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Thus we have arrived at a contradiction and proved that <strong>for</strong> RCSLMADs, each d-tuplein the domain D maps to a distinct location in memory.d∏Theorem 2. The size S <strong>of</strong> the set represented by the RCSLMAD L is given by S = u k .Pro<strong>of</strong>. The size S <strong>of</strong> the set is equal to the size <strong>of</strong> the domain D because each element inthe domain maps to a distinct memory location. The size S can simply be computed as aproduct <strong>of</strong> the upper bounds because the domain is rectangular.A loop nest may have multiple RCSLMADs and they may overlap at one or more memorylocations. If each RCSLMAD is transferred independently, then the same memory locationon the CPU may be copied to multiple copies on the GPU. If there is a read/write dependenceinvolving a memory location, then creating multiple copies on the GPU is cumbersomebecause the copies must be kept consistent. A better solution is to compute a set union <strong>of</strong>the RCSLMADs and then transfer the union to the GPU. This way there will be exactlyone copy <strong>of</strong> M on the GPU. The compiler also ensures that no other CPU thread writesto the memory locations copied to the GPU be<strong>for</strong>e the GPU execution and data transfersare complete. For correctness, it is not necessary to compute an exact union. If the unionis represented by U, then it is sufficient to compute a superset U <strong>of</strong> the union. We nowpresent the problem statement including the design criteria <strong>of</strong> an analysis to compute theunion:Problem Statement 1. Given a list {L 1 , L 2 , .., L n } <strong>of</strong> RCSLMADs defined over a d-dimensional domain D:1. Compute the set M <strong>of</strong> memory locations to be transferred to the GPU. The set M shouldbe a superset <strong>of</strong> the union U <strong>of</strong> RCSLMADs. The number <strong>of</strong> elements in M −U shouldbe as small as possible. M should be easily representable by the compiler.2. Compute the size S <strong>of</strong> the set M.3. Construct a map F with domain M. The range R <strong>of</strong> F should be a set <strong>of</strong> validmemory locations in the GPU address space. If F (m), mɛM represents the address <strong>of</strong>the memory location m on the GPU, then F should be such that F (m 1 ) ≠ F (m 2 ) ifm 1 ≠ m 2 . The Range R should be as small as possible. The computational complexityand memory requirement <strong>for</strong> computing F should be as low as possible.5.2 General solutionConsider the case where the list contains a single RCSLMAD L defined over a domain D.Without loss <strong>of</strong> generality, assume that the ordering function <strong>of</strong> L is the identity function.In such a case, an exact solution <strong>for</strong> the problem 1 can be calculated by algorithm 1.k=138
Algorithm 1 solves the data transfer problem <strong>for</strong> a single RCSLMAD L defined over adomain D.1: M = {L(V )|V ɛD}.d∏2: S = u k from property <strong>of</strong> RCSLMAD.k=13: Let m = L(V ), V ɛD, then F (m) =d∑(V (k) ∗ (k=1d∏j=k+1u j )) and range R = S.To argue that the algorithm 1 does produce a correct function F , we observe that givena d-dimensional tuple D, the composite function F (L) is also an RCSLMAD defined overthe domain D, and hence each distinct tuple V is mapped to a distinct location in theGPU memory by the composite function F (L). The range R can be directly verified bycalculating the maximum and minimum values <strong>of</strong> F (L) over D.The problem <strong>of</strong> multiple RCSLMADs can be simplified in many cases by consideringthe memory intervals spanned by the RCSLMADs in the list.If the memory intervalsspanned by the RCSLMADs in the list are completely disjoint, then each RCSLMAD canbe analyzed individually without any necessity to compute the union because there can be nodependencies and no possibilities <strong>of</strong> multiple copies. If the interval spanned by two or moreRCSLMADs overlaps, then we can <strong>for</strong>m a group <strong>of</strong> RCSLMADs. A group <strong>of</strong> RCSLMADsspans over an interval given as the union <strong>of</strong> intervals <strong>of</strong> all members <strong>of</strong> the group. In generalwe can partition the list <strong>of</strong> RCSLMADs into groups <strong>of</strong> RCSLMADs where the intervalrepresented by each group is disjoint from other groups. An algorithm to compute groups<strong>of</strong> RCSLMADs is given by algorithm 2.Algorithm 2 partitions a list <strong>of</strong> RCSLMADs defined over a domain D into disjoint groups<strong>of</strong> RCSLMADs.Inputs: List {L 1 , L 2 , .., L n } <strong>of</strong> RCSLMADs defined over a domain D.Outputs: List {G 1 , G 2 , .., G n } <strong>of</strong> groups <strong>of</strong> LMADs such that the interval spanned by eachgroup is disjoint from the interval spanned by any other group.1: <strong>for</strong> each RCSLMAD L in {L 1 , L 2 , .., L n } do2: Find out the start and end addresses <strong>of</strong> R in memory.3: end <strong>for</strong>4: Construct a graph G where each RCSLMAD is a node.5: <strong>for</strong> each pair <strong>of</strong> nodes in G do6: Insert an edge if the start-to-end intervals <strong>of</strong> the two nodes overlap.7: end <strong>for</strong>8: Find the connected components in the graph G using a suitable algorithm such as adepth first search. Each connected component represents a group <strong>of</strong> RCSLMADs.9: return A list <strong>of</strong> connected components.If one or more groups contains more than one RCSLMAD, then the problem <strong>of</strong> computingthe union (or the superset) still remains. One general solution is described in algorithm 3.39
Page 3 and 4: Examining CommitteeJose Nelson Amar
Page 5 and 6: AbstractModern Graphics Processing
Page 8 and 9: 5.3.3 Multidimensional RCSLMADs wit
Page 10 and 11: List of Figures2.1
Page 12 and 13: RAMRISCSIMDSPSPMDSSETEXTPUVDVLIWbDE
Page 14 and 15: much lower than equivalent C progra
Page 16 and 17: Chapter 2AMD RV770 architecture <st
Page 18 and 19: Figure 2.1: RV770 Block Diagram. So
Page 20 and 21: 2.2 AMD CAL Programming modelVariou
Page 22 and 23: a context. The predefined names are
Page 24 and 25: 2.3 Hardware execution model2.3.1 P
Page 26 and 27: on the texture unit to bring the da
Page 28 and 29: float4 temp1 = i1[i][threadId.y/4];
Page 30: Chapter 3Python an
Page 33 and 34: Unlike C, NumPy does not impose any
Page 35 and 36: Windows. The module initialization
Page 37 and 38: 3.5.1 Type annotationsType declarat
Page 39 and 40: UnPython generates
Page 41 and 42: Python source code
Page 43 and 44: and optimizations detailed later in
Page 45 and 46: Chapter 5Array access analysisArray
Page 47 and 48: LMADs can represent a wide variety
Page 49: Figure 5.2: Visualization o
Page 53 and 54: 0 ≤ j < 5 (5.26)In the example, t
Page 55 and 56: case, assume we had a C array start
Page 57 and 58: solve for unknown
Page 59 and 60: approach, the number of</st
Page 61 and 62: Chapter 6Loop transfor</str
Page 63 and 64: Algorithm 6 perfor
Page 65 and 66: with suitable GPU code annotations.
Page 67 and 68: Table 7.3: Execution time f
Page 69 and 70: time and the execution time on the
Page 71 and 72: suite. The serial version o
Page 73 and 74: However Shedskin does not support e
Page 75 and 76: thread. Device functions are writte
Page 77 and 78: Chapter 9ConclusionsThis thesis int
Page 79: [14] Francois Labonte, Peter Mattso

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