Which Screen Size / Density Combinations Should I Support For Bitmap Background Image?
Solution 1:
You would only provide images at different resolutions if you want that piece of art to be a constant size, regardless of the screen dpi it is displayed on. An icon or a button would be an example of this.
For the background images described here, you don't care what the image dpi is -- you only care what the x * y dimensions are. So you don't have to produce the cross-product of sizes vs. dpi. You only need consider the 4 screen size categories.
And within the 4 screen size categories, you only need store one of the larger sizes (xlarge or large) and let the framework scale that up or down as needed. You can also do the scaling programmatically, to ensure that you don't change the aspect ratio of your background, and you don't crop it.
See also Android game working on all screen sizes which will hopefully attract a better answer.
Solution 2:
I would advise providing images for ldpi, mdpi, hdpi and optionally xhdpi (depending on your target users). That will allow you to cover the most commonly used resolutions just fine.
If you end up with the feeling that your application is getting too large (either by deciding to add all possible image sizes or other reasons), you can also look into allowing your application to be moved to an SD card. That way, storage won't be much of an issue anymore. (http://developer.android.com/guide/appendix/install-location.html)
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