Proof of Optimality Proof of Optimality of BTHof BTHnn

Definitions and Definitions and NotationNotation

• sourcesource, , targettarget, , auxiliaryauxiliary• D, DD, Dnn, P , P

• packet-movepacket-move• Small(Small(nn), Big(), Big(nn))• p.t.pp.t.p

Lemma 1 -Lemma 1 - Consider any packet-move Consider any packet-move P of DP of Dnn, which preserves the initial , which preserves the initial order between disks order between disks nn and and nn-1 and -1 and

such that disk such that disk nn never moves to never moves to auxiliaryauxiliary. Then, P contains a . Then, P contains a distinguished move of disk distinguished move of disk nn..

Lemma 1 -Lemma 1 - Consider any packet-move Consider any packet-move P of DP of Dnn, which preserves the initial , which preserves the initial order between disks order between disks nn and and nn-1 and -1 and

such that disk such that disk nn never moves to never moves to auxiliaryauxiliary. Then, P contains a . Then, P contains a distinguished move of disk distinguished move of disk nn..

Lamma 2 -Lamma 2 - For any For any nn>>kk+1 and any +1 and any packet-move P of Dpacket-move P of Dnn, which preserves , which preserves the initial order between disks the initial order between disks nn and and nn-1 and such that disk -1 and such that disk nn never moves never moves to to auxiliaryauxiliary, P contains four disjoint , P contains four disjoint

packet-moves of Small(packet-moves of Small(nn-1).-1).

Lamma 2 -Lamma 2 - For any For any nn>>kk+1 and any +1 and any packet-move P of Dpacket-move P of Dnn, which preserves , which preserves the initial order between disks the initial order between disks nn and and nn-1 and such that disk -1 and such that disk nn never moves never moves to to auxiliaryauxiliary, P contains four disjoint , P contains four disjoint

packet-moves of Small(packet-moves of Small(nn-1).-1).

Lamma 3 -Lamma 3 - For any For any nn>>kk, if a packet-, if a packet-move P of Dmove P of Dnn contains a move of disk contains a move of disk nn

to auxiliary, then P contains three to auxiliary, then P contains three disjoint packet-moves of Small(disjoint packet-moves of Small(nn).).

Lamma 3 -Lamma 3 - For any For any nn>>kk, if a packet-, if a packet-move P of Dmove P of Dnn contains a move of disk contains a move of disk nn

to auxiliary, then P contains three to auxiliary, then P contains three disjoint packet-moves of Small(disjoint packet-moves of Small(nn).).

Lamma 4 -Lamma 4 - The length of any p.t.p The length of any p.t.p packet-move of D composed of 2packet-move of D composed of 2ll+1 +1

packet-moves of D is at least (2packet-moves of D is at least (2ll+2)|D|-+2)|D|-11

Evan Odd1

2

3

4

Odd

2l∙|D| =(2l+2)|D|-1+2∙(|D|-1) +1

Lamma 4 -Lamma 4 - The length of any p.t.p The length of any p.t.p packet-move of D composed of 2packet-move of D composed of 2ll+1 +1

packet-moves of D is at least (2packet-moves of D is at least (2ll+2)|D|-+2)|D|-11

Lamma 5 - Lamma 5 - For any For any l l ≥0 and ≥0 and nn≥1, let ≥1, let P be a p.t.p packet-move of DP be a p.t.p packet-move of Dnn

containing 2containing 2ll+1 disjoint packet-moves +1 disjoint packet-moves of Dof Dnn. Then . Then

|P|≥ 2|P|≥ 2ll∙b∙bnn +2∙b +2∙bnn-1-1+1= 2+1= 2ll∙b∙bnn+a+ann

and this bound is tight.and this bound is tight.

Evan Odd

Odd

Proof is by a complete induction

Basis: n≤kBasis: n≤k

Lamma 4 -Lamma 4 - The length of any p.t.p packet-move of The length of any p.t.p packet-move of D composed of 2D composed of 2ll+1 packet-moves of D is at least +1 packet-moves of D is at least

(2(2ll+2)|D|-1+2)|D|-1

|P|≥(2l+2)n-1 = 2l∙bn+2∙bn-

1+1= 2ln+2(n-1)+1

Induction step: we Induction step: we suppose that the claim suppose that the claim

holds for all lesser values holds for all lesser values of of nn and for all and for all ll

|P|=|(P|Small(n))|+|(P|Big(n))|

|(P|Big(n))|≥(2l+2)k-1

Case1Case1During P’, disk During P’, disk nn never moves to never moves to

auxiliaryauxiliary(P’)(P’)Lamma 2 -Lamma 2 - For any For any nn>>kk+1 and any packet-move P +1 and any packet-move P

of Dof Dnn, which preserves the initial order between , which preserves the initial order between disks disks nn and and nn-1 and such that disk n never moves to -1 and such that disk n never moves to auxiliaryauxiliary, P contains four disjoint packet-moves of , P contains four disjoint packet-moves of

Small(Small(nn-1).-1).=2∙bn-1-2k+2|(P|Small(n))|≥2∙bn-1-

2k+2+4lbn-k

|(P|Big(n))|≥(2l+2)k-1

|P|=|(P|Small(n))|+|(P|Big(n))|≥2l∙bn+2∙bn-1+1

|(P’|Small(n))|≥4∙bn-k-1+|(P’|

n-k)|=4∙bn-k-

1+2

=2l∙bn+2∙bn-1-(2l+2)k+2

=2∙bn-1-2k+2+2l(bn-k)

Case2Case2During P’ contains a move of disk During P’ contains a move of disk nn to to

auxiliaryauxiliary(P’)(P’)

|(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

Lamma 3 -Lamma 3 - For any For any nn>>kk, if a packet-move P of D, if a packet-move P of Dnn contains contains a move of disk a move of disk nn to auxiliary, then P contains three disjoint to auxiliary, then P contains three disjoint

packet-moves of Small(packet-moves of Small(nn).).

|(P|Big(n))|≥(2l+2)k-1

|P|=|(P|Small(n))|+|(P|Big(n))|≥2l∙bn+2∙bn-1+1

Lamma 5 - Lamma 5 - For any For any ll≥0 and ≥0 and nn≥1, let P be a p.t.p ≥1, let P be a p.t.p packet-move of Dpacket-move of Dnn containing 2 containing 2ll+1 disjoint packet-+1 disjoint packet-

moves of Dmoves of Dnn. Then . Then

|P|≥ 2|P|≥ 2ll∙b∙bnn +2∙b +2∙bn-1n-1+1= 2+1= 2ll∙b∙bnn+a+ann

|BTHn|=an

|P|≥2l∙bn+an