Download presentation

Presentation is loading. Please wait.

Published byNathan Bradford Modified over 4 years ago

1
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 1 Joint work with: An Introduction to Parallel Computation and Ρ-Completeness Theory by Raymond Greenlaw Jim Hoover Computing Science Department University of Alberta Larry Ruzzo Department of Computer Science University of Washington

2
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 2 Outline Introduction Parallel Models of Computation Basic Complexity Evidence that NC Ρ Ρ-Complete Problems Comparator Circuit Class Open Problems

3
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 3 Outline Introduction Parallel Models of Computation Basic Complexity Evidence that NC Ρ Ρ-Complete Problems Comparator Circuit Class Open Problems

4
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 4 Introduction Sequential computation: Feasible ~ n O(1) time (polynomial time). Parallel computation: Feasible ~ n O(1) time and ~ n O(1) processors.

5
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 5 Introduction Goal of parallel computation: Develop extremely fast ((log n) O(1) ) time algorithms using a reasonable number of processors. Speedup equation: (best sequential time) (number of processors) Allow a polynomial number of processors. (parallel time).

6
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 6 Introduction Roughly, A problem is feasible if it can be solved by a parallel algorithm with worst case time and processor complexity n O(1). A problem is feasible highly parallel if it can be solved by an algorithm with worst case time complexity (log n) O(1) and processor complexity n O(1). A problem is inherently sequential if it is feasible but has no feasible highly parallel algorithm for its solution.

7
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 7 Outline Introduction Parallel Models of Computation Basic Complexity Evidence that NC Ρ Ρ-Complete Problems Comparator Circuit Class Open Problems

8
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 8 Parallel Models of Computation Parallel Random Access Machine Model Boolean Circuit Model Circuits and PRAMs

9
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 9 Parallel Random Access Machine RAM Processors Global Memory Cells P0P0 P1P1 P2P2 C0C0 C1C1 C2C2

10
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 10 Boolean Circuit Model 1 01 1 0 ANDOR AND OR NOT

11
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 11 Circuits and PRAMS Theorem: A function f from {0,1}* to {0,1}* can be computed by a logarithmic space uniform Boolean circuit family { n } with depth ( n ) є (logn) O(1) and size ( n ) є n O(1) if and only if f can be computed by a CREW-PRAM M on inputs of length n in time t(n) є (logn) O(1) using p(n) є n O(1).

12
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 12 Outline Introduction Parallel Models of Computation Basic Complexity Evidence that NC Ρ Ρ-Complete Problems Comparator Circuit Class Open Problems

13
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 13 Basic Complexity Decision, Function, and Search Problems Complexity Classes Reducibility Completeness

14
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 14 Decision, Function, and Search Problems 44 44 33366 Spanning Tree-D Given: An undirected graph G = (V,E) with weights from N labeling edges in E and a natural number k. Problem: Is there a spanning tree of G with cost less than or equal to k ? Spanning Tree-F Given: Same (no k ). Problem: Compute the weight of a minimum cost spanning tree. Spanning Tree-S Given: Same. Problem: Find a minimum cost spanning tree.

15
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 15 Complexity Classes Definitions: P is the set of all languages L that are decidable in sequential time n O(1). NC is the set of all languages L that are decidable in parallel time (logn) O(1) and processors n O(1). FP is the set of all functions from {0,1}* to {0,1}* that are computable in sequential time n O(1). FNC is the set of all functions from {0,1}* to {0,1}* that are computable in parallel time (logn) O(1) and processors n O(1). NC k, k 1, is the set of all languages L such that L is recognized by a uniform Boolean circuit family { n } with size( n ) = n O(1) and depth ( n ) = O((logn) k ).

16
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 16 Definitions: A language L is many-one reducible to a language L, written L P L, if there is a function f such that x є L if and only if f(x) є L. L is P many-one reducible to L, written L P L, if the function f is in FP. For k 1, L is NC k many-one reducible to L, written L NC k L, if the function f is in FNC k. L is NC many-one reducible to L, written L NC L, if the function f is in FNC. Many-One Reducibility m m m m

17
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 17 Many-One Reducibility m Lemma: The reducibilities m, P, NC k (k 1), and NC are transitive. –If L є P and L P L, L NC k L (k > 1), or L NC L then L є P. –If L є NC k (k > 1), and L NC k L then L є NC k. –If L є NC and L NC L then L є NC. m m m m m m m

18
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 18 Turing Reducibility Definition: L is NC Turing reducible to L, written L NC L, if and only if the L-oracle PRAM on inputs of length n uses time (logn) O(1) and processors n O(1). Lemma: The reducibility NC is transitive. –If L NC L then L NC L. –If L NC L and L є NC then L є NC. –If L NC L and L є P then L є P. T T m T T T

19
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 19 Completeness Definitions: A language L is P-hard under NC reducibility if L NC L for every L є P. A language L is P-complete under NC reducibility if L є P and L is P-hard. Theorem: If any P-complete problem is in NC then NC equals P. T

20
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 20 Outline Introduction Parallel Models of Computation Basic Complexity Evidence that NC Ρ Ρ-Complete Problems Comparator Circuit Class Open Problems

21
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 21 Evidence That NC P General Simulations Are Not Fast Fast Simulations Are Not General Natural Approaches Provably Fail

22
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 22 Evidence That NC P Gaps in Simulations Model Resource 1 Resource 2 Max R2 C NC Min R2 C P DTM DTM ATM PDA UC PRAM Time = n O(1) 2 Space = n O(1) 2 Space = n O(1) Size = n O(1) Procs = n O(1) Space Reversals log(Treesize) log(Time) Depth Time log n (log n) O(1) (log n) O(1) (log n) O(1) (log n) O(1) (log n) O(1) n O(1) n O(1) n O(1)

23
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 23 Outline Introduction Parallel Models of Computation Basic Complexity Evidence that NC Ρ Ρ-Complete Problems Comparator Circuit Class Open Problems

24
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 24 P-Complete Problems There are approximately 175 P-complete problems (500 with variations). Categories: –Circuit complexity –Graph theory –Searching graphs –Combinatorial optimization and flow –Local optimality –Logic –Formal languages –Algebra –Geometry –Real analysis –Games –Miscellaneous

25
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 25 Circuit Value Problem Given: An encoding of a Boolean circuit, inputs x 1,…,x n, and a designated output y. Problem: Is output y of TRUE on input x 1,…,x n ? Theorem: [Ladner 75] The Circuit Value Problem is P-complete under NC 1 reductions. m

26
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 26 P-Complete Variations of CVP –Topologically Ordered [Folklore] –Monotone [Goldschlager 77] –Alternating Monotone Fanin 2, Fanout 2 [Folklore] –NAND [Folklore] –Topologically Ordered NOR [Folklore] –Synchronous Alternating Monotone Fanout 2 CVP [Greenlaw, Hoover, and Ruzzo 87] –Planar [Goldschlager 77]

27
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 27 NAND Circuit Value Problem Given: An encoding of a Boolean circuit that consists solely of NAND gates, inputs x 1,…,x n, and a designated output y. Problem : Is output y of TRUE on input x 1,…,x n ? Theorem: The NAND Circuit Value Problem is P-complete.

28
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 28 NAND Circuit Value Problem Proof: Reduce AM2CVP to NAND CVP. Complement all inputs. Relabel all gates as NAND. 01001 1 1 OR AND OR 1 1 10110 1 1 NAND 1 0

29
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 29 Graph Theory –Lexicographically First Maximal Independent Set [Cook 85] –Lexicographically First ( + 1)-Vertex Coloring [Luby 84] –High Degree Subgraph [Anderson and Mayr 84] –Nearest Neighbor Traveling Salesman Heuristic [Kindervater, Lenstra, and Shmoys 89]

30
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 30 Lexicographically First Maximal Independent Set Theorem: [Cook 85] LFMIS is P-complete. Proof: Reduce TopNOR CVP to LFMIS. Add new vertex 0. Connect to all false inputs. 12345 NOR 9 8 7 6 11000 0 1 12345 67 0 8 9 0 0 X X X X

31
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 31 Searching Graphs –Lexicographically First Depth-First Search Ordering [Reif 85] –Stack Breadth-First Search [Greenlaw 92] –Breadth-Depth Search [Greenlaw 93]

32
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 32 Context-Free Grammar Empty Given: A context-free grammar G=(N,T,P,S). Problem: Is L(G) empty? Theorem: [Jones and Laaser 76], [Goldschlager 81], [Tompa 91] CFGempty is P-complete. Proof: Reduce Monotone CVP to CFGempty. Given construct G=(N,T,P,S) with N, T, S, and P as follows:

33
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 33 Context-Free Grammar Empty N = {i | v i is a vertex in } T = {a } S = n, where v n is the output of. P as follows: 1. For input v i, i a i f value of v i is 1, 2. i jk i f v i v j Λ v k, and 3. i j | k if v i v j V v k. Then the value of v i is 1 if and only if i, where є {a } +. *

34
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 34 CFGempty Example x 1 = 0, x 2 = 0, x 3 = 1, and x 4 = 1. G = (N, T, S, P), where N = {1, 2, 3, 4, 5, 6, 7 } T = {a } S = 7 P = {3 a, 4 a, 5 1 | 2, 6 34, 7 56} 1234 56 7 x3x3 x4x4 x1x1 x2x2 OR AND

35
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 35 Outline Introduction Parallel Models of Computation Basic Complexity Evidence that NC Ρ Ρ-Complete Problems Comparator Circuit Class Open Problems

36
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 36 Comparator Circuit Class Definition: [Cook 82] Comparator Circuit Value Problem (CCVP) Given: An encoding of a circuit composed of comparator gates, plus inputs x 1,…,x n, and a designated output y. Problem: Is output y of TRUE on input x 1,…,x n ? Definition: CC is the class of languages NC reducible to CCVP.

37
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 37 CC-Complete Problems –Lexicographically First Maximal Matching [Cook 82] –Telephone Connection [Ramachandran and Wang 91] –Stable Marriage [Mayr and Subramanian 92]

38
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 38 Outline Introduction Parallel Models of Computation Basic Complexity Evidence that NC Ρ Ρ-Complete Problems Comparator Circuit Class Open Problems

39
An Introduction to Parallel Computation and Ρ-Completeness Theory Ray Greenlaw, Jim Hoover, and Larry Ruzzo 39 Open Problems Find an NC algorithm or classify as P-complete. –Edge Ranking –Edge-Weighted Matching –Restricted Lexicographically First Maximal Independent Set –Integer Greatest Common Divisor –Modular Powering

Similar presentations

OK

Name Convolutional codes Tomashevich Victor. Name- 2 - Introduction Convolutional codes map information to code bits sequentially by convolving a sequence.

Name Convolutional codes Tomashevich Victor. Name- 2 - Introduction Convolutional codes map information to code bits sequentially by convolving a sequence.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on do's and don'ts of group discussion questions Ppt on hdfc bank Ppt on expansion of british rule in india Ppt on bond length of hbr Ppt on the road not taken theme Ppt on assembly line balancing Download ppt on types of houses Ppt on hard gelatin capsule manufacturers Ppt on supply chain management of a mule Free ppt on network topology