データ構造とアルゴリズム avl木...
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-
AVL ()
2005 10 3
AVL
-
2 = O(n) = O(log n)AVLO(log n) 2
AVL
-
(balanced search tree)
O(log n)
AVL (2)2-3B
AVL
-
AVL
AAVL (AVL-tree)
2 1
AVL
-
AVL 1
AVL?
1
2
3
4
AVL
-
AVL 1
AVL?
1
2
3
4
AVL
-
AVL 2
AVL?
1 5 7 9 11 13
6 10 14
4 12
8
2
3 15
AVL
-
AVL 2
AVL?
1 5 7 9 11 13
6 10 14
4 12
8
2
3 15
AVL
-
AVL 3
AVL?
8
32 6
5
4 71
141512
1310
119
16
AVL
-
AVL 3
AVL?
8
32 6
5
4 71
141512
1310
119
16
AVL
-
AVL 4
AVL? ()
5
2
1 4
3
6 9
7
8
10
11
12
AVL
-
AVL 4
AVL? ()
5
2
1 4
3
6 9
7
8
10
11
12
AVL
-
AVL 5
AVL?
1 9 11
10
12
8
2
3 15
4
5
6
7
14
13
AVL
-
AVL 5
AVL?
1 9 11
10
12
8
2
3 15
4
5
6
7
14
13
AVL
-
AVL 5
AVL? (13)
1 9 11
10
12
8
2
3 15
4
5
6
7
14
13
AVL
-
AVL
AVLAVL
8
32 6
5
4 71
141512
1310
119
16
AVL
-
AVL
AVLAVL
8
32 6
5
4 71
141512
1310
119
16
AVL
-
AVL
AVLO(log n)
f (h) = hAVL
h 2
f (h) = 1 + f (h 1) + f (h 2)
(f (0) = 1, f (1) = 2)
AVL
-
AVL
AVLO(log n)
f (h) = hAVL
h 2
f (h) = 1 + f (h 1) + f (h 2)
(f (0) = 1, f (1) = 2)
AVL
-
AVL
AVLO(log n)
f (h) = hAVL
h 2
f (h) = 1 + f (h 1) + f (h 2)
(f (0) = 1, f (1) = 2)
AVL
-
AVL ()
f (h) =15
(
1 +
5
2
)h+3
(
1
5
2
)h+3
1
f (h) n
h = O(log n)
[]
AVL
-
AVL ()
f (h) =15
(
1 +
5
2
)h+3
(
1
5
2
)h+3
1
f (h) n
h = O(log n)
[]
AVL
-
AVL ()
f (h) =15
(
1 +
5
2
)h+3
(
1
5
2
)h+3
1
f (h) n
h = O(log n)
[]
AVL
-
AVL
Member, Min2InsertDelete2
AVL
-
AVL
Member, Min2InsertDelete2
AVL
-
AVL
Member, Min2InsertDelete2
AVL
-
AVL
Member, Min2InsertDelete2
AVL
-
y s(y)
s(y) =
L > E = R <
L
ER
E
E
L
RE
L
L
R
EEE
E E
8
32 6
5
4 71
141512
1310
119
16
AVL
-
y s(y)
s(y) =
L > E = R <
E E
E
E
E EE E
E
LL L
LR
R
R
1
23
4
6
7
5
8
1214
15
1310
9 11
16
AVL
-
u (single right rotation)
u
vu
v
AVL
-
v (single left rotation)
u
v
u
v
AVL
-
u (double left rotation)
u
w
v
w
v
u
AVL
-
v (double right rotation)
u
w
v
w
u
v
AVL
-
1
E E
E
E
E EE
E
LL L
LR
R
R
E
E16
1
23
4
6
7
5
8
1214
15
1310
9 11
16
17
AVL
-
1
E E
E
E
E EE
E
LL L
LR
R
R
E
E17
1
23
4
6
7
5
8
1214
15
1310
9 11
16
17
AVL
-
1
E E
E
E
E EE
E
LL L
LR
R
R
E
E17
1
23
4
6
7
5
8
1214
15
1310
9 11
16
1717E ()
AVL
-
1
E E
E
E
E EE
E
LL L
LR
R
R
E
E17
1
23
4
6
7
5
8
1214
15
1310
9 11
16
17
AVL
-
1
E E
E
E
E EE
E
LL L
LR
R
R
E
E17
1
23
4
6
7
5
8
1214
15
1310
9 11
16
16 116E 16R
AVL
-
1
E E
E
E
E EE
E
LL L
LR
R
R
R
E17
1
23
4
6
7
5
8
1214
15
1310
9 11
16
16 116E 16R
AVL
-
1
E E
E
E
E EE
E
LL L
LR
R
R
R
E17
1
23
4
6
7
5
8
1214
15
1310
9 11
16
15 115R 15
AVL
-
1
E E
E
E
E EE
E
LL L
LR
R E
E
E171
23
4
6
7
5
8
1214
1310
9 11
15
16
16R 1514
AVL
-
2
E E
E
E
E EE E
E
LL L
LR
R
R
1
23
4
6
7
5
8
1214
15
1310
9 11
16
16
AVL
-
2
E E
E
E
E EE
E
LL L
LR
R
R
E1
23
4
6
7
5
8
1214
15
1310
9 11
16
16
AVL
-
2
E E
E
E
E EE
E
LL L
LR
R
R
E1
23
4
6
7
5
8
1214
15
1310
9 11
16
16 15
AVL
-
2
E E
E
E
E EE
E
LL L
LR
R
R
E1
23
4
6
7
5
8
1214
15
1310
9 11
16
15 115R 15E
AVL
-
2
E E
E
E
E EE
E
LL L
LR
R
E
E1
23
4
6
7
5
8
1214
15
1310
9 11
16
15 115R 15E
AVL
-
2
E E
E
E
E EE
E
LL L
LR
R
E
E1
23
4
6
7
5
8
1214
15
1310
9 11
16
14 114 L 14
AVL
-
2
E E
E
E
E EE
E
LL L
LR
R
E
E1
23
4
6
7
5
8
1214
15
1310
9 11
16
15E 14
AVL
-
2
E
E
E
E
LL
R
R
E
EEEE
E
EE
1
23
4
6
7
5
8
1412
10
9 11 1513
15E 14
AVL
-
2
E
E
E
E
LL
R
R
E
EEEE
E
EE
1
23
4
6
7
5
8
1412
10
9 11 1513
18E 8 L
AVL
-
2
E
E
E
L
LL
R
R
E
EEEE
E
EE
1
23
4
6
7
5
8
1412
10
9 11 1513
18E 8 L
AVL
-
2
E
E
E
L
LL
R
R
E
EEEE
E
EE
1
23
4
6
7
5
8
1412
10
9 11 1513
8
AVL
-
InsertDelete
()
AVL
-
InsertDelete
()
AVL
-
E
1. v( 1)
1-1. v Lv vE
1-2. vEv vR
1-3. vRv
AVL
-
E
1. v( 1)
1-1. v Lv vE
1-2. vEv vR
1-3. vRv
AVL
-
E
1. v( 1)
1-1. v Lv vE
1-2. vEv vR
1-3. vRv
AVL
-
E
1. v( 1)
1-1. v Lv vE
1-2. vEv vR
1-3. vRv
AVL
-
()
1-3. vR1-3-1. v uR
v
vu
v R
R
uE
E
AVL
-
()
1-3. vR1-3-2. v u L
v
v
u v
w
u
R
L
R
L
w E
E
AVL
-
()
1-3. vR1-3-2. v u L
v
v
u v
w
u
R
L
L
L
w E
R
AVL
-
()
E
2. v( 1)
2-1. vRv vE
2-2. vEv v L
2-3. v Lv
AVL
-
()
E
2. v( 1)
2-1. vRv vE
2-2. vEv v L
2-3. v Lv
AVL
-
()
E
2. v( 1)
2-1. vRv vE
2-2. vEv v L
2-3. v Lv
AVL
-
()
E
2. v( 1)
2-1. vRv vE
2-2. vEv v L
2-3. v Lv
AVL
-
()
2-3. v L2-3-1. v u L
v
v
vu
L
L
u
E
E
AVL
-
()
2-3. v L2-3-2. v uR
v
wv
u u v
wL
L
R E
E
R
AVL
-
()
2-3. v L2-3-2. v uR
v
wv
u u v
wR
L
R L
E
E
AVL
-
1. v( 1)
1-1. v LvE
1-2. vEvR
1-3. vRv
AVL
-
1. v( 1)
1-1. v LvE
1-2. vEvR
1-3. vRv
AVL
-
1. v( 1)
1-1. v LvE
1-2. vEvR
1-3. vRv
AVL
-
1. v( 1)
1-1. v LvE
1-2. vEvR
1-3. vRv
AVL
-
()
1-3. vR1-3-1. v uE
v
vu
v R
E
u
L
R
AVL
-
()
1-3. vR1-3-2. v uR
v
vu
v R
R
uE
E
AVL
-
()
1-3. vR1-3-3. v u L
v
v
u v
w
u
R
L
L
E
w E
R
AVL
-
()
1-3. vR1-3-3. v u L
v
v
u v
w
u
R
L
E
w E
E
E
AVL
-
()
1-3. vR1-3-3. v u L
v
v
u v
w
u
R
L
L
w E
E
R
AVL
-
2. v( 1)
2-1. vRvE
2-2. vEv L
2-3. v Lv
AVL
-
2. v( 1)
2-1. vRvE
2-2. vEv L
2-3. v Lv
AVL
-
2. v( 1)
2-1. vRvE
2-2. vEv L
2-3. v Lv
AVL
-
2. v( 1)
2-1. vRvE
2-2. vEv L
2-3. v Lv
AVL
-
()
2-3. v L2-3-1. v uE
v
v
vu
L
E
u
L
R
AVL
-
()
2-3. v L2-3-2. v u L
v
v
vu
L
L
u
E
E
AVL
-
()
2-3. v L2-3-3. v uR
v
wv
u u v
wL
L
R E
E
R
AVL
-
()
2-3. v L2-3-3. v uR
v
wv
u u v
wE
L
R E
E
E
AVL
-
()
2-3. v L2-3-3. v uR
v
wv
u u v
wR
L
R L
E
E
AVL
-
AVL
MemberMin2O() = O(log n)InsertDeleteO() = O(log n)O() O() = O(log n)
AVL
-
AVL
MemberMin2O() = O(log n)InsertDeleteO() = O(log n)O() O() = O(log n)
AVL
-
AVLAVL = (log n)()
AVL
-
AVLAVL = (log n)()
AVL
-
AVLAVL = (log n)()
AVL
-
AVLAVL = (log n)()
AVL
-
AVLAVL = (log n)()
AVL
-
(1)
{1, 2, 3, 4} 2
2?AVL?AVL? 2AVLAVL?
AVL
-
(2)
22
AVL
-
(3)
1-3 2-3wE?
AVL
-
(4)
AVL (A)(F)AVL?
2
31
05
8
7
9
(A) 6 (B) 1 (C) 4(D) 3 (E) 8 (F) 1
AVL
-
(5)
AB x
a A b Ba < x < bAAVLBAVL hA, hBA {x} BAVL
O(1 + |hA hB |)?(hA hB)
AVL