'a,Collections.Generic.IComparer<'a>> }
member x.Min = x.items.MinimumElement
member x.Items = seq {for item in x.items do yield item}
member x.Length = x.items.Count
member x.IsEmpty = x.items.IsEmpty
static member FromList(list, sortFunction) =
let comparer = FComparer<'a>.Create(sortFunction)
{new 'a SortedFList with
items = Tagged.Set<'a>.Create(comparer,list)}
static member FromListWithDefaultComparer(list) =
SortedFList<'a>.FromList(list,compare)
static member Empty(sortFunction) =
SortedFList<'a>.FromList([],sortFunction)
static member EmptyWithDefaultComparer() =
SortedFList<'a>.Empty(compare)
member x.Add(y) =
{x with items = x.items.Add(y)}

member x.Remove(y) =
{x with items = x.items.Remove(y)}

Please note that this implementation stores every item only once. Next time I will show a version which allows to store repeated items.

Tags:

F#,

immutable set,

immutable sorted list,

red-black trees
## Saturday, 1. November 2008

Nov 01

In the first part of this series I showed a naïve algorithm for the Damerau-Levenshtein distance which needs O(m*n) space. In the last post I improved the algorithm to use only O(m+n) space. This time I will show a more functional implementation which uses only immutable F#-Lists and works still in O(m+n) space. This version doesn’t need any mutable data.

/// Calcs the damerau levenshtein distance.
let calcDL (a:'a array) (b: 'a array) =
let n = a.Length + 1
let m = b.Length + 1
let processCell i j act l1 l2 ll1 =
let cost =
if a.[i-1] = b.[j-1] then 0 else 1
let deletion = l2 + 1
let insertion = act + 1
let substitution = l1 + cost
let min1 =
deletion
|> min insertion
|> min substitution
if i > 1 && j > 1 &&
a.[i-1] = b.[j-2] && a.[i-2] = b.[j-1] then
min min1 <| ll1 + cost
else
min1
let processLine i lastL lastLastL =
let processNext (actL,lastL,lastLastL) j =
match actL with
| act::actRest ->
match lastL with
| l1::l2::lastRest ->
if i > 1 && j > 1 then
match lastLastL with
| ll1::lastLastRest ->
(processCell i j act l1 l2 ll1 :: actL,
l2::lastRest,
lastLastRest)
| _ -> failwith "can't be"
else
(processCell i j act l1 l2 0 :: actL,
l2::lastRest,
lastLastL)
| _ -> failwith "can't be"
| [] -> failwith "can't be"
let (act,last,lastLast) =
[1..b.Length]
|> List.fold_left processNext ([i],lastL,lastLastL)
act |> List.rev
let (lastLine,lastLastLine) =
[1..a.Length]
|> List.fold_left
(fun (lastL,lastLastL) i ->
(processLine i lastL lastLastL,lastL))
([0..m-1],[])
lastLine.[b.Length]
let damerauLevenshtein(a:'a array) (b:'a array) =
if a.Length > b.Length then
calcDL a b
else
calcDL b a

I admit the code is still a little messy but it works fine. Maybe I will find the time to cleanup a bit and post a final version.

Tags:

alignment,

Damerau,

damerau-levenshtein,

dna,

dynamic programming,

edit distance,

F#,

Levenshtein algorithm,

Levenshtein distance