Using Access Probabilities In Address Lookups

Jim Washburn & Katerina Argyraki

EE384Y Class Project

Access probabilities are not uniform

• Lookup table contains set of keys Ki

• Access probabilities Pi

• Access times Ai

• Avg access time Ei = i { AiPi}

Key

Nexthop

Header

iP

Where is accounting info stored?

Key

Count

Nexthop

Header

Key2Count

Nexthop

HeaderKey2

• Per key

• Per nexthop

Lookup table operations

• add(key)• delete(key)• search(key)• compile( )

– When to compile? – What happens to searches while

compiling ? – How long to maintain accounting info?

Approach to the problem

• Amortize the optimization• Incrementally optimize at search

time• Perform optimization step with low

probability per access– keeps cost low!

Base structure: BST on intervals

• Can do prefix matches or exact matches • Allows many different tree configurations

for the same set of keys– can be dynamically reconfigurable

• Permits localized incremental operations– affect just two nodes

BST rotation operation

C

BA

y

x

z

CB

A

x

z

y

right

left

Only nodes x and y are written

Self-adjusting heuristics

• single rotation on every access– too conservative?

• move-to-root on every access– too aggressive?

• move-up on every access– “middle ground” solution– but how many nodes?

Low-cost heuristics

• single rotation with low probability p– decrease the amount of writes

• move-to-root with probability proportional to node depth– favor nodes that are deep down the tree

• move-up with counters– per node counter keeps # of hits– a node is never moved above another

node with higher counter

BST as Markov Chain

• Tree can be in any of its possible configurations, with probabilities given by the stationary distribution.

B

A

A

B

A

B

… A

B

Markov calculations (1)

Optimal

SingleRotatio

n

Move to

Root

Expected Access time

1.3283 1.4643 1.4633

Expected time to

first passage to

optimal

NA 59.9 39.7

Done with 60 randomly selected access distributions, for a 3 node treeProbability of optimization step set to 0.1

Markov Calculations (2)

Simulation results (1)

avg OPT SR MTRSRR

MTRR

MUP

access’s

4.56

10.35

9.38

7.62

5.1 5.87

reads4.56

9.214.82

7.51

4.95 5.7

writes 0 1.134.55

0.11

0.15 0.16

• Scenario 1: 99% hit on 7 rules – .5, .25, .12, .6, .3, .1

• Average performance (the good news):

Simulation results (2)

worstOPT

SRR

MTRR

MUP

access’s

12 57 42 35

reads 12 57 21 31

writes 0 2 21 14

• Scenario 1: 99% hit on 7 rules– .5, .25, .12, .6, .3, .1

• Worst-case performance (the bad news):

Simulation results (3)

avg OPT SRRMTR

RMUP

access’s

6.79

14.93

7.6 7.6

reads6.79

14.75

7.34 7.37

writes 0 0.17 0.26 0.23

• Scenario 2: 60% hit on 12 rules– .13, .09, .06, .05, .04, .03, .02, .01, .01, .

01

• Average performance:

Simulation results (4)

worstOPT

SRR

MTRR

MUP

access’s

10 74 42 29

reads 10 72 22 25

writes 0 2 21 14

• Scenario 2: 60% hit on 12 rules– .13, .09, .06, .05, .04, .03, .02, .01, .01, .

01

• Worst-case performance:

Simulation results (5)

avgOPT

SRRMTR

RMUP

access’s

722.8

59.32 9.61

reads 722.6

58.97 9.3

writes 0 0.2 0.34 0.31

• Scenario 3: uniform distribution– The worst-case for our heuristics!

• Average performance:

Simulation results (6)

worstOPT

SRR

MTRR

MUP

access’s

9 62 40 32

reads 9 61 22 22

writes 0 2 20 16

• Scenario 3: uniform distribution– The worst-case for our heuristics!

• Worst-case performance:

Bounding the tree depth

• Check before moving a node– If move will increase depth beyond max,

don’t do it

• How to check?– Keep extra info per node e.g., node depth– Must not introduce extra accesses for

maintenance!– Impossible with single rotation or move

up

MTRR with bounded worst-case

• Augmented BST– Each node knows

• the depth of its longest branch on the left• the depth of its longest branch on the right

– Rotating a node up, requires updating all its ancestors up to the root

– With move-to-root we do that anyway

Simulation results (1)

avg OPTMTR

RMTR

B

access’s

4.56

5.1 6.47

reads4.56

4.95 6.24

writes 0 0.15 0.23

• Scenario 1: 99% hit on 7 rules – .5, .25, ,12, .6, .3, .1

• Average performance:

Simulation results (2)

worstOPT

MTRR

MTRB

access’s

9 42 40

reads 9 22 20

writes 0 21 20

• Scenario 1: 99% hit on 7 rules – .5, .25, ,12, .6, .3, .1

• Worst-case performance:

Conclusions

• We can optimize without having to maintain per-key accounting state, or periodically recompiling the entire data structure.

• Average access time results comparable to compiled results, which are close to lower bound which is determined by entropy.

• Worst case results not as good, need more heuristics to bound the tree depth