1. SIMD and Associative Computing Computational Models and Algorithms

2. 2 Associative Computing Topics Introduction –References for Associative Computing –Motivation for the MASC model –The MASC and ASC Models –A Language Designed for the ASC Model –Two ASC Algorithms and Programs ASC and MASC Algorithm Examples –ASC version of Prim’s MST Algorithm –ASC version of QUICKHULL –MASC version of QUICKHULL.

3. 3 Associative Computing References Note: Below KSU papers are available on the website: http://www.cs.kent.edu/~parallel/ (Click on the link to “papers” in left column) 1.Maher Atwah, Johnnie Baker, and Selim Akl, An Associative Implementation of Classical Convex Hull Algorithms, Proc of the IASTED International Conference on Parallel and Distributed Computing and Systems, 1996, 435-438 2.Johnnie Baker and Mingxian Jin, Simulation of Enhanced Meshes with MASC, a MSIMD Model, Proc. of the Eleventh IASTED International Conference on Parallel and Distributed Computing and Systems, Nov. 1999, 511-516.

4. 4 Associative Computing References 3.Mingxian Jin, Johnnie Baker, and Kenneth Batcher, Timings for Associative Operations on the MASC Model, Proc. of the 15th International Parallel and Distributed Processing Symposium, (Workshop on Massively Parallel Processing, San Francisco, April 2001. 4.Jerry Potter, Johnnie Baker, Stephen Scott, Arvind Bansal, Chokchai Leangsuksun, and Chandra Asthagiri, An Associative Computing Paradigm, Special Issue on Associative Processing, IEEE Computer, 27(11):19-25, Nov. 1994. (Note: MASC is called ‘ASC’ in this article.) 5.Jerry Potter, Associative Computing - A Programming Paradigm for Massively Parallel Computers, Plenum Publishing Company, 1992.

5. 5 Associative Computers Associative Computer: A SIMD computer with a few additional features supported in hardware. These additional features can be supported (less efficiently) in traditional SIMDs in software. The name “associative” is due to its ability to locate items in the memory of PEs by content rather than location.

6. 6 Associative Models The ASC model (for ASsociative Computing) gives a list of the properties assumed for an associative computer. The MASC (for Multiple ASC) Model Supports multiple SIMD (or MSIMD) computation. Allows model to have more than one Instruction Stream (IS) –The IS corresponds to the control unit of a SIMD. ASC is the MASC model with only one IS. –The one IS version of the MASC model is sufficiently important to have its own name.

7. 7 ASC & MASC are KSU Models Several professors and their graduate students at Kent State University have worked on models The STARAN and the ASPRO fully support the ASC model in hardware. The MPP supports ASC, partly in hardware and partly in software. –Prof. Batcher was chief architect or consultant Dr. Potter developed a language for ASC Dr. Baker works on algorithms for models, applications, and architectures to support models Dr. Walker has students who worked on a hardware design to support the ASC and MASC models. Dr. Batcher and Dr. Potter are currently not actively working on ASC/MASC models but still provide advice.

8. 8 Motivation The STARAN Computer (Goodyear Aerospace, early 1970’s) and later the ASPRO provided an architectural model for associative computing embodied in the ASC model. ASC extends the data parallel programming style to a complete computational model. ASC provides a practical model that supports massive parallelism. MASC provides a hybrid data-parallel, control parallel model that supports associative programming. Descriptions of these models allow them to be compared to other parallel models

9. 9 The ASC Model IS C E L L N E T W O R K PEMemory Cells    PEMemory PEMemory

10. 10 Basic Properties of ASC Instruction Stream –The IS has a copy of the program and can broadcast instructions to cells in unit time Cell Properties –Each cell consists of a PE and its local memory –All cells listen to the IS –A cell can be active, inactive, or idle Inactive cells listen but do not execute IS commands until reactivated Idle cells contain no essential data and are available for reassignment Active cells execute IS commands synchronously

11. 11 Basic Properties of ASC Responder Processing –The IS can detect if a data test is satisfied by any of its responder cells in constant time (i.e., any-responders property). –The IS can select an arbitrary responder in constant time (i.e., pick-one property).

12. 12 Constant Time Global Operations (on a parallel variable with a value in each PE) –Logical OR and AND of binary values –Maximum and minimum of numbers –Associative searches Communications –There are two networks PE communications (or cell) network IS broadcast/reduction network (which could be implemented as two separate networks) Basic Properties of ASC

13. 13 Basic Properties of ASC –The PE communications network is normally supported by an interconnection network E.g., a 2D mesh –The broadcast/reduction network(s) are normally supported by two networks (broadcast & reduction) but sometimes combined. See posted paper by Jin, Baker, & Batcher (listed in associative references) Control Features –PEs and the IS and the networks all operate synchronously, using the same clock

14. 14 Non-SIMD Properties of ASC Observation: The ASC properties that are unusual for SIMDs are the constant time operations: –Constant time responder processing Any-responders? Pick-one –Constant time global operations Logical OR and AND of binary values Maximum and minimum value of numbers Associative Searches Typically, SIMDs cannot perform any of these operations in constant time – but have efficient executions of them. These timings are justified by implementations using a resolver in the paper by Jin, Baker, & Batcher (listed in associative references and posted).

15. 15 1 Busy- idle Dodge Ford Make Subaru Color PE1 PE2 PE3 PE4 PE5 PE6 PE7 red blue white red Year 1994 1996 1998 1997 Model Price On lot 1 1 0 0 0 0 1 0 1 1 0 0 1 IS Typical Data Structure for ASC Model Make, Color – etc. are fields the programmer establishes Various data types are supported. Some examples will show string data, but they are not supported in the ASC simulator.

16. 16 Dodge Ford Make Subaru Color PE1 PE2 PE3 PE4 PE5 PE6 PE7 red blue white red Year 1994 1996 1998 1997 Model Price On lot 1 1 0 0 0 0 1 Busy- idle 1 0 1 1 0 0 1 IS The Associative Search IS asks for all cars that are red and on the lot. PE1 and PE7 respond by setting a mask bit in their PE.

17. 17 Characteristics of Associative Programming Consistent use of style of programming called data parallel programming Consistent use of global associative searching and responder processing Usually, frequent use of the constant time global reduction operations: AND, OR, MAX, MIN Broadcast of data using IS bus allows the use of the PE network to be restricted to parallel data movement.

18. 18 Associative Programming Characteristics Tabular representation of data – think 2D arrays Use of searching instead of sorting Use of searching instead of pointers Use of searching instead of the ordering provided by linked lists, stacks, queues Promotes an highly intuitive programming style that promotes high productivity Uses structure codes (i.e., numeric representation) to represent data structures such as trees, graphs, embedded lists, and matrices. Examples of the above are given in –Ref: Nov. 1994 IEEE Computer article. –Also, see “Associative Computing” book by Potter.

19. 19 Languages Designed for the ASC Professor Potter has created several languages for the ASC model. ASC is a C-like language designed for ASC model ACE is a higher level language than ASC that uses natural language syntax; e.g., plurals, pronouns. Anglish is an ACE variant that uses an English-like grammar (e.g., “their”, “its”) An OOPs version of ASC for the MASC was discussed (by Potter and his students), but never designed. Language References: –ASC Primer – Copy available on parallel lab website www.cs.kent.edu/~parallel/ www.cs.kent.edu/~parallel/ –“Associative Computing” book by Potter [11] – some features in this book were never fully implemented in ASC Compiler

20. 20 Algorithms and Programs Implemented in ASC A wide range of algorithms implemented in ASC without the use of the PE network: –Graph Algorithms minimal spanning tree shortest path connected components –Computational Geometry Algorithms convex hull algorithms (Jarvis March, Quickhull, Graham Scan, etc) Dynamic hull algorithms

21. 21 ASC Algorithms and Programs (not requiring PE network) –String Matching Algorithms all exact substring matches all exact matches with “don’t care” (i.e., wild card) characters. –Algorithms for NP-complete problems traveling salesperson 2-D knapsack. –Data Base Management Software associative data base relational data base

22. 22 ASC Algorithms and Programs (not requiring a PE network) –A Two Pass Compiler for ASC – not the one we will be using. This compiler uses ASC parallelism. first pass optimization phase –Two Rule-Based Inference Engines for AI An Expert System OPS-5 interpreter PPL (Parallel Production Language interpreter) –A Context Sensitive Language Interpreter (OPS-5 variables force context sensitivity) –An associative PROLOG interpreter

23. 23 Associative Algorithms & Programs (using a network) There are numerous associative programs that use a PE network; –2-D Knapsack ASC Algorithm using a 1-D mesh –Image processing algorithms using 1-D mesh –FFT (Fast Fourier Transform) using 1-D nearest neighbor & Flip networks –Matrix Multiplication using 1-D mesh –An Air Traffic Control Program (using Flip network connecting PEs to memory) Demonstrated using live data at Knoxville in mid 70’s. All but first were developed in assembler at Goodyear Aerospace

24. Two Algorithm Examples for ASC The remainder of the slides are optional They illustrate two optimal algorithms for ASC – described at the data structures level. –Minimal Spanning Tree Provides an ASC version of the sequential Prim’s algorithm (sometimes called Dijkstra-Prim algorithm) –Convex Hull Provides an ASC version of the Quickhull sequential algorithm The implementation of these algorithms are simpler than implementation of their sequential version. 24

25. 25 Example 1 - MST A graph has nodes labeled by some identifying letter or number and arcs which are directional and have weights associated with them. Such a graph could represent a map where the nodes are cities and the arc weights give the mileage between two cities. A B C D E 3 52 5 4

26. 26 The MST Problem The MST problem assumes the weights are positive, the graph is connected, and seeks to find the minimal spanning tree, – i.e. a subgraph that is a tree 1, that includes all nodes (i.e. it spans), and –where the sum of the weights on the arcs of the subgraph is the smallest possible weight (i.e. it is minimal). Why would an algorithm solving this problem be useful? Note: The solution may not be unique. 1 A tree is a set of points called vertices, pairs of distinct vertices called edges, such that (1) there is a sequence of edges called a path from any vertex to any other, and (2) there are no circuits, that is, no paths starting from a vertex and returning to the same vertex.

27. 27 ASC-MST Algorithm Preliminaries Next, a “data structure” level presentation of Prim’s algorithm for the MST is given. The data structure used is illustrated in the next two slides. –This example is from the Nov. 1994 IEEE Computer paper cited in the references. There are two types of variables for the ASC model, namely –the parallel variables (i.e., ones for the PEs) –the scalar variables (ie., the ones used by the IS). –Scalar variables are essentially global variables. Can replace each with a parallel variable with this scalar value stored in each entry.

28. 28 ASC-MST Algorithm Preliminaries (cont.) In order to distinguish between them here, the parallel variables names end with a “$” symbol. Each step in this algorithm takes constant time. One MST edge is selected during each pass through the loop in this algorithm. Since a spanning tree has n-1 edges, the running time of this algorithm is O(n) and its cost is O(n 2 ). –Definition of cost is (running time)  (number of processors) Since the sequential running time of the Prim MST algorithm is O(n 2 ) and is time optimal, this parallel implementation is cost optimal. –Cost & optimality will be covered in parallel algorithm performance evaluation chapter (See Ch 7 of Quinn)

29. 29 Graph used for Data Structure Figure 6 in [Potter, Baker, et. al.] a bc d e f 2 8 9 6 3 3 4 7 2

30. 30 Data Structure for MST Algorithm

31. 31 Algorithm: ASC-MST-PRIM(root) 1.Initialize candidates to “waiting” 2.If there are any finite values in root’s field, 3. set candidate$ to “yes” 4. set parent$ to root 5. set current_best$ to the values in root’s field 6. set root’s candidate field to “no” 7.Loop while some candidate$ contain “yes” 8. for them 9. restrict mask$ to mindex(current_best$) 10. set next_node to a node identified in the preceding step 11. set its candidate to “no” 12. if the value in their next_node’s field are less than current_best$, then 13. set current_best$ to value in next_node’s field 14. set parent$ to next_node 15. if candidate$ is “waiting” and the value in its next_node’s field is finite 16. set candidate$ to “yes” 17. set parent$ to next_node 18. set current_best to the values in next_node’s field

32. 32 Comments on ASC-MST Algorithm The three preceding slides are Figure 6 in [Potter, Baker, et.al.] IEEE Computer, Nov 1994]. Preceding slide gives a compact, data-structures level pseudo-code description for this algorithm –Pseudo-code illustrates Potter’s use of pronouns (e.g., them, its) and possessive nouns. –The mindex function returns the index of a processor holding the minimal value. –This MST pseudo-code is much shorter and simpler than data-structure level sequential MST pseudo- codes e.g., see one of Baase’s textbooks cited in references Algorithm given in Baase’s books is identical to this parallel algorithm, except for a sequential computer Next, a more detailed explanation of the algorithm in preceding slide will be given next.

33. 33 Tracing 1 st Pass of MST Algorithm on Figure 6 (Put below chart & Figure 6 on board)

34. 34 Algorithm: ASC-MST-PRIM Initially assign any node to root. All processors set –candidate$ to “wait” –current-best$ to  –the candidate field for the root node to “no” All processors whose distance d from their node to root node is finite do –Set their candidate$ field to “yes –Set their parent$ field to root. –Set current_best$ = d.

35. 35 Algorithm: ASC-MST-PRIM (cont. 2/3) While the candidate field of some processor is “yes”, –Restrict the active processors whose candidate field is “yes” and (for these processors) do Compute the minimum value x of current_best$. Restrict the active processors to those with current_best$ = x and do –pick an active processor, say node y. »Set the candidate$ value of node y to “no” –Set the scalar variable next-node to y.

36. 36 Algorithm: ASC-MST-PRIM (cont. 3/3) –If the value z in the next_node column of a processor is less than its current_best$ value, then »Set current_best$ to z. »Set parent$ to next_node –For all processors, if candidate$ is “waiting” and the distance of its node from next_node y is finite, then Set candidate$ to “yes” Set current_best$ to the distance of its node from y. Set parent$ to y

37. 37 Quickhull Algorithm for ASC Reference: –[Maher, Baker, Akl, “An Associative Implementation of Classical Convex Hull Algorithms” ] Review of Sequential Quickhull Algorithm –Suffices to find the upper convex hull of points that are on or above the line Select point h so that the area of triangle weh is maximal. Proceed recursively with the sets of points on or above the lines and. w e h

38. 38 Previous Illustration w e h

39. 39 Example for Data Structure p1, w p7 p2 P3, e p4 p5 P6, h

40. 40 Data Structure for Preceding Example

41. 41 Algorithms & Assumption Basic algorithms exist for the following problems in Euclidean geometry for plane: –Determine whether a third point lies on, above, or below the line determined by two other points. –Compute the area of a triangle determined by three points. Standard Assumption –Three arbitrary points do not all lie on the same line. Reference: Introduction to Algorithms by Cormen, Leisterson, Rivest, (& Stein), McGraw Hill, Chapter on Computational Geometry.

42. 42 ASC Quickhull Algorithm (Upper Convex Hull) ASC-Quickhull( planar-point-set ) 1.Initialize: ctr = 1, area$ = 0, hull$ = 0 2.Find the PE with the minimal x-coord$ and let w be its point$ a)Set its hull$ value to 1 3.Find the PE with the PE with maximal x-coord$ and let e be its point$ a)Set its hull$ to 1 4.All PEs set their left-pt to w and right-pt to e. 5.If the point$ for a PE lies above the line a)Then set its job$ value to 1 b)Else set its job$ value to 0

43. 43 ASC Quickhull Algorithm (cont) 6.Loop while parallel job$ contains a nonzero value a)The IS makes its active cell those with a maximal job$ value. b)Each (active) PE computes and stores the area of triangle (left-pt$, right-pt$, point$ ) in area$ c)Find the PE with the maximal area$ and let h be its point. Set its hull$ value to 1 d)Each PE whose point$ is above sets its job$ value to ++ctr sets its right-pt to h e)Each PE whose point$ is above sets its job$ to ++ctr sets its left-pt to h f) Each PE with job$ < ctr -2 sets its job$ value to 0

44. 44 Highest Job Order Assigned to Points Above Lines  1 2 6 7 3 5 4

45. 45 Order that Triangles are Computed  1 5 7 6 2 3 4

46. 46 Performance of ASC-Quickhull Average Case: Assume either of the following: –For some integer k>1, on average 1/k of the points above each line being processed are eliminated each round. For example, consider k = 3, as one of three different areas are eliminated each round –O(lg n) points are on the convex hull. For randomly generated points, the number of convex hull points is very close to lg(n) points.

47. 47 Performance of ASC-Quickhull (cont) Either of above assumptions imply the average running time is O(lg n). –For example, each pass through algorithm loop produces one convex hull point. The average cost is O(n lg n) Worst Case: Running time is O(n). Cost is O(n 2 ) Recall: The definition of cost is Cost = (running time)  (nr. of processors)

48. 48 Additional Comments on ASC Convex Hull Algorithm The full “convex hull” algorithm requires that an order (e.g., clockwise) list of convex hull points be returned. –Preceding algorithms for ASC and MASC can be extended to handle this. This detail is omitted here to keep the algorithms simpler. –More information can be found in the paper “An Associative Implementation of Classical Convex Hull Algorithms” by Atwah, Baker, and Akl and in Maher Atwah’s master’s thesis at KSU.