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Bottom-Up Splay TreesAnalysis

Actual and amortized complexity of join is

O(1).

Amortized complexity of search, insert, delete,and split is O(log n).

Actual complexity of each splay tree operation

is the same as that of the associated splay. Sufficient to show that the amortized

complexity of the splay operation is O(log n).

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Potential Function

size(x) = #nodes in subtree whose root is x. rank(x) = floor(log2 size(x)).

P(i) = Sx is a tree node rank(x).

P(i) is potential afterith operation.

size(x) and rank(x) are computed afterithoperation.

P(0) = 0.

When join and split operations are done,number of splay trees > 1 at times.

P(i) is obtained by summing over all nodes in

all trees.

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Example

size(x) is in red.

20

10

6

8

40

30

1

rank(x) is in blue.

Potential = 5.

2

3

1

2

6

0

1

1

0

1

2

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Root Rank

rank(root) = floor(log2 n).

20

10

6

8

40

30

1

2

3

1

2

6

0

1

1

0

1

2

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Insert

On every leaf to root path ranksare non-decreasing.

At most floor(log2 n)+1 different

ranks on such a path.

Insert may increase the rank of

only the last node in each sequence

of equal rank nodes.

When you insert, potentialincreases by up to floor(log2 n)+1. 0

1

1

4

4

4

7

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Join

Rank of root is floor(log2 n).

Other ranks unchanged.

Potential increases by floor(log2 n).

S

m

B

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Splay Step Amortized Cost

Ifq = null orq is the root, do nothing (splay

is over). DP = 0.

amortized cost = actual cost +DP

= 0.

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Splay Step Amortized Cost

Ifq is at level 2, do a one-level move andterminate the splay operation.

p

q

a b

c

b c

a

q

p

r(x) = rank ofxbefore splay step.

r(x) = rank ofx after splay step.

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Splay Step Amortized Cost

p

q

a b

c

b c

a

q

p

DP = r(p) + r(q) r(p)r(q)

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2-Level Move (case 1)

DP = r(gp) + r(p) + r(q) r(gp) r(p)r(q)

p

q

a b

c

gp

d

c d

b

p

gp

q

a

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2-Level Move (case 1)

r(q) = r(gp)

r(p)

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2-Level Move (case 1)

DP = r(gp) + r(p) + r(q) r(gp) r(p)r(q)

r(q) = r(gp)

r(gp)

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2-Level Move (case 1)

A more careful analysis reveals that

DP

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2-Level Move (case 1)

amortized cost = actual cost +DP

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2-Level Move (case 2)

Similar to Case 1.

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Splay Operation

When q != null and q is not the root, zero or

more 2-level splay steps followed by zero or

one 1-level splay step.

Let r(q) be rank ofq just after last 2-level

splay step.

Let r(q) be rank ofq just after1-level

splay step.

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Splay Operation

Amortized cost of all 2-level splay steps is

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Actual Cost Of Operation

Sequence

Actual cost of an n operation sequence

=O(actual cost of the associated n splays).

actual_cost_splay(i) = amortized_cost_splay(i)DP

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Actual Cost Of Operation

Sequence

actual_cost_splay(i) = amortized_cost_splay(i)DP