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SIMD and Associative Computing Computational Models and Algorithms

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2 Associative Computing Topics Introduction –References –SIMD computing & Architecture –Motivation for the MASC model –The MASC and ASC Models –A Language Designed for the ASC Model –List of Algorithms and Programs designed for ASC An ASC Algorithm Examples –ASC version of Prim’s MST Algorithm

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Comment on Slides Included Some of these slides will be covered only lightly or else left for students to read. –The emphasis here is to provide an introduction to material covered, not a deep understanding. –Inclusion of these slides will provide a better survey of this material. –This material is a useful background for the Air Traffic Control example and projects we expect to use in this course. 3

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4 Associative Computing References Note: Below KSU papers are available on the website: (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, 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.

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5 Associative Computing References 3.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 (Note: MASC is called ‘ASC’ in this article.) 4.Jerry Potter, Associative Computing - A Programming Paradigm for Massively Parallel Computers, Plenum Publishing Company, 1992.

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SIMD slides from Chapter 2

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7 Alternate Names for SIMDs Recall that all active processors of a true SIMD computer must simultaneously access the same memory location. The value in the i-th processor can be viewed as the i-th component of a vector. SIMD machines are sometimes called vector computers [Jordan,et.al.] or processor arrays [Quinn 94,04] based on their ability to execute vector and matrix operations efficiently.

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8 SIMD Architecture Has only one control unit. Scientific applications have data parallelism

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9 Data/instruction Storage Front end computer –Also called the control unit –Holds and runs program –Data manipulated sequentially Processor array –Data manipulated in parallel

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10 Processor Array Performance Performance: work done per time unit Performance of processor array –Speed of processing elements –Utilization of processing elements

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11 Performance Example processors Each adds a pair of integers in 1 sec (1 microsecond or one millionth of second or second.) What is the performance when adding two 1024-element vectors (one per processor)?

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12 Performance Example processors Each adds two integers in 1 sec What is the performance when adding two vectors of length 600? Since 600 > 512, 88 processor must add two pairs of integers. The other 424 processors add only a single pair of integers.

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13 Example of a 2-D Processor Interconnection Network in a Processor Array Each VLSI chip has 16 processing elements. Each PE can simultaneously send a value to a neighbor. PE = processor element

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14 SIMD Execution Style The traditional (SIMD, vector, processor array) execution style ([Quinn 94, pg 62], [Quinn 2004, pgs 37-43]: –The sequential processor that broadcasts the commands to the rest of the processors is called the front end or control unit (or sometimes host). –The front end is a general purpose CPU that stores the program and the data that is not manipulated in parallel. –The front end normally executes the sequential portions of the program. –Each processing element has a local memory that can not be directly accessed by the control unit or other processing elements.

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15 SIMD Execution Style –Collectively, the individual memories of the processing elements (PEs) store the (vector) data that is processed in parallel. Called the parallel memory –When the front end encounters an instruction whose operand is a vector, it issues a command to the PEs to perform the instruction in parallel. –Although the PEs execute in parallel, some units can be allowed to skip any particular instruction.

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16 Masking in Processor Arrays All the processors work in lockstep except those that are masked out (by setting mask register). The conditional if-then-else is different for processor arrays than sequential version –Every active processor tests to see if its data meets the negation of the boolean condition. –If it does, it sets its mask bit so those processors will not participate in the operation initially. –Next the unmasked processors, execute the THEN part. –Afterwards, mask bits (for original set of active processors) are flipped and unmasked processors perform the ELSE part.

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17 if (COND) then A else B

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18 if (COND) then A else B

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19 if (COND) then A else B

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20 SIMD Machines An early SIMD computer designed for vector and matrix processing was the Illiac IV computer –Initial development at the University of Illinois –Moved to NASA Ames, completed in 1972 but not fully functional until –See Jordan et. al., pg 7 and Wikipedia The MPP, DAP, the Connection Machines CM-1 and CM-2, MasPar MP-1 and MP-2 are examples of SIMD computers –See Akl pg 8-12 and [Quinn, 94] The CRAY-1 and the Cyber-205 use pipelined arithmetic units to support vector operations and are sometimes called a pipelined SIMD –See [Jordan, et al, p7], [Quinn 94, pg 61-2], and [Quinn 2004, pg37).

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21 SIMD Machines Quinn [1994, pg 63-67] discusses the CM-2 Connection Machine (with 64K PEs) and a smaller & updated CM-200. Our Professor Batcher was the chief architect for the STARAN and the MPP (Massively Parallel Processor) and an advisor for the ASPRO –ASPRO is a small second generation STARAN used by the Navy in surveillance planes. Professor Batcher is best known architecturally for the MPP, which is at the Smithsonian Institute & currently displayed at a D.C. airport.

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22 Today’s SIMDs Many SIMDs are being embedded in sequential machines. Others are being build as part of hybrid architectures. Others are being build as special purpose machines, although some of them could classify as general purpose. Much of the recent work with SIMD architectures is proprietary. –Often the fact that a parallel computer is SIMD is not mentioned by company building them.

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23 ClearSpeed’s Inexpensive SIMD ClearSpeed is producing a COTS (commodity off the shelf) SIMD Board Not a traditional SIMD as the hardware doesn’t synchronize every step. –PEs are full CPUs – Hardware design supports efficient synchronization This machine is programmed like a SIMD. The U.S. Navy has observed that their machines process radar a magnitude faster than others. There is quite a bit of information about this at and

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24 Special Purpose SIMDs in the Bioinformatics Arena Parcel –Acquired by Celera Genomics in 2000 –Products include the sequence supercomputer GeneMatcher, which has a high throughput sequence analysis capability Supports over a million processors –GeneMatcher was used by Celera in their race with U.S. government to complete the description of the human genome sequencing TimeLogic, Inc –Has DeCypher, a reconfigurable SIMD

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25 Advantages of SIMDs Reference: [Roosta, pg 10] Less hardware than MIMDs as they have only one control unit. –Control units are complex. Less memory needed than MIMD –Only one copy of the instructions need to be stored –Allows more data to be stored in memory. Less startup time in communicating between PEs.

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26 Advantages of SIMDs (cont) Single instruction stream and synchronization of PEs make SIMD applications easier to program, understand, & debug. –Similar to sequential programming Control flow operations and scalar operations can be executed on the control unit while PEs are executing other instructions. MIMD architectures require explicit synchronization primitives, which create a substantial amount of additional overhead.

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27 Advantages of SIMDs (cont) During a communication operation between PEs, –PEs send data to a neighboring PE in parallel and in lock step –No need to create a header with routing information as “routing” is determined by program steps. –the entire communication operation is executed synchronously –SIMDs are deterministic & have much more predictable running time. Can normally compute a tight (worst case) upper bound for the time for communications operations. Less complex hardware in SIMD since no message decoder is needed in the PEs – MIMDs need a message decoder in each PE.

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28 SIMD Shortcomings (with some rebuttals) Claims are from our textbook [i.e., Quinn 2004]. –Similar statements are found in [Grama, et. al]. Claim 1: Not all problems are data-parallel –While true, most problems seem to have a data parallel solution. –In [Fox, et.al.], the observation was made in their study of large parallel applications at national labs, that most were data parallel by nature, but often had points where significant branching occurred.

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29 SIMD Shortcomings (with some rebuttals) Claim 2: Speed drops for conditionally executed branches –MIMDs processors can execute multiple branches concurrently. –For an if-then-else statement with execution times for the “then” and “else” parts being roughly equal, about ½ of the SIMD processors are idle during its execution With additional branching, the average number of inactive processors can become even higher. With SIMDs, only one of these branches can be executed at a time. This reason justifies the study of multiple SIMDs (or MSIMDs).

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30 SIMD Shortcomings (with some rebuttals) Claim 2 (cont): Speed drops for conditionally executed code –In [Fox, et.al.], the observation was made that for the real applications surveyed, the MAXIMUM number of active branches at any point in time was about 8. –The cost of the extremely simple processors used in a SIMD are extremely low Programmers used to worry about ‘full utilization of memory’ but stopped this after memory cost became insignificant overall.

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31 SIMD Shortcomings (with some rebuttals) Claim 3: Don’t adapt to multiple users well. –This is true to some degree for all parallel computers. –If usage of a parallel processor is dedicated to a important problem, it is probably best not to risk compromising its performance by ‘sharing’ –This reason also justifies the study of multiple SIMDs (or MSIMD). –SIMD architecture has not received the attention that MIMD has received and can greatly benefit from further research.

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32 SIMD Shortcomings (with some rebuttals) Claim 4: Do not scale down well to “starter” systems that are affordable. –This point is arguable and its ‘truth’ is likely to vary rapidly over time –ClearSpeed currently sells a very economical SIMD board that plugs into a PC.

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33 SIMD Shortcomings (with some rebuttals) Claim 5: Requires customized VLSI for processors and expense of control units in PCs has dropped. Reliance on COTS (Commodity, off-the-shelf parts) has dropped the price of MIMDS Expense of PCs (with control units) has dropped significantly However, reliance on COTS has fueled the success of ‘low level parallelism’ provided by clusters and restricted new innovative parallel architecture research for well over a decade.

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34 SIMD Shortcomings (with some rebuttals) Claim 5 (cont.) There is strong evidence that the period of continual dramatic increases in speed of PCs and clusters is ending. Continued rapid increases in parallel performance in the future will be necessary in order to solve important problems that are beyond our current capabilities Additionally, with the appearance of the very economical COTS SIMDs, this claim no longer appears to be relevant.

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Slides from Associative Computing – Part 1

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36 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.

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37 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.

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38 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 –He received both the Eckert-Mauchly Award and the Seymour Cray Computer Engineering Award 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.

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39 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. –STARAN built to support Air Traffic Control. –ASPRO built to support Air Defense Systems 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

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40 The ASC Model IS C E L L N E T W O R K PEMemory Cells PEMemory PEMemory

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41 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

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42 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).

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43 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

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44 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

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45 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).

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46 1 Busy- idle Dodge Ford Make Subaru Color PE1 PE2 PE3 PE4 PE5 PE6 PE7 red blue white red Year Model Price On lot 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.

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47 Dodge Ford Make Subaru Color PE1 PE2 PE3 PE4 PE5 PE6 PE7 red blue white red Year Model Price On lot Busy- idle 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.

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48 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)

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49 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.

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50 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.

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51 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. Examples of the above are given in –Ref: Nov IEEE Computer article in references –Also, see “Associative Computing” book by Potter.

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52 Languages Designed for the ASC Professor Potter has created several languages for the ASC model. The most important of these is called ASC, 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. Language References: –ASC Primer – Copy available on parallel lab website –“Associative Computing” book by Potter [11] – some features in this book were never fully implemented in ASC Compiler

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53 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

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54 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

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55 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

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56 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

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57 Example 1 – An ASC algorithm for 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

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58 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). 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.

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Recalling Prim’s MST Sequential Algorithm The next 12 slides are included to recall Prim’s MST sequential algorithm These slides are reference slides for students and will not be covered in class. 59

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60 An Example (Prim’s MST Sequential Algorithm) DE HI FC G BA 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

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61 An Example – Step 0 DE HI FC G BA 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.

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62 An Example – Step 0 DE HI FC G BA 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.

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63 An Example – Step 1 DE HI FC G BA Select an arbitrary node to place in V1, say A. Put into V2, all nodes incident with A.

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64 An Example – Step 2 DE HI FC G BA 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.

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65 An Example – Step 3 DE HI FC G BA 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.

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66 An Example – Step 4 DE HI FC G BA 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.

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67 An Example – Step 5 DE HI FC G BA Update the candidate nodes and edges by including all that are incident to those that are in V1 and colored red.

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68 An Example – Step 6 DE HI FC G BA Select I as its edge is minimal. Mark node and edge as red.

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69 An Example – Step 7 DE HI FC G BA 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.

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70 An Example – after several more passes, C is added & we have … DE HI FC G BA Note that when CH is added, GH is dropped as CH has less weight. Candidate edge BC is also dropped since 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.

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71 An Example – the final solution DE HI FC G BA 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.

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72 An ASC MST Algorithm vs the Sequential Prim’s MST Algorithm 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, et. al. 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 ASC algorithm given here basically follows the preceding outline provided for Prim’s MST, using pseudo-code based on the ASC language. A pointer to the ASC manual will be posted on the course web site. –The ASC pseudo-code used for algorithms will require using only a few ASC language commands.

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73 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 upcoming slides. –This example is from the paper, “ASC: An Associative Paradigm”, listed in the references and on the class website under the online 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 with a vector with each vector entry stored in its PE.

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74 ASC-MST Algorithm Preliminaries (cont.) In order to distinguish between them here, the parallel variables names end with a “$” symbol. –This convention is optional and not part of ASC language 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 ). –Recall, 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.

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75 Graph used for Data Structure Figure 6 in [Potter, Baker, et. al.] a bc d e f

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76 MST Algorithm Data Structure for Figure 6 (Data Structure Before Execution) Data Structure for MST Algorithm

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77 Shorter Version of 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

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78 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 in website references Algorithm given in Baase’s books is identical to this parallel algorithm, except it is for a sequential computer Next, a more detailed explanation of the algorithm in preceding slide will be given next.

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79 Algorithm: ASC-MST-PRIM (A more detailed presentation) 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.

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80 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.

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81 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

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82 Trace of 1 st Pass of MST Algorithm for Figure 6

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ASC Quickhull Algorithm A Second ASC Algorithm Example

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84 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

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85 Previous Illustration w e h

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

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87 Data Structure for Preceding Example

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88 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.

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89 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

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90 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

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91 Highest Job Order Assigned to Points Above Lines

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92 Order that Triangles are Computed

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93 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.

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94 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)

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