IEEE Systems, Man and Cybernetics Magazine - April 2020 - 15

	 1	Input-Output: A, B, E, Pn + 1, d n + 1, Fn + 1, G, H
	2	Pn + 1 = " Pn + 1,w 2 ,;   d n + 1 = " d n + 1, z 2 ,;   Fn + 1 = " Fn + 1,v 2 ,
	3	G = " G, Pos ^w 2 h,

	 4	if w 2 = A 1 then
	 5		 C =  LongestSubsequence ^Pn, Pn + 1 h (algorithm S11)
	 6		 B = Pn \C
	 7		 while length ^H h 2 length ^B h do
	 8			 delete H 1
	 9		 end
	10		 E = Pn + 1 \ " n + 1 , \C
	11	 else
	12		 E = " E,w 2 ,
	13	 end
	14	 delete w 2 from A

Algorithm S14. Procedure M3.
Input : z 3 -  minimum distance from G3
				 w 3 -  closest point index from G3
				 v 3 -  MST connection index from G3
	 1	Input-Output: A, B, E, Pn + 1, d n + 1, Fn + 1, G, H
	2	Pn + 1 = " Pn + 1,w 3 ,;   d n + 1 = " d n + 1, z 3 ,; Fn + 1 = " Fn + 1,v 3 ,
	3	G = " G, Pos ^w 3 h,

	 4	if w 3 = A 1 then
	 5		 C =  LongestSubsequence ^Pn, Pn + 1 h (algorithm S11)
	 6		 B = Pn \C
	 7		 while length ^H h 2 length ^B h do
	 8			 delete H 1
	 9		 end
	10		 E = Pn + 1 \ " n + 1 , \C
	11	 else
	12		 E = " E,w 3 ,
	13	 end

Algorithm S16. Decremental VAT
(dec-VAT).
Input : D )n - n # n VAT reordered dissimilarity matrix for X n
				 Pn -  VAT reordering indices of D )n
				 d n -  MST cut magnitude order of D )n
				 Fn -  MST connection indices of D )n
				 x p -  point to remove
Output: D *n - 1 - ^n - 1h # ^n - 1 h  VAT reordered dissimilarity
  matrix for X n - 1
				 Pn - 1 -  VAT reordering indices of D )n -1
				 d n - 1 - MST cut magnitude order of D )n -1
				 Fn - 1 - MST connection indices of D )n -1
	 1	Find the position i of x p in Pn
	2	i = arg ^Pn = p h

	 3	Find the data points ^J h and their indices  ^ I h  in
Pn , that are connected to x p in the MST of D )n
	4	I = arg ^Fn = i h; J = Pn
I

	 5	if J = 4  then
	 6			^D )n - 1, Pn - 1, d n - 1, Fn - 1 h = LeafNodeRemove ^D )n, Pn,
  d n, Fn, i h 
(algorithm S17)
	 7	else
	 8			 if i = Pn then
	 9					 Pn - 1 = " Pn ,; d n - 1 = 4; Fn - 1 = " 1 ,
	10					 A = " Pn , Pn ,f, Pn ,
	11					 C = " Pn , Pn ,
	12					 B = Pn \C
	13					 E = 4
	14					 G = {2}
	15					 H = " Fn , Fn ,f, Fn ,
	16					 k = arg ^J = Pn h; delete J k
	17			 else
	18					 Pn - 1 = " Pn , Pn ,f, Pn ,; d n - 1 = " d n , d n ,f, d n ,;
Fn - 1 = " Fn , Fn ,f, Fn ,
	19					 A = " Pn , Pn ,f, Pn ,
	20					 C = " Pn , Pn ,f, Pn ,
	21					 B = Pn \C
	22					 E = 4
	23					 G = " 1, 2,f, i - 1 ,
	24					 H = " Fn , Fn ,f, Fn ,
	25			 end
1

2

3

4

1

2

3

Algorithm S15. Incremental iVAT
(inc-iVAT).
Input : D )n + 1 - ^n + 1h # ^n + 1 h  inc-VAT reordered
  -dissimilarity matrix for X n + 1
				 D nl) - n # n  iVAT dissimilarity matrix for X n
				i -  insertion index of the new data point x n+1
  in Pn + 1
Output: D nl)+ 1 - ^n + 1h # ^n + 1 h inc-iVAT dissimilarity
  matrix for X n + 1
1	 c = " 1, 2,f, i - 1 ,
2	 D nl)+ 1 = D nl )
cc

cc

3	for r ! i to n + 1 do
4		 j = argmin " D )n + 1 ,

4

n

2

1

2

1

i+1

i-1

2

i+1

1

	14	 delete w 3 from A

n

i+2

2

i+2

1

2

i-2

i-1

n

i

n

	26			 while A ! 4 do
	27					 if  B 1 = J 1 then
	28						 ^A, B, E, Pn - 1, d n - 1, Fn - 1, G, H, J h =
  SpecialInsert ^A, B, E, Pn - 1, d n - 1, Fn - 1, G, H, J,
D )n, Pn, d n, Fn h (algorithm S18)
		
	29					 else
	30						 ^A, B, E, Pn - 1, d n - 1, Fn - 1, G, H, h =
  DecInsert ^A, B, E, Pn - 1, d n - 1, Fn - 1, G, H, D n) , Pn,
d n, Fn h 
		
(algorithm S19)
	31					 end
	32			 end
	33			 D )n - 1 = D )n
	34	 end
G, G

rm

1 # m # r-1
)
)
n +1r j
n +1r j

5		 D l = D
6		 c = " 1, 2,f, r - 1 , \ " j ,
7		 D nl)+1 = max {D n) + 1 , D nl)+1 }
8		 D nl)+1 = D nl)+1
9	end
rc

rc

rj

jc

cr

	

Ap ri l 2020

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE	

15



IEEE Systems, Man and Cybernetics Magazine - April 2020

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