IEEE Systems, Man and Cybernetics Magazine - April 2020 - 14

Algorithm S9. Incremental VAT
(inc-VAT).
Input : D l)n - n # n VAT reordered dissimilarity matrix for
  X n, n $ 2
				 Pn -  VAT reordering indices of D )n
				 d n -  MST cut magnitude order of D )n
				 Fn -  MST connection indices of D )n
				 V = " v 1,v 2,f,v n , -  distance of x n+1 from
				  " x 1, x 2,f, x N ,
Output: D )n + 1 - (n + 1) # (n + 1) 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 insertion position i for x n+1
	 2	(i, Pn + 1, d n + 1, Fn + 1) = InsertPosition (Pn, d n, Fn,V )

(algorithm S8)
	3	 A = " Pn , Pn ,f, Pn ,
	4	C = " Pn , Pn ,f, Pn ,
	5	B = Pn \C
	6	E = 4
	7	G = " 1, 2,f, i - 1 ,
	8	H = " Fn , Fn ,f, Fn ,
i

1

i

i+1

n

2

i-1

i+1

n

	 9	Reorder the remaining points after insertion
 position
	10	 while A ! 4 do
	11		 ^A, B, E, Pn + 1, d n + 1, Fn + 1, G, H h =
  IncInsert ^A, B, E, Pn + 1, d n + 1, Fn + 1, G, H, D )n, Pn, d n, Fn h
	12	
(algorithm S10)
	13	 end
	14	 D )n = D )n
	15	 Y ) = YG
	16	 Y ) = 6Y 1) ,Y 2) ,f,Y i)- 1, 0,Y )i ,Y )i + 1,f,Y )n@
	17	 D )n + 1 = Insert Y ) after i - 1th row and i - 1th column
  of D )n
G, G

Algorithm S10. IncInsert.
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
	 1	Input-Output: A, B, E, Pn + 1, d n + 1, Fn + 1, G, H
z1 = dn
	2	w 1 = B 1
v 1 = arg ^Pn + 1 = Pn
Pos(B 1) - 1

H1

_ Minimum distance, closest
bb
point index, and MST
`
hb connection index from G1
a

z 2 = min (YA) Minimum distance, closest point
index, and MST connection
	3	w 2 = A argmin(Y ) 4
v2 = i
index from G2

	 6	switch z do
	 7		 case z 1 do
^A B, E, Pn + 1, d n + 1, Fn + 1, G, H h =
	
8			,
  M1^A, B, E, Pn + 1, d n + 1, Fn + 1, G, H, z 1,w 1,v 1 h
	 9	
(algorithm S12)
	10		 end
	11		 case z 2 do
^A B, E, Pn + 1, d n + 1, Fn + 1, G, H h =
	
12			,
  M2 ^A, B, E, Pn + 1, d n + 1, Fn + 1, G, H, z 2,w 2,v 2 h
	13	
(algorithm S13)
	14		 end
	15		 case z 3 do
^A B, E, Pn + 1, d n + 1, Fn + 1, G, H h =
	
16			,
  M3 ^A, B, E, Pn + 1, d n + 1, Fn + 1, G, H, z 3,w 3,v 3 h
	17	
(algorithm S14)
	18		 end
	19	 end

Algorithm S11. LongestSubsequence
Input : Pn -  VAT reordering indices of D )n
				 Pn + 1 -  VAT reordering indices of D )n +1
Output: LongestSubsequence ^Pn, Pn + 1 h
1	for j ! 1 to n do
2		 if Pn " Pn + 1 then
3			 break
4		 end
5	end
j

6	 LongestSubsequence ^Pn, Pn + 1 h = " Pn , Pn ,f, Pn ,
1

2

Algorithm S12. Procedure M1.
Input : z 1 -  minimum distance from G1
				 w 1 -  closest point index from G1
				 v 1 -  MST connection index from G1
1	Input-Output: A, B, E, Pn + 1, d n + 1, Fn + 1, G, H
2	 Pn + 1 = " Pn + 1,w 1 ,;   d n + 1 = " d n + 1, z 1 ,;   Fn + 1 = " Fn + 1,v 1 ,
3	 G = " G, Pos ^w 1 h,
4	C =  LongestSubsequence ^Pn, Pn + 1 h  (algorithm S11)
5	 B = Pn \C
6	while length ^H h 2 length ^B h do
7		 delete H 1
8	end
9	 delete w 1 from A

A

h
z 3 = min ^D )n
( j, k) = argmin ^D )n
	4	
w3 = A j
v 3 = arg (Pn + 1 = E k )
Pos (A), Pos (E )

Pos (A), Pos (E )

	5	z = min ^z 1, z 2, z 3 h

14	

_ Minimum distance,
b
hb closest point index,
`
b and MST connection
b
index from G3
a

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2020

j-1

Algorithm S13. Procedure M2.
Input : z 2 -  minimum distance from G2
				 w 2 -  closest point index from G2
				 v 2 -  MST connection index from G2
IEEE Systems, Man and Cybernetics Magazine - April 2020

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