Bin Packing - Shelf Algorithms

I’ve recently been working on an update to the twodee library we use for Ludum Dare games. One (of many) areas I’ll be focusing on is speeding up text rendering.

Text is currently very slow because we have to create and bind a new texture, render glyphs to it, then draw geometry for each piece of text in a scene. One simple optimization is to pack frequently-used text into a single texture which will remove many (expensive) texture binds.

Packing a bunch of rectangles into a texture isn’t the easiest thing to do well. There’s a whole class of algorithms dealing with this “bin packing” problem, each with various tradeoffs. Luckily, I found a very useful paper which covers many of these algorithms (thanks Jukka Jylänki!). To get a feel for how well each of them perform, I decided to implement a few in Javascript (you can see the source here).

In production, I would use a bin packing implementation which would grow the texture image when it ran out of space, or just fail if it couldn’t find room for a new rectangle.

Here, I’m going to fix a texture size and allow overflows. This should highlight the efficiency of each algorithm through some key metrics:

  • How much of the image is filled up by text?
  • How much of the text overflowed out of the image?
  • How many words fit before an overflow?

The texture size can be adjusted with these controls:

Static examples are pretty boring, so there is some randomness in my word generation. For reproducibility, I’ve used a random number generator which accepts a seed, which can be configure here:

I borrowed Amit Patel’s random noise generator and used it to create some parameterized random text. Changing the Exponent sliders below will change the way the random distribution looks. Lower values wind up looking more like a sine wave, while higher values look more chaotic.

You can configure the distribution of font sizes in the text:

The lengths and count of strings are adjustable as well. I’m just generating words of varying lengths but the generated text could really be any type of string.

By changing the values above, the following list will update. Each algorithm will attempt to pack the same list of strings.

There are a few assumptions I made which will make my implementations differ from Jukka’s paper. It is concerned with the rotation of the nested rectangles to achieve an optimal fit. Since my text strings are generally more likely to be wider than tall (and about the same height) I’m not going to worry about it.

I’m also going to pack from the top down, as opposed to bottom up. This just makes the visualization a little nicer on this page. I could easily flip the y-coordinate to render bottom-up, like OpenGL texture coorindates.

Shelf Next Fit (SHELF-NF)

The simplest class of packing algorithms are labeled SHELF since they approach the problem by trying to fit as many rectangles onto a horizontal row (or shelf) as possible.

Shelf Next Fit is the most direct implementation of this idea:

  1. For each rectangle in the list of strings to pack:
    1. If the rectangle does not fit in the current shelf:
      1. Close the current shelf.
      2. Open a new shelf with height 0.
    2. Add the rectangle to the current shelf.
      1. If the rectangle’s height is greater than the shelf’s height:
        1. Set the shelf height to the rectangle’s height.

This is inefficient in the sense that when a shelf is closed, any remaining space is no longer considered usable. If a small rectangle which would fit in the extra space shows up later, it is placed on the current shelf instead. Hence, there’s a ragged right edge of wasted space as the word count grows.

The next several algorithms attempt to reclaim some of that space in various ways. This comes with a cost — where SHELF-NF runs in O(N) time and O(1) space, the remaining algorithms are implemented in O(N^2) time and consume O(N) space. The extra cost is required to keep track of and search through all of the prior shelves.

Shelf First Fit (SHELF-FF)

To try and reduce the wasted space, Shelf First Fit iterates over every shelf for each new word. It inserts the text into the first shelf with available space (creating a new shelf if no existing shelf has room).

My first intuition was that SHELF-FF would always be a better approach. This is incorrect — it may wind up making locally-optimal choices which reduce the global efficiency. Consider an “unlucky” case where placing a small word on a prior row aligns a lot of future words in a worse way.

Shelf Best Width Fit (SHELF-BWF)

Shelf Best Width is like SHELF-FF, but instead of choosing the first shelf with room, chooses the horizontally-smallest shelf with enough room for the word.

function heuristic(shelf, word) {
  return shelf.remainingX() - word.width;
};

In terms of efficiency I don’t see a ton of difference between SHELF-BWF and SHELF-FF, although if you set the image dimensions to be tall and narrow there are sometimes differences in the layout on the texture.

Shelf Best Height Fit (SHELF-BHF)

Shelf Best Height optimizes for lowest remaining vertical space. Since there’s not a ton of variance in rectangle heights I didn’t expect this to be very efficient. In tall and narrow images I found a few examples where it performed worse than SHELF-FF.

function heuristic(shelf, word) {
  return shelf.height - word.height;
};

Shelf Best Area Fit (SHELF-BAF)

Shelf Best Area fit optimizes for minimum remaining area after the word is placed.

function heuristic(shelf, word) {
  var shelfArea = shelf.remainingX() * shelf.height,
      wordArea = word.width * word.height;
  return shelfArea - wordArea;
};

On this text-like input SHELF-BAF performs very close to SHELF-BWF. Higher variance in font size can introduce some efficiency differences but not in a meaningful way.

Shelf Worst Width Fit (SHELF-WWF)

Shelf Worst Width Fit is the most unusual SHELF algorithm. It is based off of the idea that rectangles should be packed in shelves which would result in the most remaining space (with the exception that if a word fits exactly it should be placed right away).

function heuristic(shelf, word) {
  if (shelf.remainingX() === word.width) {
    return Number.MAX_VALUE; // Immediately pick this fit.
  }
  return shelf.remainingX() - word.width;
};

function compare(score, bestScore) {
  return score > bestScore; // Higher is better.
};

Presumably leaving more space might allow for bigger words to be placed in extra space, but in practice this seemed to consistently perform the worst of the non-SHELF-NF algorithms on the text input.

Summary

The input I’m testing is very special-purposed so these results are probably not indicative of general performance. But for text, it generally seems that SHELF-NF performs worst, followed by SHELF-WWF and SHELF-FF. The remaining algorithms tend to be pretty close, with SHELF-BWF or SHELF-BAF having the best result.

It is possible to contrive specific examples to make any specific algorithm “win”, and I’ve found it useful to view how changing the page parameters makes different algorithms stand out. I’ve added another set of controls and a summary view for your convenience in case you wish to do the same. The parameters will change the examples above so you can visually examine any given packing by scrolling up.

Of course, SHELF is not the only class of bin packing algorithm. There are several more advanced approaches which will probably have better performance. I expect to visit them in a similar update in the near future.

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