CS660: Splay Performance

CS660 Combinatorial Algorithms
Fall Semester, 1996
Splay Performance

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San Diego State University -- This page last updated Nov 14, 1996

Contents of Splay Performance

Zipf's Law

Splay Performance Slide # 1

Zipf's Law

Zipf noticed that in English the frequency of word usage follows:

where f_idenotes the frequency of the i^th most frequent word

Zipfian Probability Distribution:
Assume we have a list of n items:

The a's are ordered by frequency of access

Probability of accessing item is and

Then

But so we have

Splay Performance Slide # 2

Zipfian Probability Distribution Let for k = 1, 2, ..., n

where

P_k n = 2

k	1	2
P_k	0.6667	0.3333

n = 3

k	1	2	3
P_k	0.5455	0.2727	0.1818

n = 4

k	1	2	3	4
P_k	0.48	0.24	0.16	0.12

n = 5

k	1	2	3	4	5
P_k	0.438	0.219	0.146	0.109	0.0876

n = 6

k	1	2	3	4	5	6
P_k	0.408	0.204	0.136	0.102	0.0816	.0680

Splay Performance Slide # 3

How to Implement Zipf's Distribution
Let

Method 1
If <= rand() < then return k
Method 2
For sample of size N

let item a₁be included N * P₁ times
let item a₂be included N * P₂ times

Splay Performance Slide # 4

Exp (?) Distribution
let

Splay Performance Slide # 5

Move-to-root vs. Splay Training
Insert integers 1 - Tree Size in random order

Same random order used to build each tree

Create a Zipfian sample of size "Sample"
Access each sample item in a tree

Each tree gets different sample

Tree Size 800		Tree Size 1600
Sample 100		Sample 200
Splay	MtoR	Splay	MtoR
31	18	50	49
18	41	37	42
18	17	42	40
18	19	38	40
18	18	38	42
20	18	38	38
17	17	36	38
17	17	37	38
18	17	38	35
19	15	36	37
16	17	35	36
17	14	37	37
17	17	37	33
18	16	36	35
16	15	35	34
16	16	37	35
27	16	35	35
16	16	38	37
15	16	37	37
18	15	37	35

Splay Performance Slide # 6

Building Trees
Integers from 1 to N are inserted into a tree in random order

Each tree is created with the same random sequence

Times are in milliseconds

Size	BST	AVL	Splay	MoveToRoot
200	17	23	45	52
400	25	34	97	97
800	55	69	222	222
1600	116	146	499	490
3200	246	305	1136	1092
6400	531	650	2446	2406
12800	1155	1340	5330	5196
25600	2617	2835	11519	11246

Splay Performance Slide # 7

Accessing Nodes in a Tree
In a tree of size N, accessed N keys

Keys to access were selected according to a distribution of keys in tree

Keys access were in random order

In each tree used the same keys in the same order
Exp Distribution

Size	BST	AVL	Splay	MoveToRoot
200	9	5	9	8
400	22	11	18	16
800	30	25	35	32
1600	64	55	68	61
3200	168	122	135	122
6400	422	264	260	254
12800	707	489	512	467
25600	1555	1240	1029	979

Zipf Distribution

Size	BST	AVL	Splay	MoveToRoot
200	7	5	28	28
400	14	12	69	64
800	30	24	148	140
1600	69	60	326	318
3200	141	120	697	682
6400	344	265	1521	1482
12800	713	618	3341	3213
25600	1676	1251	7197	6972

CS660 Combinatorial Algorithms Fall Semester, 1996 Splay Performance

Contents of Splay Performance

Zipf's Law

CS660 Combinatorial Algorithms
Fall Semester, 1996
Splay Performance