C-Perfect Hashing Schemes for Arrays, with Applications to Parallel Memories G. Cordasco 1, A. Negro 1, A. L. Rosenberg 2 and V. Scarano 1 1 Dipartimento

c-Perfect Hashing Schemes for Arrays, with Applications to

Parallel MemoriesG. Cordasco1, A. Negro1, A. L. Rosenberg2 and V. Scarano1

1 Dipartimento di Informatica ed Applicazioni ”R.M. Capocelli” Università di Salerno, 84081, Baronissi (SA) – Italy

2 Dept.of Computer Science University of Massachusetts at Amherst

Amherst, MA 01003, USA

Workshop on Distributed Data and Structures 2003

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Summary

The Problem The Motivation and some examples Our results Conclusion


3

The problemMapping nodes of a data structure on a parallel memory system in such a way that data can be efficiently accessed by templates.

Data structures (D): array, trees, hypercubes… Parallel memory systems (PM):

PP-1

P0

P1

M0

M1

MM -2

MM-1

PM

D


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The problem(2)Mapping nodes of a data structure on a parallel memory system in such a way that data can be efficiently accessed by templates.

Templates: distinct sets of nodes (for arrays: rows, columns, diagonals, submatrix; for trees: subtrees, simple paths, levels; etc).

Formally: Let GD be the graph that describes the data structure D. Atemplate t, for D is defined such as a subgraph of GD andeach subgraph of GD isomorphous to t will be called instanceof t-template.

Efficiently: few(or no) conflicts (i.e. few processors need to access the same memory module at the same time).


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How a mapping algorithm should be:

Efficient: the number of conflicts for each instance of the considered template type should be minimized. Versatile: should allow efficient data access by an algorithm that uses different templates. Balancing Memory Load: it should balance the number of data items stored in each memory module. Allow for Fast Memory Address Retrieval: algorithms for retrieving the memory module where a given data item is stored should be simple and fast. Use memory efficiently: for fixed size templates, it should use the same number of memory modules as the size.


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Why the versatility?

Multi-programmed parallel machines: different set of processors run different programs and access different templates. Algorithms using different templates: in manipulating arrays, for example,accessing lines (i.e. rows and columns) is common. Composite templates: some templates are “composite”, e.g. the Range-query template consisting of a path with complete subtrees attached.

A versatile algorithm is likely to perform well on that...


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Previous results: Array

Research in this field originated with strategies for mapping two-dimensional arrays into parallel memories.

Euler latin square (1778): conflict-free (CF) access to lines (rows and columns). Budnik, Kuck (IEEE TC 1971): skewing distance

CF access to lines and some subblocks. Lawrie (IEEE TC 1975): skewing distance

CF access to lines and main diagonals. requires 2N modules for N data items.


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Previous results: Array(2)

Colbourn, Heinrich (JPDC 1992): latin squares CF access for arbitrary subarrays: for r > s r × s and s × r subarrays. Lower Bound: for any CF mapping algorithm for arrays, where templates are r × s and s × r subarrays ( r > s ) and lines requires n > rs + s memory modules. Corollary: more than one-seventh of the memory modules are idle when any mapping is used that is CF for lines, r × s and s × r subarrays ( r > s ).


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Kim, Prasanna Kumar (IEEE TPDS 93): latin squares CF access for lines, and main squares (i.e.

where the top-left item ( i, j ) is such that i=0 mod and

j=0 mod ). Perfect Latin squares: main diagonals are also CF. Every subarray has at most 4 conflicts (intersects at most 4 main squares)


N N

N

N

0 1 2 3

2 3 0 1

3 2 1 0

1 0 3 2

N N


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Das, Sarkar (SPDP 94): quasi-groups (or groupoids). Fan, Gupta, Liu (IPPS 94): latin cubes.

These source offer strategies that scale with the number of memory module so that the number of available modules changes with the problem instance.



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We study templates that can be viewed as generalizing array-blocks and “paths” originating from the origin vertex <0,0>:

Chaotic array (C): A (two-dimensional) chaotic array C is an undirected graph whose vertex-set VC is a subset of N × N that is

order-closed, in the sense that for each vertex <r, s> ≠ <0, 0> of C, the set

Our results: Templates

1, , , 1cV r s r s


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Ragged array (R): A (two-dimensional) ragged array R is a chaotic array whose vertex-set VR

satisfies the following:

if <v1 , v2 > VR , then {v1} × [v2] VR;

if <v1 , 0 > VR , then [v1] × {0} VR;

Motivation pixel maps; lists of names where the table change shape dynamically; relational table in relational database.

For each n N, [n] denotes the set {0, 1, …, n -1}

Our results: Templates(2)


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Rectangular array (S): A (two-dimensional) rectangular array S is a ragged array whose vertex-set has the form [a] × [b] for some a, b N.

Our results: Templates(3)


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Our results: Templates Chaotic Arrays

Ragged Arrays

Rectangular Arrays

Rectangualar

RaggedChaotic

Arrays


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For any integer c > 0 , a c-contraction of an array A is a graph G that is obtained from A as follows.

Rename A as G(0). Set k = 0. Pick a set S of c vertices of G(k) that were vertices of A. Replace these vertices by a single vertex vS; replace all edges of G(k) that are incident to vertices of S by edges that are incident to vS. The graph so obtained is G(k+1)

Iterate step 2 some number of times; G is the final G(k).

Our results: c-contraction

G(1)A=G(0) G(2)=G


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Our results are achieved by proving a (more general) result, of independent interest, about the sizes of graphs that are “almost” perfect universal for arrays.

A graph Gc= (Vc, Ec) is c-perfect-universal for the family An if for each array A An exists a c-contraction of A that is a labeled-subgraph of Gc.

An denotes the family of arrays having n or fewer vertices.

Our results: some definitions


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The c-perfection number for An, denoted Perfc(An), is the size of the smallest graph that is c-perfect-universal for the family An

Theorem (F.R.K. Chung, A.L.Rosenberg, L.Snyder 1983)

Perf1(Cn) =

Perf1(Rn) =

Perf1(Sn) = n

Our results: c-perfection number

12 2

n n 1 2

1 32 3 3

n nn


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Theorem

For all integers c and n, letting X ambiguously denote Cn and Rn,

we have Perfc(X) .

Theorem

For all integers c and n, Perfc(Cn) .

Theorem

For all integers c and n, Perfc(Rn) .

Theorem

For all integers c and n, Perfc(Sn) = .

Our results

21n

c

11

2 2( 1)

n n

c c

.

1

1

n nn

c c

n

c


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Perfc(Cn) Perfc(Rn)

Our results:A Lower Bound on Perfc(Cn) and Perfc(Rn) 1 1

( ) 12 2( 1)n

n nSize Uc c c

.

0,2

n

)1(2

,0c

n

0,0

Un


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Perfc(Cn)

d = (n+1)/c; Vd = { v = <v1, v2> | v1 < d and v2 < d }.

For each vertex v = <v1, v2> :

if v Vd then color(v) = v1 d + v2;

if v Vd then color(v) = color(<v1 mod d, v2 mod d>);

Our results: An Upper Bound on Perfc(Cn)

21n

c

Vd

0,0

d = 3


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Perfc(Cn) Size(Vd)=

Our results: An Upper Bound on Perfc(Cn) 2

1n

c

n = 13

c = 5

d = (n+1)/c = 3

Vd

0,0


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Conclusions

A mapping strategy for versatile templates with a given “conflict tolerance” c.

c-Perfect Hashing Schemes for Binary Trees, with Applications to Parallel Memories (accepted for EuroPar 2003):

Future Work: Matching Lower and Upper Bound for Binary Trees.

( 1) /( 1) (1 1/ ) ( 1) /( 1)2 ( ) 2n c c n cc nPerf T c

Documents

C-Perfect Hashing Schemes for Arrays, with Applications to Parallel Memories G. Cordasco 1, A. Negro 1, A. L. Rosenberg 2 and V. Scarano 1 1 Dipartimento