1 / 39

Joint work with :

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

yagil
Download Presentation

Joint work with :

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. An Introduction toParallel ComputationandΡ-Completeness TheorybyRaymond Greenlaw Joint work with: Larry RuzzoDepartment of Computer ScienceUniversity of Washington Jim HooverComputing Science DepartmentUniversity of Alberta An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  4. Introduction • Sequential computation: Feasible ~ nO(1) time (polynomial time). • Parallel computation: Feasible ~ nO(1) time and~ nO(1) processors. An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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). An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 nO(1). • A problem is feasiblehighly parallelif it can be solved by an algorithm with worst case time complexity (log n)O(1) and processor complexity nO(1). • A problem is inherentlysequentialif it is feasible but has no feasible highly parallel algorithm for its solution. An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

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

  9. P0 C0 P1 C1 P2 C2 Parallel Random Access Machine RAM Processors Global Memory Cells An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  10. 0 1 1 1 0 AND OR OR AND OR NOT AND Boolean Circuit Model An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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) єnO(1)if and only iff 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 nO(1). An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

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

  14. Decision, Function, and Search Problems 4 4 Spanning Tree-DGiven: 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-FGiven: Same (no k ).Problem: Compute the weight of a minimum cost spanning tree.Spanning Tree-SGiven: Same.Problem: Find a minimum cost spanning tree. 3 6 3 6 3 4 4 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  15. Complexity Classes Definitions:P is the set of all languages L that are decidable in sequential time nO(1).NC is the set of all languages L that are decidable in parallel time (logn)O(1) and processors nO(1). FP is the set of all functions from {0,1}* to {0,1}* that are computable in sequential time nO(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 nO(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) = nO(1) and depth (n) = O((logn)k). An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  16. Many-One Reducibility Definitions:A language L is many-one reducibleto a language L’, written L PL’ , if there is a function f such that x є L if and only if f(x) є L’. L is P many-one reducibleto L’, written L  PL’ , if the function fis in FP.For k  1, L is NCk many-one reducibleto L’, written L  NC kL’, if the function fis in FNCk. L is NC many-one reducibleto L’, written L NCL’, if the function fis in FNC. m m m m An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  17. Many-One Reducibility Lemma:The reducibilities m,  P, NC k(k 1), and NC are transitive. • If L’ є P and LP L’, LNC kL’ (k >1), or L NCL’ then L є P. • If L’ є NC k (k >1), and LNC kL’ then L є NC k. • If L’ є NC and LNCL’ then L є NC . m m m m m m m m An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  18. Turing Reducibility Definition:L is NC Turing reducibleto L’, written LNCL’, if and only if the L’-oracle PRAM on inputs of length n uses time (logn)O(1) and processors nO(1). Lemma: The reducibility NC is transitive. • If LNCL’then LNCL’. • If LNCL’and L’ єNCthen L є NC. • If LNCL’and L’ єPthen L є P. T T m T T T An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  19. Completeness Definitions: A language L is P-hard under NC reducibility if L’ NCL 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  22. Evidence That NC P Gaps in Simulations An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  25. Circuit Value Problem Given: An encoding  of a Boolean circuit , inputs x1,…,xn, and a designated output y. Problem:Is output y of  TRUE on input x1,…,xn? Theorem: [Ladner 75]The Circuit Value Problem is P-complete under NC1 reductions. m An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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] An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 x1,…,xn, and a designated output y. Problem:Is output y of  TRUE on input x1,…,xn? Theorem:The NAND Circuit Value Problem is P-complete. An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

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

  30. 1 1 0 0 0 0 X 1 2 3 4 5 X X NOR NOR 1 2 3 4 5 6 7 0 6 7 X NOR 1 8 8 0 NOR 9 9 0 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. An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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] An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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: An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  33. Context-Free Grammar Empty N = {i | vi is a vertex in } T = {a} S = n, where vn is the output of . P as follows: 1. For input vi, i aif value of vi is 1, 2. i jkif vi vjΛ vk, and 3. i j | k if vi vj V vk . Then the value of vi is 1 if and only if i , where  є {a}+. * An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

  34. x1 x3 x2 x4 1 2 3 4 5 6 AND OR AND 7 CFGempty Example x1 = 0, x2 = 0, x3 = 1, and x4 = 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} An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 comparatorgates, plus inputs x1,…,xn, and a designated output y. Problem: Is output yof  TRUE on input x1,…,xn? Definition: CC is the class oflanguages NC reducible to CCVP. An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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] An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay 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 An Introduction to Parallel Computation and Ρ-Completeness TheoryRay Greenlaw, Jim Hoover, and Larry Ruzzo

More Related