Overview External"Memory Algorithms ParallelGraphAlgorithms - - PDF document

Parallelism andVLSIGroup Prof.Dr.JörgKeller DepartmentofMathematics andComputerScience

External"Parallel GraphAlgorithms

Course 01727ParallelProgramming

DepartmentofMathematics andComputerScience

Overview

External"Memory Algorithms ParallelGraphAlgorithms Application Algorithms Summary

• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

External-Memory Algorithms

External"memory alg =data set too largetoholdinmem Data onhard disk: bandwidth ok for few access patterns pagewise access latency high random access very slow Bestknown example: external sorting (mostly merge sort,mostly databases)

• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Parallel Graph Algorithms I

• Graphrepresentation:

" explicit by adjacency list " implicit by oracle

• Oracle:call black"box code with node xasparameter
• btain f(x)suchthat (x,f(x))is edge
• Advantageofexplicit representation:

graph can be changed,e.g.pointer jumping

• Advantageofimplicit representation:

able tohandlereally largegraphs without ext.memory

• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Parallel Graph Algorithms II

• Example:

given digraph Gwith outdegree 1by oracle compute strongly connected components (SCCs)

• Two variants:

" indegree alsoexactly 1(know function fis bijective) " indegree may vary (know nothing about function f)

• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Application I

• Consider finitestate machine accepting noinput

(except ininit phase)

• Stategraph hasoutdegree exactly 1
• Examples:

pseudo"random number generator stream cipher iterated blockcipher cryptographic hash chain

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Prof.Dr.J.Keller

DepartmentofMathematics andComputerScience

Application II

• Oracle=code ofstate"transition function
• Statespace typically large:264 atleast
• Strongly connected components =cycles

cycle lengths important parameter wrt security

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Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Application III

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Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Algorithms I

• Listranking asfirst excercise
• Given:Listoflength ninarray succ[1:n]

endoflistpoints toitself

• Wanted:for each listelement,distancetoendoflist
• For(t=1;t<logn;t++)

doinparallelfor i=1..n{ dist[i]+=dist[succ[i]];succ[i]=succ[succ[i]]; }

• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Algorithms II

• Pointerdoubling inexternal memory:
• Storelistofedges (i,succ[i])twice onexternal memory
• Sort one copy with i,one copy with succ[i]
• Foreach i:

findpredecessor jin2ndcopy,succ[j]=i(possibly mult.) findsuccessor succ[i]in1stcopy write (j,succ[i])todisc

• Iterate logntimes:O(n/p*(log(n))2)
• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Algorithms III

• FindSCCs ofdeg1"graphgiven by oracle f

if each SCCcontains anode with property A

• For(all nodes xwith property A){

follow path x,f(x),f(f(x))… till reach node ywith A write (x,y,distxy)todisk } do{ make copy ofedge listwith reversed edges; sort 1stcopy with x,sort 2ndcopy with y; for each (x,y)in1stcopy: write (x,y)todisk if (x,y‘)in2ndcopy }while(edges have been removed);

• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Algorithms IV

• Algorithm „shaves“ leaves from graph
• Mayiterate for long,expect distancetocycle O(sqrt(n))
• Therefore:use either if known that graph is shallow
• r use until size reduced that fits into main mem
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Prof.Dr.J.Keller

DepartmentofMathematics andComputerScience

Algorithms V

• Findmost cycles (=SCCs)for deg1"graphwith bij.oracle f
• Preprocessing:

define asmany random anchor nodes asfitonyour disks for(each anchor x){ follow path x,f(x),f(f(x)),… until reach other anchor y write todisk (x,y,distxy) }

• Converts graph with oracle into smaller graph with

adjacency list contains only cycles with ≥1anchor

• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Algorithms VI

• How todetect that node is anchor?

If assumed that deg1"graphis random: use deterministic anchors e.g.allwith 24zero bits If not:use additionally pseudorandom permutation pi where pi(x)andpi"1(x)can be computed fast

• Why somany anchors that disks are needed?

Chanceofputting anchor onto small cycle increases Findmore cycles!

• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Algorithms VII

• Compute cycles inanchor graph inseveral rounds
• Each round:

load edges (x,y)from largest x,asfitinto main mem compact alledges with (y,…)alsoinmain mem

• Remove references tothose edges inedges ondisk
• Paralleltime:O((n/m)2/p)
• Parallelism andVLSIGroup

Prof.Dr.J.Keller DepartmentofMathematics andComputerScience

Summary

• Ext"mem algorithms gain importance:

largedata sets,complex memory hierarchies

• Many ext"mem algorithms use sorting
• Currently:some better algorithms for undirected graphs
• Algorithms very complex
• Parallelism andVLSIGroup

Prof.Dr.J.Keller