  A linear time approximate maximum likelihood decoding algorithm on
tail-biting trellises is prsented, that requires exactly two rounds on the
trellis. This is an adaptation of an algorithm proposed earlier with the
advantage that it reduces the time complexity from O(mlogm) to O(m) where m is
the number of nodes in the tail-biting trellis. A necessary condition for the
output of the algorithm to differ from the output of the ideal ML decoder is
reduced and simulation results on an AWGN channel using tail-biting rrellises
for two rate 1/2 convoluational codes with memory 4 and 6 respectively are
reported
