Time Machine Backup Stuck on Preparing Backup for 12 hours

Question

Hello,
I have a 2011 iMac and am running OS X 10.7.2.  Up until a few days ago, my Time Machine backups (to a USB 2.0 external hard drive) were running fine.  Now, when Time Machine tries to back up, all it does is display the progress bar with the words "Preparing backup..." below.  This message will remain indefinitely (I've let Time Machine run for as long as 12 hours and nothing changes).
I have done a fair amount of searching online regarding this problem and have tried the steps in Pondini's troubleshooting list ( http://pondini.org/TM/D1.html ):
1.  I first tried downloading and running the Time Machine buddy widget.  I installed it but the widget was just blank.  So, I went into Console and saved one my logs instead.  My log consisted of a huge string of similar entries.  Three of these entries are shown below:
1/28/12 12:12:53.629 PM          com.apple.backupd          Error (256): fetching properties: Error Domain=NSCocoaErrorDomain Code=256 "The file “yprime.h” couldn’t be opened." UserInfo=0x7f91ae359590 {NSURL=file://localhost/Volumes/JQTimeMachineBackup/Backups.backupdb/Jason%20Qu int%E2%80%99s%20iMac/2012-01-25-234450/Macintosh%20HD/Applications/MATLAB.app/ex tern/examples/shrlib/yprime.h, NSFilePath=/Volumes/JQTimeMachineBackup/Backups.backupdb/Jason Quint’s iMac/2012-01-25-234450/Macintosh HD/Applications/MATLAB.app/extern/examples/shrlib/yprime.h, NSUnderlyingError=0x7f91ae358480 "The operation couldn’t be completed. Not a directory"}
1/28/12 12:13:20.758 PM          com.apple.backupd          Error (256): fetching properties: Error Domain=NSCocoaErrorDomain Code=256 "The file “yprime.mexmaci64” couldn’t be opened." UserInfo=0x7f91ae071010 {NSURL=file://localhost/Volumes/JQTimeMachineBackup/Backups.backupdb/Jason%20Qu int%E2%80%99s%20iMac/2012-01-25-234450/Macintosh%20HD/Applications/MATLAB.app/ex tern/examples/shrlib/yprime.mexmaci64, NSFilePath=/Volumes/JQTimeMachineBackup/Backups.backupdb/Jason Quint’s iMac/2012-01-25-234450/Macintosh HD/Applications/MATLAB.app/extern/examples/shrlib/yprime.mexmaci64, NSUnderlyingError=0x7f91ae072e40 "The operation couldn’t be completed. Not a directory"}
1/28/12 12:13:47.893 PM          com.apple.backupd          Error (256): fetching properties: Error Domain=NSCocoaErrorDomain Code=256 "The file “include” couldn’t be opened." UserInfo=0x7f91ae071010 {NSURL=file://localhost/Volumes/JQTimeMachineBackup/Backups.backupdb/Jason%20Qu int%E2%80%99s%20iMac/2012-01-25-234450/Macintosh%20HD/Applications/MATLAB.app/ex tern/include/, NSFilePath=/Volumes/JQTimeMachineBackup/Backups.backupdb/Jason Quint’s iMac/2012-01-25-234450/Macintosh HD/Applications/MATLAB.app/extern/include, NSUnderlyingError=0x7f91ae072050 "The operation couldn’t be completed. Not a directory"}
2.  I then tried performing a "full reset" of Time Machine, as explained on Pondini's page.  This didn't fix my problem.
3.  Finally, I used Disk Utility and clicked "Repair" for my external hard drive --no problems were found.  I also selected my internal hard drive (my main drive with OS X on it) and clicked "Verify"; no problems were found.
Any ideas what might be going on?  Note that I had emptied about 70 GB of files from my trash and deleted some other movies from my hard drive soon before these problems cropped up.
Thank you very much,
Jason

Answer 1

This issue has been reported a few times, and it's clearly a bug in Time Machine, but I'm not sure what triggers it. TM creates multi-linked directories, and it seems that these links sometimes get broken.
The first thing you should do is a start a new backup set on another drive. You should do that anyway, because one backup isn't enough, even if you're not having problems with it.
Was the MATLAB application one of the items you had deleted from the source volume? If so, and if you're sure you'll never want to restore it from a backup, I suggest you delete it from the TM repository by entering TM, selecting it in a snapshot, and then choosing "Delete All Backups of..." from the action menu (gear icon.) You may have to do the same with other items that are giving the same error.