Transform recursion into loop


The C# code below computes the smallest enclosing circle of a set of points but causes stack overflow exceptions due to it's recursive implementation.

<img alt="enter image description here" class="b-lazy" data-src="https://i.stack.imgur.com/UI2A1.png" data-original="https://i.stack.imgur.com/UI2A1.png" src="https://etrip.eimg.top/images/2019/05/07/timg.gif" />

It was extracted from <a href="http://www.inf.ethz.ch/personal/emo/PublFiles/SmallEnclDisk_LNCS555_91.pdf" rel="nofollow">a paper</a> which specified the algorithm as recursive.

The algorithm is <strong>not</strong> tail recursive and it is not trivially convertible to a loop structure.

The recursive function is

public static Circle MiniDiskImpl(IReadOnlyList<Point2D> points, List<Point2D> boundary) { if (!points.Any() || boundary.Count == 3) { if (boundary.Count == 0) return null; if (boundary.Count == 1) return null; if (boundary.Count == 2) { var radius = boundary[0].DistanceTo(boundary[1])/2; var center = (boundary[0] + boundary[1])*0.5; return new Circle(Plane.XY, center, radius); } return new Circle(Plane.XY,boundary[0],boundary[1],boundary[2]); } var p = points[0]; var Q = points.GetRange(1,points.Count - 1); var D = MiniDiskImpl(Q, boundary); if (D==null || D.Center.DistanceTo(p) >= D.Radius) { D = MiniDiskImpl(Q, boundary.Concat(p).ToList()); } return D; }

The user calls this function.

public static Circle MiniDisk(List<Point2D> points) { points = points.Slice(0); // Clone the list so we can manipulate in place points.Shuffle(); // sorted points cause poor algorithm performance return MiniDiskImpl(points, new List<Point2D>()); }

The question is, what is the transformation required to make the algorithm non recursive?


I just amazed myself with the following.

/// <summary> /// Find the smallest enclosing circle. See reference /// http://www.inf.ethz.ch/personal/emo/PublFiles/SmallEnclDisk_LNCS555_91.pdf /// "Smallest enclosing disks (balls and ellipsoids) EMO WELZL /// </summary> /// <returns></returns> public static class Circles { private static Return<Circle> Left (List<Point2D> points, List<Point2D> boundary) { if ( !points.Any() || boundary.Count == 3 ) { if ( boundary.Count <= 1 ) { return Return.Value<Circle>(null); } if ( boundary.Count == 2 ) { var radius = boundary[0].DistanceTo(boundary[1]) / 2; var center = ( boundary[0] + boundary[1] ) * 0.5; return Return.Value(new Circle(Plane.XY, center, radius)); } return Return.Value(new Circle(Plane.XY, boundary[0], boundary[1], boundary[2])); } var p = points[0]; var q = points.GetRange(1, points.Count - 1); return from circle in Return.Call(() => Left(q, boundary)) from result in (circle == null || circle.Center.DistanceTo(p) >= circle.Radius) ? Return.Call(() => Left(q, boundary.Concat(p).ToList())) : Return.Value(circle) select result; } public static Circle MiniDisk( List<Point2D> points ) { points = points.Slice(0); points.Shuffle(); return TrampolineRecursion.Do(()=>LeftLeft(points, new List<Point2D>())); } }

using the following implementation of the Trampoline monad I just magicked up.

namespace Weingartner.Numerics.Recursion { public class Return<T> { #region return value mode public bool HasReturnValue; public T ReturnValue; #endregion #region call and continue mode public Func<Return<T>> Call; public Func<T, Return<T>> Continue; #endregion /// <summary> /// To avoid GC pressure we use a thread static variable /// to handle the return value from functions. /// </summary> [ThreadStatic] private static Return<T> _ReturnRegister; public static Return<T> R() { if (_ReturnRegister == null) { _ReturnRegister = new Return<T>(); } return _ReturnRegister; } public Return<T> Bind(Func<T, Return<T>> fn) { if (HasReturnValue) return Return.Call(() => Return.Value(ReturnValue), fn); if(Continue!=null) return Return.Call(Call, t => Continue(t).SelectMany(fn)); return Return.Call(Call, fn); } public Return<T> SelectMany(Func<T, Return<T>> fn) { return Bind(fn); } public Return<T> SelectMany( Func<T, Return<T>> f, Func<T, T, T> g) { return Bind(x => f(x).Bind(y => Return.Value(g(x, y)))); } public Return<T> Select(Func<T, T> fn) { if (HasReturnValue) return Return.Value(fn(ReturnValue)); if(Continue!=null) return Return.Call(Call, t => Continue(t).Select(fn)); return Return.Call(Call, t => Continue(t).Select(fn)); } public T Run() { var stack = new Stack<Func<T,Return<T>>>(); if(Continue!=null) stack.Push(Continue); stack.Push(r => Call()); var @return = Return.Value(default(T)); while ( stack.Count > 0 ) { @return = stack.Peek()(@return.ReturnValue); stack.Pop(); if (!@return.HasReturnValue) { if(@return.Continue!=null) stack.Push(@return.Continue); var t = @return; stack.Push(r => t.Call()); } } return ReturnValue = @return.ReturnValue; } } public class Return { public static Return<T> Value<T>( T t ) { var r = Return<T>.R(); r.HasReturnValue = true; r.ReturnValue = t; return r; } public static Return<T> Call<T> (Func<Return<T>> call, Func<T, Return<T>> @continue = null) { var r = Return<T>.R(); r.HasReturnValue = false; r.Call = call; r.Continue = @continue; return r; } } public static class TrampolineRecursion { public static T Do<T>(Func<Return<T>> call) { return Return.Call(call).Run(); } } }


When you use recursion, you make use of the call stack to perform something, you can always transform a recursive algorithm into it's iterative version by dropping your own objects to an actual stack, see DFS example on wiki: <a href="http://en.wikipedia.org/wiki/Depth-first_search" rel="nofollow">http://en.wikipedia.org/wiki/Depth-first_search</a>


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