Parallel Recursive State Compression for Free

PARALLEL RECURSIVE STATE COMPRESSION FOR FREE

ALFONS LAARMAN

JOINT WORK WITH:

MICHAEL WEBER

JACO VAN DE POL

12/7/2011

SPIN 2011

• Motivation

• Previous work

• Problem Description

• Background

• Tree Compression

• Contribution

• Concurrent Tree Compression

• An Incremental Algorithm

• Analysis of Compression Ratios

• Experiments

• Conclusion / Future Work

12/7/2011Parallel Recursive State Compression for Free 2

OVERVIEW

(Enumerative) reachability as a basis for many verification problems

S: open set (stack or queue)

DB: closed set

MOTIVATIONPREVIOUS WORK

12/7/2011 4

PREVIOUS WORKMOTIVATION

Parallel Recursive State Compression for Free

Worker 1 Worker 2

Worker 3Worker 4re

less h

Synchronous Random Polling

post()

[FMCAD 2010]

12/7/2011 5

PROPERTIES OF THE LOCKLESS HASH TABLEMOTIVATION

• Monotonically growing (no remove() operation)

• find_or_put(vector) : boolean

• Without resizing

• Avoid scalability problem

• Stable indices: find_or_put(vector) : (index, boolean)

• Fixed-sized integer vectors (limitation of LTSmin)

12/7/2011 6

MEMORY BECOMES THE PROBLEM (AGAIN)MOTIVATION

• The hash set has to store up to 10^9 state vectors of length up to 1 kbyte

• With 16 cores: 1GB/sec of new state vectors

• In asynchronous systems, transitions are typically local:

• States are very similar

Solution:

• Compress the DB

Cf. COLLAPSE in SPIN

• Exploit locality with incremental algorithms

++++++------+---------++-------+------------------+-+----++++++------+---------------------++--+++------------++-----------------++--------+--------++-----------+-----++--------------+--++-----------------+-+-------+--------+-++---------+-----+-++------------+--+-++---+-----------+-+-

• Motivation

• Previous work

• Background

• Contribution

• Experiments

OVERVIEW

12/7/2011 8

MEMORY-EFFICIENT STORAGEBACKGROUND

12/7/2011 9

TREE COMPRESSIONBACKGROUND

Store a table of vectors as a fixed-shaped tree of tables with tuples

12/7/2011 10

TREE COMPRESSIONBACKGROUND

12/7/2011 11

TREE COMPRESSION ALGORITHMBACKGROUND

returns (index, bool)

Used before in distributed model checking – Blom et al.

CONFLICTING REQUIREMENTSPROBLEM DESCRIPTION

Goal: scalable concurrent tree database

Unknown table sizes requires resizing breaks stable indices

• Motivation

• Previous work

• Background

• Contribution

• Experiments

OVERVIEW

INTRODUCE MAXIMAL SHARINGCONCURRENT TREE COMPRESSION

Idea: Merge all hash tables into a single non-resizing hash table.

This maintains an injection Tk : Natk Nat

T2(4,1) = 1

T6(3,5,5,4,1,3) = 7

T3(4,1,3) = 4

Idea: Merge all hash tables into a single non-resizing hash table.

Problem: T6 (….) = T2 (3,5)

Solution: add an isRoot tag to nodes

For correct boolean result

12/7/2011 17

THE ALGORITHMCONCURRENT TREE COMPRESSION

INCREMENTAL TREE INSERTIONCONCURRENT TREE COMPRESSION

Hash table

1 (random) memory access of k integers (k-1) (random)

accesses of two integers

Next problem:

12/7/2011 19

INCREMENTAL TREE INSERTIONCONCURRENT TREE COMPRESSION

Exploits locality (log2(k) accesses in many cases)

post( ) =

• Motivation

• Previous work

• Background

• Contribution

• Experiments

OVERVIEW

WORST AND BEST CASE INPUTSANALYSIS OF COMPRESSION RATIOS

Inputs Tuples in tree nodes Ratio

Worst case:

Best case:

12/7/2011 22

MEMORY USAGEEXPERIMENTS

[BEEM database]

12/7/2011 23

MEMORY USAGEEXPERIMENTS

• Always compression with tree

• In most cases, compression is

close to optimal (see median)

• Always higher ratios than

COLLAPSE

10 100 10000.01

SPIN COL-LAPSETree com-pressionoptimal 8/kmedian 9.6/k

state length (byte)

12/7/2011 24

RUNTIMES COMPARED TO HASH TABLEEXPERIMENTS

0.05 0.50 5.00 50.00 500.000.05

500.00Sequential runtime (sec)16 core runtime (sec)x = y

Hash table (sec)

• Performance close to HT

• Due to incremental algorithm

12/7/2011 25

SCALABILITY EXPERIMENTS (AGGREGATE SPEEDUPS)EXPERIMENTS

With paying careful attention to data layout, we designed a parallel on-

the-fly compression method, that features:

• Impressive compression ratios

• Good performance/scalability

• For the full BEEM benchmark database

CONCLUSION

FUTURE WORK

Support dynamic-sized vectors

Overcome hardware limitations (2 billion vectors due to CAS)

use compact hash table (Cleary)

Static reordering of vector slots to improve compression ratios

Also Multi-Core LTL Model Checking (ATVA 2011 / PDMC 14:00)

Implemented in LTSmin (open source): http://fmt.cs.utwente.nl/tools/ltsmin/

AN INTUITIONANALYSIS OF COMPRESSION RATIOS

Parallel Recursive State Compression for Free

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