I understand the four different type os FIR filters however i don´t understand why do linear-phase FIR filters offer opportunities for efficient realization.
Linear phase FIR filters should have some sort of symmetry (even or odd) in their coefficients which will reduce the number of independent multiplications by about two yielding an efficient realization compared to a nonsymmetric FIR filter of the same length.
However, linear phase FIR filters are important not because of their computational efficiency but for their linear phase property.
Linear phase FIR systems have poles only at the origin and zeros are conjugate symmetric. So, we can see system function as a cascade of different systems. It is easier to see this by using signal flow graph. On the delay line, a new coefficient is added just adding delay and multiplication of the coefficient. No extra addition is required. You can simply add new filter coefficients and that gives you opportunities. Also, FIR filters are cheaper than IIR filters.