Via this article http://www.pmean.com/08/RegressionAndAnova.html. It was written that linear least squares ,which is used to calculate regression, is used to calculate ANOVA, but there are other methods to calculate ANOVA. The article referred to this method as the regression algorithm, but, just to be clear I am not so much looking for an algorithm as much as a general concept used to calculate the anova table. I do not consider linear least squares as an algorithm, it is a general concept. Hence, What other general concepts are used to find the ANOVA table? What is the advantages of using the linear least square concept in calculating ANOVA?.
The standard general concept to find the ANOVA table was method of moments (MM); this approach relied strongly on normality assumptions and it is sub-optimal in comparison with standard least-squares (LS) procedures. In addition if the data are unbalanced a lot of ANOVA's theoretical concepts are seriously perplexed and this makes the estimation even more questionable. The user @Placidia has given two excellent answers on the matter here and here. MM is sometimes still used in one-way and two-way ANOVA because of its ease to code it up conceptually but essentially all $n$-way ANOVA routines now rely on LS. I have seen some (usually clustering) algorithms using MM because they are easier to code up and faster to converge; they serve as good hot-start points for optimization tasks.
(This post is an over-sized comment.)