I need an algorithm that can fit n rectangles of any size in a larger one minimizing its area - optimization

Question

I need an algorithm that would take n rectangles of any sizes, and calculate a rectangle big enough to fit them all, minimizing its area so the wasted area is minimum, and also returning the position of all the smaller rectangles within.
The specific task I need this to implement on is in a sprite sheet compiler that would take individual PNG files and make a large PNG with all the images in it, so individual frames can be blitted from this surface at run time.
A nice to have feature would be that it aims to a specific given width/height ratio, but it's not mandatory.
I'd prefer simple, generic code I can port to another language.

Answer 1

This is what I put together for my own needs. The T parameter is whatever object you want associated with the results (think of it like the Tag property). It takes a list of sizes and returns a list of Rects that are arranged
static class LayoutHelper
{
/// <summary>
/// Determines the best fit of a List of Sizes, into the desired rectangle shape
/// </summary>
/// <typeparam name="T">Holder for an associated object (e.g., window, UserControl, etc.)</typeparam>
/// <param name="desiredWidthToHeightRatio">the target rectangle shape</param>
/// <param name="rectsToArrange">List of sizes that have to fit in the rectangle</param>
/// <param name="lossiness">1 = non-lossy (slow). Greater numbers improve speed, but miss some best fits</param>
/// <returns>list of arranged rects</returns>
static public List<Tuple<T, Rect>> BestFitRects<T>(double desiredWidthToHeightRatio,
List<Tuple<Size, T>> rectsToArrange, int lossiness = 10)
{
// helper anonymous function that tests for rectangle intersections or boundary violations
var CheckIfRectsIntersect = new Func<Rect, List<Rect>, double, bool>((one, list, containerHeight) =>
{
if (one.Y + one.Height > containerHeight) return true;
return list.Any(two =>
{
if ((one.Top > two.Bottom) ||
(one.Bottom < two.Top) ||
(one.Left > two.Right) ||
(one.Right < two.Left)) return false; // no intersection
return true; // intersection found
});
});
// helper anonymous function for adding drop points
var AddNewPotentialDropPoints = new Action<SortedDictionary<Point, object>, Rect>(
(potentialDropPoints, newRect) =>
{
// Only two locations make sense for placing a new rectangle, underneath the
// bottom left corner or to the right of a top right corner
potentialDropPoints[new Point(newRect.X + newRect.Width + 1,
newRect.Y)] = null;
potentialDropPoints[new Point(newRect.X,
newRect.Y + newRect.Height + 1)] = null;
});
var sync = new object();
// the outer boundary that limits how high the rectangles can stack vertically
var containingRectHeight = Convert.ToInt32(rectsToArrange.Max(a => a.Item1.Height));
// always try packing using the tallest rectangle first, working down in height
var largestToSmallest = rectsToArrange.OrderByDescending(a => a.Item1.Height).ToList();
// find the maximum possible container height needed
var totalHeight = Convert.ToInt32(rectsToArrange.Sum(a => a.Item1.Height));
List<Tuple<T, Rect>> bestResults = null;
// used to find the best packing arrangement that approximates the target container dimensions ratio
var bestResultsProximityToDesiredRatio = double.MaxValue;
// try all arrangements for all suitable container sizes
Parallel.For(0, ((totalHeight + 1) - containingRectHeight) / lossiness,
//new ParallelOptions() { MaxDegreeOfParallelism = 1},
currentHeight =>
{
var potentialDropPoints = new SortedDictionary<Point, object>(Comparer<Point>.Create((p1, p2) =>
{
// choose the leftmost, then highest point as earlier in the sort order
if (p1.X != p2.X) return p1.X.CompareTo(p2.X);
return p1.Y.CompareTo(p2.Y);
}));
var localResults = new List<Tuple<T, Rect>>();
// iterate through the rectangles from largest to smallest
largestToSmallest.ForEach(currentSize =>
{
// check to see if the next rectangle fits in with the currently arranged rectangles
if (!potentialDropPoints.Any(dropPoint =>
{
var workingPoint = dropPoint.Key;
Rect? lastFittingRect = null;
var lowY = workingPoint.Y;
var highY = workingPoint.Y - 1;
var boundaryFound = false;
// check if it fits in the current arrangement of rects
do
{
// create a positioned rectangle out of the size dimensions
var workingRect = new Rect(workingPoint,
new Point(workingPoint.X + currentSize.Item1.Width,
workingPoint.Y + currentSize.Item1.Height));
// keep moving it up in binary search fashion until it bumps the higher rect
if (!CheckIfRectsIntersect(workingRect, localResults.Select(a => a.Item2).ToList(),
containingRectHeight + (currentHeight * lossiness)))
{
lastFittingRect = workingRect;
if (!boundaryFound)
{
highY = Math.Max(lowY - ((lowY - highY) * 2), 0);
if (highY == 0) boundaryFound = true;
}
else
{
lowY = workingPoint.Y;
}
}
else
{
boundaryFound = true;
highY = workingPoint.Y;
}
workingPoint = new Point(workingPoint.X, lowY - (lowY - highY) / 2);
} while (lowY - highY > 1);
if (lastFittingRect.HasValue) // found the sweet spot for this rect
{
var newRect = lastFittingRect.Value;
potentialDropPoints.Remove(dropPoint.Key);
// successfully found the best location for the new rectangle, so add it to the pending results
localResults.Add(Tuple.Create(currentSize.Item2, newRect));
AddNewPotentialDropPoints(potentialDropPoints, newRect);
return true;
}
return false;
}))
{
// this only occurs on the first square
var newRect = new Rect(0, 0, currentSize.Item1.Width, currentSize.Item1.Height);
localResults.Add(Tuple.Create(currentSize.Item2, newRect));
AddNewPotentialDropPoints(potentialDropPoints, newRect);
}
});
// layout is complete, now see if this layout is the best one found so far
var layoutHeight = localResults.Max(a => a.Item2.Y + a.Item2.Height);
var layoutWidth = localResults.Max(a => a.Item2.X + a.Item2.Width);
var widthMatchingDesiredRatio = desiredWidthToHeightRatio * layoutHeight;
double ratioProximity;
if (layoutWidth < widthMatchingDesiredRatio)
ratioProximity = widthMatchingDesiredRatio / layoutWidth;
else
ratioProximity = layoutWidth / widthMatchingDesiredRatio;
lock (sync)
{
if (ratioProximity < bestResultsProximityToDesiredRatio)
{
// this layout is the best approximation of the desired container dimensions, so far
bestResults = localResults;
bestResultsProximityToDesiredRatio = ratioProximity;
}
}
});
return bestResults ?? new List<Tuple<T, Rect>>() {Tuple.Create(rectsToArrange[0].Item2,
new Rect(new Point(0, 0), new Point(rectsToArrange[0].Item1.Width, rectsToArrange[0].Item1.Height))) };
}
}