[ ] Find a furthest point from M, f0 = MaxPoint[SpS((x-M)o(x-M))]. Do M (round) gap analysis on SpS((x-M)o(x-M)). f2 x' Do f0 (round) gap analysis on SpS((x-f0)o(x-f0)). d0≡(M-f0)/|M-f0|. f1 x Do d0 (linear) gap analysis on SpS((x-f0)od0). x-f0 Find a furthest pt from f0, f1MaxPoint[SpS((x-f0)o(x-f0))]). d1≡(f1-f0)/|f1-f0|. Do d1 (linear) gap analysis on SpS((x-f1)od1). Do f1 (round) gap on SpS((x-f1)o(x-f1)). d2 ((x-f0)od1)d1 d1 ≡ space perpendicular to d1. The projection of x-f0 onto d1 is ≡ - x' (x-f0) ((x-f0)od1)d1 d1 x'ox' [ - (x-f0) ((x-f0)od1)d1 = ] o f0 d1 (x-f0) o ((x-f0)od1) - (d1od1) + ((x-f0)od1)2 = (x-f0) o ((x-f0)od1)2 - 2 (1) + = So, x'ox' = |x'|2 = (x-f0)o(x-f0) - ((x-f0)od1)2 (also from Pythagorus) SpS(x'ox') = SpS[(x-f0)o(x-f0)) - SpS[(x-f0)od1]*SpS[(x-f0)od1]. Let f2MaxPoint[SpS(x'ox')] and d2≡f2'/|f2'| = f2-(f2od1)d1 |f2-(f2od1)d1| Note that d2od1= [(f2od1)-(f2od1)](d1od1) |f2-(f2od1)d1| =0 and |d2|=1 x''≡ x-f0 - ((x-f0)od1)d1 - ((x-f0)od2)d2 SpS(x''ox'')= SpS[(x-f0)o((x-f0)] - SpS[(x-f0)od1]*SpS[(x-f0)od1] - SpS[(x-f0)od2]*SpS[(x-f0)od2] Let f3MaxPoint(SpS(x"ox"), d3=f3"/|f3"| (Verify x'' ≡ (x-f0) - (x-f0)od1 d1 - (x-f0)od2 d2 is orthog to d1 and d2 x''od1= (x-f0)od1- (x-f0)od1 (d1od1) - (x-f0)od2 (d2od1) x''od1= (x-f0)od1- (x-f0)od1 ( 1 ) - (x-f0)od2 ( 0 ) = 0 x''od2= (x-f0)od2- (x-f0)od1 (d1od2) - (x-f0)od2 (d2od2) x''od2= (x-f0)od2- (x-f0)od1 ( 0 ) - (x-f0)od2 ( 1 ) = 0 SpS(x(k)ox(k)) = SpS[(x-f0)o(x-f0)] - SpS[(x-f0)od1]*SpS[(x-f0)od1) - ... - SpS[(x-f0)odk-1]*SpS[(x-f0)odk-1] = SpS[ x(k-1)ox(k-1) ] - SpS[(x-f0)odk-1]*SpS[(x-f0)odk-1] fkMaxPoint(SpS(x(k)ox(k))) and dk≡fk'/|fk'|. Do fk (round) gap analysis on SpS((x-fk)o(x-fk)). Do dk (linear) gap analysis on SpS((x-fk)odk).

SL SW PL PW set 51 35 14 2 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 set 49 30 14 2 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 set 47 32 13 2 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 set 46 31 15 2 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 set 50 36 14 2 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 set 54 39 17 4 0 1 1 0 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 set 46 34 14 3 0 1 0 1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 set 50 34 15 2 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 set 44 29 14 2 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 set 49 31 15 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 set 54 37 15 2 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 0 set 48 34 16 2 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 set 48 30 14 1 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 set 43 30 11 1 0 1 0 1 0 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 set 58 40 12 2 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 set 57 44 15 4 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 set 54 39 13 4 0 1 1 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 set 51 35 14 3 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 set 57 38 17 3 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 set 51 38 15 3 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 set 54 34 17 2 0 1 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 set 51 37 15 4 0 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 set 46 36 10 2 0 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 set 51 33 17 5 0 1 1 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 set 48 34 19 2 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 0 set 50 30 16 2 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 set 50 34 16 4 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 set 52 35 15 2 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 set 52 34 14 2 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 set 47 32 16 2 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 set 48 31 16 2 0 1 1 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 set 54 34 15 4 0 1 1 0 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 set 52 41 15 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 set 55 42 14 2 0 1 1 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 set 50 32 12 2 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 set 55 35 13 2 0 1 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 1 0 set 44 30 13 2 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 set 51 34 15 2 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 set 50 35 13 3 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 1 1 set 45 23 13 3 0 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 0 0 1 1 set 44 32 13 2 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 set 50 35 16 6 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 set 51 38 19 4 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 set 48 30 14 3 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 set 51 38 16 2 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 set 46 32 14 2 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 set 53 37 15 2 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 0 set 50 33 14 2 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 ver 70 32 47 14 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 ver 64 32 45 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 ver 69 31 49 15 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 1 ver 55 23 40 13 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 ver 65 28 46 15 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 ver 57 28 45 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 1 ver 63 33 47 16 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 0 0 ver 49 24 33 10 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 ver 66 29 46 13 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 ver 52 27 39 14 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 ver 50 20 35 10 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 ver 59 30 42 15 0 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 1 1 1 1 ver 60 22 40 10 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 ver 61 29 47 14 0 1 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 1 0 ver 56 29 36 13 0 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 0 0 0 0 1 1 0 1 ver 67 31 44 14 1 0 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 ver 56 30 45 15 0 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 ver 58 27 41 10 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 ver 62 22 45 15 0 1 1 1 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 ver 56 25 39 11 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 ver 59 32 48 18 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 ver 61 28 40 13 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 ver 63 25 49 15 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 1 1 1 1 ver 61 28 47 12 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 0 0 ver 64 29 43 13 1 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 1 ver 66 30 44 14 1 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 0 ver 68 28 48 14 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 ver 67 30 50 17 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 ver 60 29 45 15 0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 ver 57 26 35 10 0 1 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 ver 55 24 38 11 0 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 ver 55 24 37 10 0 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 ver 58 27 39 12 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 0 1 1 0 0 ver 60 27 51 16 0 1 1 1 1 0 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 0 0 ver 54 30 45 15 0 1 1 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 ver 60 34 45 16 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 ver 67 31 47 15 1 0 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 1 1 ver 63 23 44 13 0 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 ver 56 30 41 13 0 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 ver 55 25 40 13 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 ver 55 26 44 12 0 1 1 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 ver 61 30 46 14 0 1 1 1 1 0 1 0 1 1 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 SL SW PL PW ver 61 30 46 14 0 1 1 1 1 0 1 0 1 1 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 ver 58 26 40 12 0 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 ver 50 23 33 10 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 ver 56 27 42 13 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1 ver 57 30 42 12 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 1 1 0 0 ver 57 29 42 13 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 1 ver 62 29 43 13 0 1 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 1 ver 51 25 30 11 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 1 ver 57 28 41 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 vir 63 33 60 25 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 vir 58 27 51 19 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 vir 71 30 59 21 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 1 vir 63 29 56 18 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 vir 65 30 58 22 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 vir 76 30 66 21 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 vir 49 25 45 17 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 vir 73 29 63 18 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 0 1 0 vir 67 25 58 18 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 vir 72 36 61 25 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 vir 65 32 51 20 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 0 vir 64 27 53 19 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 0 1 0 0 1 1 vir 68 30 55 21 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 0 1 0 1 0 1 vir 57 25 50 20 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 vir 58 28 51 24 0 1 1 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0 0 vir 64 32 53 23 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 vir 65 30 55 18 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 0 1 0 0 1 0 vir 77 38 67 22 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 vir 77 26 69 23 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1 1 vir 60 22 50 15 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 vir 69 32 57 23 1 0 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 vir 56 28 49 20 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 vir 77 28 67 20 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 0 vir 63 27 49 18 0 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1 0 0 1 0 vir 67 33 57 21 1 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 0 1 0 1 vir 72 32 60 18 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 vir 62 28 48 18 0 1 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 vir 61 30 49 18 0 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 0 1 0 0 1 0 vir 64 28 56 21 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 vir 72 30 58 16 1 0 0 1 0 0 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 0 0 0 vir 74 28 61 19 1 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 1 0 0 1 1 vir 79 38 64 20 1 0 0 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 vir 64 28 56 22 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 vir 63 28 51 15 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 vir 61 26 56 14 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 vir 77 30 61 23 1 0 0 1 1 0 1 0 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 1 vir 63 34 56 24 0 1 1 1 1 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 0 vir 64 31 55 18 1 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 0 1 0 vir 60 30 18 18 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 vir 69 31 54 21 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 vir 67 31 56 24 1 0 0 0 0 1 1 0 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 0 vir 69 31 51 23 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 vir 68 32 59 23 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 1 1 vir 67 33 57 25 1 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 0 0 1 vir 67 30 52 23 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 1 1 1 vir 63 25 50 19 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 vir 65 30 52 20 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 vir 62 34 54 23 0 1 1 1 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 0 1 1 1 vir 59 30 51 18 0 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 t1 20 30 37 12 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 t2 58 5 37 12 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 1 t3 58 30 2 12 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 t4 58 30 37 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 t12 20 5 37 12 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 1 t13 20 30 2 12 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 t14 20 30 37 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 t23 58 5 2 12 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 t24 58 5 37 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 t34 58 30 2 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 t123 20 5 2 12 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 t124 20 5 37 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 t134 20 30 2 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 t234 58 5 2 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 tall 20 5 2 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 b1 90 30 37 12 1 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 b2 58 60 37 12 0 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 b3 58 30 80 12 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 1 b4 58 30 37 40 0 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 b12 90 60 37 12 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 b13 90 30 80 12 1 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 1 b14 90 30 37 40 1 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 b23 58 60 80 12 0 1 1 1 0 1 0 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 b24 58 60 37 40 0 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 b34 58 30 80 40 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 b123 90 60 80 12 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 b124 90 60 37 40 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 b134 90 30 80 40 1 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 b234 58 60 80 40 0 1 1 1 0 1 0 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 ball 90 60 80 40 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 Before adding the new tuples: MINS 43 20 10 1 MAXS 79 44 69 25 MEAN 58 30 37 12 same after additions.

Round gap outliers w Dis(x,M ) (gap>4; singletons/doubletons): {ball} Round gap outliers w Dis(x,f0=ball) (gap>4; singletons/doubletons): {ball}, {t13,t134}, {t123,tall}. gp>4 SortedD(x,f0) 1 0 ball 0 28 b123 ... 0 97 set 1 98 t34 0 103 t12 0 104 t23 0 107 t124 1 108 t234 0 113 t13 1 116 t134 0 122 t123 0 125 tall Linear gap ols D(x,f0=bal) gap>4; single/double): none Round gap outliers w Dis(x,f0=ball) (gap>4; singletons/doubletons): {tall}, {t123}, {t134,t13}, {b134}, {b123}, {ball} gp >4 SortedD(x,f0) 1 0 tall 1 12 t123 0 25 t134 1 28 t13 0 33 set ... 0 108 b13 1 110 b234 0 115 b134 1 119 b123 0 125 ball gp>4 SortedD(x,M) ... 0 62 b134 1 63 b123 0 69 ball Linear gap outliers w (x-f1)od1 (gap>4; singletons/doubletons): {bal}, {b123}, {b134}, {b234}, {b13}, {t12,t23}, {t13}, {t134},{t123},{tall} g>4 Sorted 1 -254. ball 1 -238. b123 1 -231. b134 1 -222. b234 1 -214. b13 0 -206. b124 1 -206. b23 0 -199. vir 0 -198. b34 1 -196. vir 0 -192. vir ... 0 -83.4 set 1 -83.3 set 0 -76.0 t12 1 -74.9 t23 0 -69.0 t124 1 -68.0 t234 1 -56.4 t13 1 -49.5 t134 1 -36.6 t123 1 -29.6 tall

The mathematics of SpS's: Every SpS is a functional on the set X (function from X to R1). Every real functioinal on X is an SpS on X. A vertical functional is a functional expressed vertically as a two column table, vf( X-column, R1-value-column ). A pTree functional is a vertical functional in which the value bitslices are expressed as basic pTrees in a PTreeSet. The simplest examples of a vertical functional is a coordinate projections, ek:XR1 where ek( x≡(x1,...xn) ) = xk Others include distance(x,p) where distance is any distance and p is a fixed point (e.g., such as M or f0 or f1 or ...) ( same as length(x-p) ) length(x) length2(x)≡xox Given any hyperplane, H, with orthonormal basis, o1,...ok, projoh(x) = oh-component of the projection of x on H (e.g., H is the space perpendicular to the line from f0 to f1).

p x y 1 6 36 2 7 39 3 8 41 4 9 34 5 9 38 6 10 42 7 12 34 8 12 38 9 13 35 10 13 40 11 19 38 12 25 38 13 22 22 14 26 16 15 26 25 16 29 11 17 31 18 18 32 26 19 34 11 20 34 23 21 35 20 22 37 10 23 37 23 24 38 13 25 38 21 26 39 24 27 40 9 28 42 9 29 38 39 30 38 42 31 39 44 32 41 41 33 41 45 34 42 39 35 42 43 36 44 43 37 45 40 No gaps (ct=0_intervals) on the furthest-to-Mean line, but 3 ct=1 intevals. Declare p=p12, p16, p18 anomaly if pofM is far enough from the bddry pts of its interval? VOM (34, 35) Mean, M Round 2 is straight forward. So, 1. Given gaps, find ct=k_intervals. 2. Find good gaps (dot prod with a constant vector for linear gaps?) For rounded gaps, use xox? Note: in this example, vom works better than mean. 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 10 20 30 40 50 If data shifted, len doesn't work. xofM is independent of pt placement wrt the origin. Length based gapping is dependent.

Thin interval finder on the fM line using the scalar pTreeSet, PTreeSet(xofM) (the pTree slices of these projection lengths) Looking for Width24_Count=1_ThinIntervals or W16_C1_TIs 1 z1 z2 z7 2 z3 z5 z8 3 z4 z6 z9 4 za 5 M 6 7 8 zf 9 zb a zc b zd ze c 0 1 2 3 4 5 6 7 8 9 a b c d e f X x1 x2 z1 1 1 z2 3 1 z3 2 2 z4 3 3 z5 6 2 z6 9 3 z7 15 1 z8 14 2 z9 15 3 za 13 4 zb 10 9 zc 11 10 zd 9 11 ze 11 11 zf 7 8 xofM 11 27 23 34 53 80 118 114 125 110 121 109 83 p6 1 p5 1 p4 1 p3 1 p2 1 p1 1 p0 1 p6' 1 p5' 1 p4' 1 p3' 1 p2' 1 p1' 1 p0' 1 f= &p5' 1 C=3 p5' C=2 p5 C=8 &p4' 1 C=1 p4' p4 C=2 C=0 C=6 p6' 1 C=5 p6 C10 W=24 C=1 [000 0000, 000 1111] =[0,16). z1ofM=11 is 5 units from 16, so z1 not declared an anomaly. W=24 C=1 [010 0000 , 010 1111] =[32,48). z4ofM=34 is within 2 of 32, so z4 is not declared an anomaly. W=24 C=1 [0110000, 0111111] =[48, 64). z5ofM=53 is 19 from z4ofM=34 (>24) but 11 from 64. The next interval [64,80) is empty and it's 27 from 80 (>24) so z5 is an anomaly and we make a cut through z5. W=24 C=0 [100 0000 , 100 1111]=[64, 80). Ordinarily we cut thru the midpoint of C=0 intervals, but in this case it's unnecessary since it would duplicate the z5 cut just made. Here we started with xofM distances. The same process works starting with any distance based ScalarPTreeSet, e.g., xox, etc.

Defining gaps: Any scalar pTreeSet where the scalar is a distance, can be used for gap based FAUST Clustering / Anomaly _Detection or FAUST Classification. Certainly the dot product with any fixed vector works (gaps in the projections along the line generated by the vector). E.g., use vectors fM; fM/|fM|; or in general, a*fM (a constant); (where M is a medoid (mean or vector of medians) and f is a "furthest point" from M). fF; fF/|fF|; or in general, a*fF (a constant); (where F is a "furthest point" from f). ek where ek - (0 0 0 ... 1 0 0 0 ...) (1 in the kth position) V1=(-b a 0 0 0 ...) where V = (a b c d e ...) is any one of the vectors above. (gives us a vector orthogonal to V) V2=(a b C 0 0 ...) where C=-(a2 + b2)/c (vector orthogonal to V and to V1); etc. (Vk for all k=1...n forming a orthogonal basis with V But also, if one takes the ScalarPTreeSet of all vector lengths (or squares of lengths to make it easy) that is also a ScalarPTreeSet and the gaps are radial gaps as one proceeds out from the origin. One can note that this is just the column of xox values, so it is dot product generated also. If one takes just the ScalarPTreeSet of all ith coordinate values (V=ei above), that works as well. In this case we get gaps in the value distribution of the ith coordinates. This was used, for instance in coordinate-wise (non-Oblique) FAUST. PTreeSet 11 27 23 34 53 80 118 114 125 110 121 109 83 p6 1 p5 1 p4 1 p3 1 p2 1 p1 1 p0 1 p6' 1 p5' 1 p4' 1 p3' 1 p2' 1 p1' 1 p0' 1 Take a fixed vector, y0, the ScalarptreeSet (SpS) of all vector lengths (or squares of lengths) of vectors, x-y0, is also a ScalarPTreeSet that works and the gaps are radial gaps as one proceeds out from the point, y0. Note that this is just the column of xoy0 values.

APPENDIX: FAUST=Fast, Accurate Unsupervised and Supervised Teaching (Teaching big data to reveal info) FAUST CLUSTER-fmg (furthest-to-mean gaps for finding round clusters): C=X (e.g., X≡{p1, ..., pf}= 15 pix dataset.) While an incomplete cluster, C, remains find M ≡ Medoid(C) ( Mean or Vector_of_Medians or? ). Pick fC furthest from M from S≡SPTreeSet(D(x,M) .(e.g., HOBbit furthest f, take any from highest-order S-slice.) If ct(C)/dis2(f,M)>DT (DensThresh), C is complete, else split C where P≡PTreeSet(cofM/|fM|) gap > GT (GapThresh) End While. Notes: a. Euclidean and HOBbit furthest. b. fM/|fM| and just fM in P. c. find gaps by sorrting P or O(logn) pTree method? C2={p5} complete (singleton = outlier). C3={p6,pf}, will split (details omitted), so {p6}, {pf} complete (outliers). That leaves C1={p1,p2,p3,p4} and C4={p7,p8,p9,pa,pb,pc,pd,pe} still incomplete. C1 is dense ( density(C1)= ~4/22=.5 > DT=.3 ?) , thus C1 is complete. Applying the algorithm to C4: In both cases those probably are the best "round" clusters, so the accuracy seems high. The speed will be very high! {pa} outlier. C2 splits into {p9}, {pb,pc,pd} complete. 1 p1 p2 p7 2 p3 p5 p8 3 p4 p6 p9 4 pa 5 6 7 8 pf 9 pb a pc b pd pe c d e f 0 1 2 3 4 5 6 7 8 9 a b c d e f M0 8.3 4.2 M1 6.3 3.5 f1=p3, C1 doesn't split (complete). M f M4 1 2 p2 p5 p1 3 p4 p6 p9 4 p3 p8 p7 5 pf pb 6 pe pc 7 pd pa 8 1 2 3 4 5 6 7 8 9 a b c d e f Interlocking horseshoes with an outlier X x1 x2 p1 1 1 p2 3 1 p3 2 2 p4 3 3 p5 6 2 p6 9 3 p7 15 1 p8 14 2 p9 15 3 pa 13 4 pb 10 9 pc 11 10 pd 9 11 pe 11 11 pf 7 8 D(x,M0) 2.2 3.9 6.3 5.4 3.2 1.4 0.8 2.3 4.9 7.3 3.8 3.3 1.8 1.5 C1 C2 C3 C4 M1 M0

FAUST Oblique PR = P(X dot d)<a d-line D≡ mRmV = oblique vector. d=D/|D| Separate classR, classV using midpoints of means (mom) method: calc a View mR, mV as vectors (mR≡vector from origin to pt_mR), a = (mR+(mV-mR)/2)od = (mR+mV)/2 o d (Very same formula works when D=mVmR, i.e., points to left) Training ≡ choosing "cut-hyper-plane" (CHP), which is always an (n-1)-dimensionl hyperplane (which cuts space in two). Classifying is one horizontal program (AND/OR) across pTrees to get a mask pTree for each entire class (bulk classification) Improve accuracy? e.g., by considering the dispersion within classes when placing the CHP. Use 1. the vector_of_median, vom, to represent each class, rather than mV, vomV ≡ ( median{v1|vV}, 2. project each class onto the d-line (e.g., the R-class below); then calculate the std (one horizontal formula per class; using Md's method); then use the std ratio to place CHP (No longer at the midpoint between mr [vomr] and mv [vomv] ) median{v2|vV}, ... ) dim 2 vomR vomV r   r vv r mR   r      v v v v       r    r      v mV v      r    v v     r         v                     v2 v1 d-line dim 1 d a std of these distances from origin along the d-line

1. MapReduce FAUST. Current_Relevancy_Score =9. Killer_Idea_Score=2 1. MapReduce FAUST Current_Relevancy_Score =9 Killer_Idea_Score=2 Nothing comes to minds as to what we would do here.  MapReduce.Hadoop is a key-value approach to organizing complex BigData.  In FAUST PREDICT/CLASSIFY we start with a Training TABLE and in FAUST CLUSTER/ANOMALIZER  we start with a vector space. Mark suggests (my understanding), capturing pTreeBases as Hadoop/MapReduce key-value bases? I suggested to Arjun developing XML to capture Hadoop datasets as pTreeBases. The former is probably wiser. A wish list of great things that might result would be a good start. 2.  pTree Text Mining: Current_Relevancy_Score =10  Killer_Idea_Score=9   I I think Oblique FAUST is the way to do this.  Also there is the very new idea of capturing the reading sequence, not just the term-frequency matrix (lossless capture) of a corpus. 3. FAUST CLUSTER/ANOMALASER: Current_Relevancy_Score =9               Killer_Idea_Score=9   No No one has taken up the proof that this is a break through method.  The applications are unlimited! 4.  Secure pTreeBases: Current_Relevancy_Score =9            Killer_Idea_Score=10     This seems straight forward and a certainty (to be a killer advance)!  It would involve becoming the world expert on what data security really means and how it has been done by others and then comparing our approach to theirs.  Truly a complete career is waiting for someone here! 5. FAUST PREDICTOR/CLASSIFIER: Current_Relevancy_Score =9             Killer_Idea_Score=10 No one done a complete analysis of this is a break through method.  The applications are unlimited here too! 6.  pTree Algorithmic Tools: Current_Relevancy_Score =10                 Killer_Idea_Score=10 This is Md’s work.  Expanding the algorithmic tool set to include quadratic tools and even higher degree tools is very powerful.  It helps us all! 7.  pTree Alternative Algorithm Impl: Current_Relevancy_Score =9               Killer_Idea_Score=8 This is Bryan’s work.  Implementing pTree algorithms in hardware/firmware (e.g., FPGAs) - orders of magnitude performance improvement? 8.  pTree O/S Infrastructure: Current_Relevancy_Score =10                    Killer_Idea_Score=10 This is Matt’s work.  I don’t yet know the details, but Matt, under the direction of Dr. Wettstein, is finishing up his thesis on this topic – such changes as very large page sizes, cache sizes, prefetching,…  I give it a 10/10 because I know the people – they do double digit work always! From: Arjun.Roy@my.ndsu.edu] Sent: Thurs, Aug 09 Dear Dr. Perrizo, Do you think a map reduce class of FAUST algorithms could be built into a thesis? If the ultimate aim is to process big data, modification of existing P-tree based FAUST algorithms on Hadoop framework could be something to look on? I am myself not sure how far can I go but if you approve, then I can work on it. From: Mark to:Arjun Aug 9 From industry perspective, hadoop is king (at least at this point in time). I believe vertical data organization maps really well with a map/reduce approach –   these are complimentary as hadoop is organized more for unstructured data, so these topics are not mutually exclusive. So from industry side I’d vote hadoop… from Treeminer side text (although we are very interested in both) From: msilverman@treeminer.com Sent: Friday, Aug 10 I’m working thru a list of what we need to get done – it will include implementing anomaly detection which is now on my list for some time.  I tried to establish a number of things such that even if we had some difficulties with some parts we could show others (w/o digging us too deep). Once I get this I’ll get a call going.  I have another programming resource down here who’s been working with me on our production code who will also be picking up some of the work to get this across the finish line, and a have also someone who was a director at our customer previously assisting us in packaging it all up so the customer will perceive value received… I think Dale sounded happy yesterday.