1. SIMD, Associative, and Multi-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”) 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.) –First reading assignment 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 and architectures to support models Dr. Walker is working with 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 (across PEs) –Logical OR and AND of binary values –Maximum and minimum of numbers –Associative searches Communications –There are at least two real or virtual 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 a broadcast and a reduction network (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 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 MASC Model Basic Components –An array of cells, each consisting of a PE and its local memory –A PE interconnection network between the cells –One or more Instruction Streams (ISs) –An IS network MASC is a MSIMD model that supports –both data and control parallelism –associative programming Memory PE Interconnection Network IS Network PE Instruc- tion Stream (IS) Instruc- tion Stream (IS) Instruc- tion Stream (IS)

18. 18 MASC Basic Properties Each cell can listen to only one IS Cells can switch ISs in unit time, based on the results of a data test. Each IS and the cells listening to it follow rules of the ASC model. Control Features: –The PEs, ISs, and networks all operate synchronously, using the same clock –Restricted job control parallelism is used to coordinate the interaction of the multiple ISs.

19. 19 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.

20. 20 Characteristics of Associative Programming 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. We’ll see examples of the above. –Ref: Nov. 1994 IEEE Computer article. –Also, see “Associative Computing” book by Potter.

21. 21 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

22. 22 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

23. 23 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

24. 24 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

25. 25 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

26. 26 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

27. 27 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.

28. 28 An Example DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 As we will see, the algorithm is simple. The ASC program is quite easy to write. A SISD solution is a bit messy because of the data structures needed to hold the data for the problem

29. 29 An Example – Step 0 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 We will maintain three sets of nodes whose membership will change during the run. The first, V1, will be nodes selected to be in the tree. The second, V2, will be candidates at the current step to be added to V1. The third, V3, will be nodes not considered yet.

30. 30 An Example – Step 0 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 V1 nodes will be in red with their selected edges being in red also. V2 nodes will be in light blue with their candidate edges in light blue also. V3 nodes and edges will remain white.

31. 31 An Example – Step 1 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 Select an arbitrary node to place in V1, say A. Put into V2, all nodes incident with A.

32. 32 An Example – Step 2 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 Choose the edge with the smallest weight and put its node, B, into V1. Mark that edge with red also. Retain the other edge-node combinations in the “to be considered” list.

33. 33 An Example – Step 3 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 Add all the nodes incident to B to the “to be considered list”. However, note that AG has weight 3 and BG has weight 6. So, there is no sense of including BG in the list.

34. 34 An Example – Step 4 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 Add the node with the smallest weight that is colored light blue and add it to V1. Note the nodes and edges in red are forming a subgraph which is a tree.

35. 35 An Example – Step 5 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 Update the candidate nodes and edges by including all that are incident to those that are in V1 and colored red.

36. 36 An Example – Step 6 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 Select I as its edge is minimal. Mark node and edge as red.

37. 37 An Example – Step 7 DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 Add the new candidate edges. Note that IF has weight 5 while AF has weight 7. Thus, we drop AF from consideration at this time.

38. 38 An Example – after several more passes we have … DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 Note that when CH is added, GH is dropped as CH has less weight. Also, BC is dropped for the same type of reasoning (i.e., it would form a back edge between two nodes already in the MST). When there are no more nodes to be considered, i.e. no more in V3, we obtain the final solution.

39. 39 An Example – the final solution DE HI FC G BA 8 6 5 3 3 2 2 2 1 6 1 4 2 47 The subgraph is clearly a tree – no cycles and connected. The tree spans – i.e. all nodes are included. While not obvious, it can be shown that this algorithm always produces a minimal spanning tree. The algorithm is known as Prim’s Algorithm for MST.

40. 40 The ASC Program vs a SISD solution in, say, C, C++, or Java First, think about how you would write the program in C or C++. The usual solution uses some way of maintaining the sets as lists using pointers or references. –See solutions to MST in Algorithms texts by Baase listed in the posted references. In ASC, pointers and references are not even supported as they are not needed and their use is likely to result in inefficient SIMD algorithms The implementation of MST in ASC, basically follows the outline that I provided to the problem, but first, we need to learn something about the language ASC. The ASC manual (or a pointer to it) will be posted on the course web site.

41. 41 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.

42. 42 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)

43. 43 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

44. 44 Data Structure for MST Algorithm

45. 45 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

46. 46 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.

47. 47 Tracing 1 st Pass of MST Algorithm on Figure 6 (Put below chart & Figure 6 on board)

48. 48 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.

49. 49 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.

50. 50 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

51. 51 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

52. 52 Previous Illustration w e h

53. 53 Example for Data Structure p1, w p7 p2 P3, e p4 p5 P6, h

54. 54 Data Structure for Preceding Example

55. 55 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

56. 56 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 e)Each PE whose point$ is above sets its job$ to ++ctr f) Each PE with job$ < ctr -2 sets its job$ value to 0.

57. 57 Highest Job Order Assigned to Triangles  1 2 6 7 3 5 4

58. 58 Order that Triangles are Found  1 5 7 6 2 3 4

59. 59 Performance of ASC-Quickhull (cont) 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 –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. 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 ) –Definition of cost is (running time)  (nr. of processors)

60. 60 Master/Slave IS Control Structure for MASC Model Instruction Streams –a manager forks and joins tasks manages the job pool & idle IS pool –≥2 worker ISs execute tasks A work pool of tasks –Data parallel tasks that are ready to be assigned to idle worker instruction streams

61. 61 Master/Slave IS Control Structure for the MASC Model The master IS is connected to each worker IS by an IS broadcast/reduction network. The master IS maintains a pool of unassigned jobs and a pool of idle ISs A job consists of a task to be performed and the idle PEs which will perform this job. When the job and IS pools are nonempty, the master IS will assign a job to an idle worker IS. An active IS will return any jobs it creates that need to be reassigned to the master IS.

62. 62 MASC Quickhull Algorithm MASC Modification of ASC Quickhull Algorithm: 1.Initially, the master IS executes the initialization phase of the ASC Quicksort, using all the PEs. Alternately, it could assign a worker IS to do this. 2.The master IS maintains the scalar variable “ctr”. Whenever the job pool and the IS pool are both nonempty, the master IS will assign a job to an IS. Additionally, it will give the IS its value for ctr and then will increment its own ctr variable by 2. 3.Each IS computes the steps in the loop in ASC- Quickhull, and returns the any jobs it creates to the master IS along with their job number. 4.The algorithm will terminate when there are no more jobs in the job pool and all ISs are idle.

63. 63 Approximate Order MASC Quickhull Processes Triangles (Assuming sufficient ISs)  0 1 2 2 1 2 2

64. 64 Analysis for MASC Quickhull Average Case: Assumptions: –The remaining unidentified hull points are roughly evenly distributed among the partitions in each recursive level. –O(lg n) Instruction Streams are available. –There are O(lg n) convex hull points The time for master IS to assign a task to an IS is a small constant and this time will be included with the time required to execute the task. The average running time is O(lg lg n) and the average running time is O(n lg lg n). O(lg lg n) increases so slowly that it is essentially a constant for practical values of n. Worst Case O(n) –Happens if all points are hull points & only one new job is created each time. –Also O(n) if preceding statement is roughly true.

65. 65 Comments on Previous Slide The splitting of jobs can be represented as a binary tree. Assume there are at most  lg n  convex hull points If at each hull recursive level, “the remaining unidentified hull points are roughly evenly distributed among the nodes” the binary tree is roughly complete (or full). To simplify calculations, we assume above binary tree is complete and let m be the number of leaves. – The height h of the binary tree is lg m. –Since there are m-l interior nodes, there are m-1 and n = 2m-1. –Since calculation at each node produces one convex hull point, 2m-1 =  lg n  and m = (  lg n  +1)/2. –If m ISs are available, all jobs at each recursive level of the tree can be executed simultaneous and running time is k = lg m  lg (lg n)/2 When the binary “jobs tree” is roughly complete and there are O(lg n) convex hull points, the running time for this algorithm O(lg lg n).

66. 66 Additional Comments on MASC Quickhull For one million randomly generated points, this algorithm would require a maximum of  lg n  = 20 ISs on any level. –Note that 33.5 million randomly generated points only requires 25 ISs at each level, or 5 more than required for 1 million Even if there  (lg n) ISs are available for this algorithm, there is likely to be occasions during execution when the IS pool is empty. However, this algorithm will provide a speedup, even if only a small number k of ISs are available. –The complexity of the running time will still be O(lg n). –The actual running time should be close to k times faster than for one IS. –There will be a small loss of efficiency due to IS interactions. –This algorithm works, whether or not sufficient ISs are available.

67. 67 Additional Comments on ASC and MASC Algorithms 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.

68. END OF CHAPTER

69. 69 Tracing MST Algorithm on Figure 6 The following slides are to be used to trace the first pass of the MST algorithm on Figure 6 Print several copies of next slide for each student prior to covering detailed MST algorithm. This will allow students to copy dynamic trace of algorithm during class.

70. 70 Tracing 1 st Pass of MST Algorithm on Figure 6