Proof of Optimality of
BTHn
Definitions and Notation
source, target, auxiliary
• D, Dn, P
• packet-move
• Small(n), Big(n)
• p.t.p
•
Lemma 1 - Consider any packet-move P of Dn,
which preserves the initial order between disks n
and n-1 and such that disk n never moves to
auxiliary. Then, P contains a distinguished move of
disk n.
Lemma 1 - Consider any packet-move P of Dn,
which preserves the initial order between disks n
and n-1 and such that disk n never moves to
auxiliary. Then, P contains a distinguished move of
disk n.
Lamma 2 - For any n>k+1 and any packet-move
P of Dn, which preserves the initial order between
disks n and n-1 and such that disk n never moves to
auxiliary, P contains four disjoint packet-moves of
Small(n-1).
Lamma 2 - For any n>k+1 and any packet-move
P of Dn, which preserves the initial order between
disks n and n-1 and such that disk n never moves to
auxiliary, P contains four disjoint packet-moves of
Small(n-1).
Lamma 3 - For any n>k, if a packet-move P of
Dn contains a move of disk n to auxiliary, then P
contains three disjoint packet-moves of Small(n).
Lamma 3 - For any n>k, if a packet-move P of
Dn contains a move of disk n to auxiliary, then P
contains three disjoint packet-moves of Small(n).
Lamma 4 - The length of any p.t.p packet-move
of D composed of 2l+1 packet-moves of D is at
least (2l+2)|D|-1
Evan
1
2
3
4
Odd
Odd
2l∙|D|+2∙(|D|-1) +1 =(2l+2)|D|-1
Lamma 4 - The length of any p.t.p packet-move
of D composed of 2l+1 packet-moves of D is at
least (2l+2)|D|-1
Lamma 5 - For any l ≥0 and n≥1, let P be a p.t.p
packet-move of Dn containing 2l+1 disjoint
packet-moves of Dn. Then
|P|≥ 2l∙bn +2∙bn-1+1= 2l∙bn+an
and this bound is tight.
Odd
Evan
Odd
Proof is by a complete induction
Basis: n≤k
Lamma 4 - The length of any p.t.p packet-move of D composed
of 2l+1 packet-moves of D is at least (2l+2)|D|-1
|P|≥(2l+2)n-1= 2ln+2(n-1)+1 = 2l∙bn+2∙bn-1+1
Induction step: we suppose that
the claim holds for all lesser
values of n and for all l
|P|=|(P|Small(n))|+|(P|Big(n))|
|(P|Big(n))|≥(2l+2)k-1
Case1
During P’, disk n never moves to auxiliary(P’)
Lamma 2 - For any n>k+1 and any packet-move P of Dn, which
preserves the initial order between disks n and n-1 and such that disk
n never moves to auxiliary, P contains four disjoint packet-moves of
Small(n-1).
|(P’|Small(n))|≥4∙bn-k-1+|(P’|n- =4∙bn-k-1+2 =2∙bn-1-2k+2
k)| -2k+2+4lb =2∙b -2k+2+2l(b -k)
|(P|
)|≥2∙b
Small(n)
n-1
n-k
n-1
=2l∙bn+2∙bn-1-(2l+2)k+2
|(P|Big(n))|≥(2l+2)k-1
|P|=|(P|Small(n))|+|(P|Big(n))|≥2l∙bn+2∙bn-1+1
n
Case2
During P’ contains a move of disk n to auxiliary(P’)
Lamma 3 - For any n>k, if a packet-move P of Dn contains a move of disk n
to auxiliary, then P contains three disjoint packet-moves of Small(n).
|(P’|Small(n))|≥3∙bn-k+4l∙bn-k≥(4l+2)bn-k+2∙bn-k-1+1≥
4l∙bn-k+4∙bn-k-1+3=2l∙bn+2∙bn-1-(2l+2)k+3
|(P|Big(n))|≥(2l+2)k-1
|P|=|(P|Small(n))|+|(P|Big(n))|≥2l∙bn+2∙bn-1+1
Lamma 5 - For any l≥0 and n≥1, let P be a p.t.p packet-move of Dn
containing 2l+1 disjoint packet-moves of Dn. Then
|P|≥ 2l∙bn +2∙bn-1+1= 2l∙bn+an
|BTHn|=an
|P|≥2l∙bn+an