Algorithm Engineering „Parallele Algorithmen“

Algorithm Engineering

„Parallele Algorithmen“

Stefan Edelkamp

Übersicht

 Parallele Externe Suche

 Parallele Verspätete Duplikatselimination

 Parallele Expansion

 Verteilte Sortierung

 Parallele Strukturierte Duplikatselimination

 Disjunkte Duplikatserkennungsbereiche

 ”Schlöser”

 Parallele Algorithmen

 Matrix-Multiplikation

 List Ranking

 Euler Tour

Verteilte Suche

 Distributed setting provides more space.

 Experiments show that internal time dominates I/O.

Exploiting Independence

 Since each state in a Bucket is independent of the other – they can be expanded in

parallel.

 Duplicates removal can be distributed on different

processors.

 Bulk (Streamed) transfers much better than single ones.

Distributed Queue for Parallel Best- First Search

P0

P1

P2

<15,34, 0, 100>

<g, h, start byte, size>

<15,34, 20, 100>

TOP

<15,34, 40, 100>

<15,34, 60, 100>

Multiple Processors - Multiple Disks Variant

Sorted

buffers w.r.t the hash val Sorted Files

P1 P2 P3 P4

Divide w.r.t the hash ranges Sorted

buffers from every

processor Sorted File

h₀ ….. h_k-1 h_k ….. h_l-1

Parallel External A*

Parallel External A*

Distributed Heuristic Evaluation

 Assume one child processor for each tile one master processor

B₃ B₁ B₂

B₈

B₄ B₅ B₆ B₇ B₉ B₁₀ B₁₁ B₁₂ B₁₃ B₁₄ B₁₅ B₀

B₃ B₁ B₂

B₈

B₄ B₅ B₆ B₇ B₉ B₁₀ B₁₁ B₁₂ B₁₃ B₁₄ B₁₅ B₀

Distributed Pattern Database Search

 Only pattern databases that include the client tile need to be loaded on the client

 Because multiple tiles in pattern, from birds eye PDB loaded multiple times

 In 15-Puzzle with corner and fringe PDB this saves RAM in the order of factor 2 on each machine, compared to loading all

 In 36-Puzzle with 6-tile pattern databases this saves RAM in the order of factor 6 on each machine, compared to loading all

 Extends to additive pattern databases

Distributed

Heuristic Evaluation

Same bottleneck in external-memory search

Bottleneck: Duplicate detection

 Duplicate paths cause parallelization overhead

A

C D

BB

C DDDD

Internal memory External memory vs.

fast slow

A

Disjoint duplicate-detection scopes

B₁

B₀ B₄

B₀ B₁ B₂ B₃

B₈

B₄ B₅ B₆ B₇ B₉ B₁₀ B₁₁ B₁₂ B₁₃ B₁₄ B₁₅ B₀ B₁

B₄

B₃ B₂

B₇

B₂

B₃ B₇

B₁₂ B₈

B₁₃ B₁₄ B₁₅ B₁₁ B₈

B₁₂ B₁₃ B₁₅ B₁₁ B₁₄

Finding disjoint duplicate-detection scopes

B₁

B₀ B₄

0 0 0 0

0

0 0 0 0

0 0 1

0 0 0 0

0 1 1

0 2

1

B₂

B₃ B₇

0 1 0

B₈

B₁₂ B₁₃ B₁₅ B₁₁ B₁₄

1

2 2

01 2

2 2

2 1 2

2

2 2

2

0 1

1 1

0

1 0

2

3 3

2 B₁

B₅ B₆ B₄ B₉

2

3

3

4 3

3

Implementation of Parallel SDD

 Hierarchical organization of hash tables

 One hash table for each abstract node

 Top-level hash func. = state-space projection func.

 Shared-memory management

 Minimum memory-allocation size m

 Memory wasted is bounded by O(m#processors)

 External-memory version

 I/O-efficient order of node expansions

 I/O-efficient replacement strategy

Benötigt nur ein Mutex

“Schloss”

B₃ B₁ B₂

B₈

B₄ B₅ B₆ B₇ B₉ B₁₀ B₁₁ B₁₂ B₁₃ B₁₄ B₁₅ B₀

Parallelle Matrix-

Multiplication

Parallele Matrix

Multiplication

Exklusives Schreiben

Parallele Kopien

Fazit Matrix

Multiplication

Paralleles List Ranking

List

Ranking

Erster Algorithmus

Prinzip

Komplexität

Verbesserungen

Strategie

Unabhängige Mengen

2-Färbung

Reduktion

Restauration

Beispiel

Variablen

Beispiel (ctd.)

Pseudo Code

Nächster Schritt

Analyse

Backup

Algo

Algo

Speicher

Analyse

Ausblick:

Randomisiert in O(n) whp?

Probleme mit DFS

Idee Euler Tour

Parallel DFS

DFS

Nummern

Allgemein

Allgemein

Allgemein

Beispiel

Ein Zyklus oder

mehrere?

Korrektheit

Korrektheit

Beispiel

Konstruktion Euler

Tour

Fazit Euler Touren

GPU Architektur

Effektivität

Hierarchischer Speicher

Hash-based Partitioning

BFS

Kernel Functions