computing the quartet distance between general trees

70 APPENDIX A. PREPROCESSING FOR THE SUB-CUBIC ALGORITHMAnd two more arrays:R ′′ [i ] =C ′ ∑d v[j ] = I ′ [i , j ]M ′ =∑d vj =1i=1∑d v d v ′∑i=1 j =1I ′ [i , j ]I [i , j ](C [j ] − I [i , j ])(A.6)(A.7)(A.8)C ′′ ∑d v[j ] = I [i , j ](R[i ] − I [i , j ])(A.9)i=1Only one of the two following arrays will need calculation, depending on which of thetwo expressions for counting different butterflies is used, Eq. (7.9) or Eq. (7.10):R ′′′ [i ] =∑d vj =1I [i , j ] 2C ′′′ ∑d v[j ] = I [i , j ] 2With the approach taken in the article [14], the following table needs calculation.i=1(A.10)(A.11)I ′′′ [i , j ] =∑d vd v ′∑k=1,k≠i l=1,l≠jI [i ,l]I [k, j ]I [k,l]However, if calculated naively, the sub-cubic time bound promised will be broken. Thesolution is instead to calculate either I1 ′′ = I I T and then I1 ′′′ = I ′′ I , which is basically1dI 1 ′′v ′[i ,k] = ∑j =1I [i , j ]I [k, j ](A.12)I 1 ′′′d v[i , j ] = ∑k=1I [k, j ]I 1 ′′ [i ,k], (A.13)or otherwise calculate I2 ′′ = I T I and then I2 ′′′ = I I ′′ , which is basically2I 2 ′′d v[j,l] = ∑i=1I [i , j ]I [i ,l](A.14)dI 2 ′′′v ′[i , j ] = ∑I [i ,l]I 2 ′′ [l,k]. (A.15)l=1The choice is essential to the algorithm and is explained in further detail in Sec. 7.1.4.1.