Until this century, relative dating was the only technique for identifying the age of a truly ancient object.
By examining the object's relation to layers of deposits in the area, and by comparing the object to others found at the site, archaeologists can estimate when the object arrived at the site.
Accelerator mass spectrometry is not dependent upon the radioactive decay.
What you’re doing is measuring all of the carbon isotopes in the sample – the 12, 13 and 14 – the accelerator operates like a giant mass spectrometer.
AMS is faster and needs a much smaller sample, but is more expensive.