I know a few examples exist online but between a very abstract paper like this one and a javascript implementation on GitHub I was wondering does one could write an algorithm from scratch using a naive approach.
Let's start with a simple example, the geometric series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64… would give us something like this
My guess would be to use a recursive algorithm to create our treemap. First we need to order all the elements from biggest to smallest. Then we start to divide the area for each elements starting with the first element which takes half the available space. After that we divide the remaining area with the rest of the elements. We repeat this process until we have all the pieces.
Now if we take another more realistic example
Here every time we need to slice the area we need to make sure it won't be too small or won't look so great. If the element represent 50% or more of the total it won't be a problem but what about only 10% or 4%? I think we should set a minimum threshold for our slice, let say 25% for now. We will experiment with this value once we finish our algorithm.
So what happen if the largest value is below our 25% threshold? I think we should include more elements in the slice until we reach at least 25%. For example if we have 14%, 8% and 5% the total is 27%, so our first slice will be 27% of the available space. Then we need to distribute the 3 elements in that slice. This is in essence a subset of our original problem so we can repeat the process just for that slice.
By the way what orientation the first slice should be? I think we should always slice on the longest side of the area rectangle. If we have a square then it doesn't matter really.
Next, how do we divide the first slice? We have 3 elements: 14%, 8% and 5%. If we look for a solution from the start we now have to following proportions: 51.8%, 29.6% and 18.5%. We can take a new slice only for the first element. And we repeat the process for the last 2 elements. Which is now 61.5% and 38.5%. At each step we need to evaluate if the slice will be horizontal or vertical depending on the the shape of the area we have.
Finally, all we have to do is repeat this until all the elements are placed!
Here is my implementation in LinqPad in 3 parts
Slice calculation
public Slice<T> GetSlice<T>(IEnumerable<Element<T>> elements, double totalSize,
double sliceWidth)
{
if (!elements.Any()) return null;
if (elements.Count() == 1) return new Slice<T>
{ Elements = elements, Size = totalSize };
var sliceResult = GetElementsForSlice(elements, sliceWidth);
return new Slice<T>
{
Elements = elements,
Size = totalSize,
SubSlices = new[]
{
GetSlice(sliceResult.Elements, sliceResult.ElementsSize, sliceWidth),
GetSlice(sliceResult.RemainingElements, 1 - sliceResult.ElementsSize,
sliceWidth)
}
};
}
private SliceResult<T> GetElementsForSlice<T>(IEnumerable<Element<T>> elements,
double sliceWidth)
{
var elementsInSlice = new List<Element<T>>();
var remainingElements = new List<Element<T>>();
double current = 0;
double total = elements.Sum(x => x.Value);
foreach (var element in elements)
{
if (current > sliceWidth)
remainingElements.Add(element);
else
{
elementsInSlice.Add(element);
current += element.Value / total;
}
}
return new SliceResult<T>
{
Elements = elementsInSlice,
ElementsSize = current,
RemainingElements = remainingElements
};
}
public class SliceResult<T>
{
public IEnumerable<Element<T>> Elements { get; set; }
public double ElementsSize { get; set; }
public IEnumerable<Element<T>> RemainingElements { get; set; }
}
public class Slice<T>
{
public double Size { get; set; }
public IEnumerable<Element<T>> Elements { get; set; }
public IEnumerable<Slice<T>> SubSlices { get; set; }
}
public class Element<T>
{
public T Object { get; set; }
public double Value { get; set; }
}
Generating rectangles using leaf slice (slice with only one element in it)
public IEnumerable<SliceRectangle<T>> GetRectangles<T>(Slice<T> slice, int width,
int height)
{
var area = new SliceRectangle<T>
{ Slice = slice, Width = width, Height = height };
foreach (var rect in GetRectangles(area))
{
// Make sure no rectangle go outside the original area
if (rect.X + rect.Width > area.Width) rect.Width = area.Width - rect.X;
if (rect.Y + rect.Height > area.Height) rect.Height = area.Height - rect.Y;
yield return rect;
}
}
private IEnumerable<SliceRectangle<T>> GetRectangles<T>(
SliceRectangle<T> sliceRectangle)
{
var isHorizontalSplit = sliceRectangle.Width >= sliceRectangle.Height;
var currentPos = 0;
foreach (var subSlice in sliceRectangle.Slice.SubSlices)
{
var subRect = new SliceRectangle<T> { Slice = subSlice };
int rectSize;
if (isHorizontalSplit)
{
rectSize = (int)Math.Round(sliceRectangle.Width * subSlice.Size);
subRect.X = sliceRectangle.X + currentPos;
subRect.Y = sliceRectangle.Y;
subRect.Width = rectSize;
subRect.Height = sliceRectangle.Height;
}
else
{
rectSize = (int)Math.Round(sliceRectangle.Height * subSlice.Size);
subRect.X = sliceRectangle.X;
subRect.Y = sliceRectangle.Y + currentPos;
subRect.Width = sliceRectangle.Width;
subRect.Height = rectSize;
}
currentPos += rectSize;
if (subSlice.Elements.Count() > 1)
{
foreach (var sr in GetRectangles(subRect))
yield return sr;
}
else if (subSlice.Elements.Count() == 1)
yield return subRect;
}
}
public class SliceRectangle<T>
{
public Slice<T> Slice { get; set; }
public int X { get; set; }
public int Y { get; set; }
public int Width { get; set; }
public int Height { get; set; }
}
Drawing the rectangles in WinForm
public void DrawTreemap<T>(IEnumerable<SliceRectangle<T>> rectangles, int width,
int height)
{
var font = new Font("Arial", 8 );
var bmp = new Bitmap(width, height);
var gfx = Graphics.FromImage(bmp);
gfx.FillRectangle(Brushes.Blue, new RectangleF(0, 0, width, height));
foreach (var r in rectangles)
{
gfx.DrawRectangle(Pens.Black,
new Rectangle(r.X, r.Y, r.Width - 1, r.Height - 1));
gfx.DrawString(r.Slice.Elements.First().Object.ToString(), font,
Brushes.White, r.X, r.Y);
}
var form = new Form() { AutoSize = true };
form.Controls.Add(new PictureBox()
{ Width = width, Height = height, Image = bmp });
form.ShowDialog();
}
And finally to generate a Treemap in LinqPad
void Main()
{
const int Width = 400;
const int Height = 300;
const double MinSliceRatio = 0.35;
var elements = new[] { 24, 45, 32, 87, 34, 58, 10, 4, 5, 9, 52, 34 }
.Select (x => new Element<string> { Object = x.ToString(), Value = x })
.OrderByDescending (x => x.Value)
.ToList();
var slice = GetSlice(elements, 1, MinSliceRatio).Dump("Slices");
var rectangles = GetRectangles(slice, Width, Height)
.ToList().Dump("Rectangles");
DrawTreemap(rectangles, Width, Height);
}
References:
