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...t colors. If we choose k to match the number of machine registers, then we can map a k-coloring of G into a feasible register assignment for the underlying code. Because graph coloring is NP-complete =-=[21]-=-, the compiler uses a heuristic method to search for a coloring; it is not guaranteed to find a k-coloring for all k-colorable graphs. If a k-coloring is not discovered, some values are spilled; that ...

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...ring Values. To find values, we construct the procedure’s static single assignment (SSA) graph, a representation that transforms the code so that each use of a value references exactly one definitio=-=n [13].-=- To achieve this goal, the construction technique inserts special definitions called φ-nodes at those points where control-flow paths join and different values merge. We actually use the pruned SSA g...

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...roblems to graph coloring dates from the early 1960’s [28] (see Section 7). The first implementation of a graph coloring register allocator was done by Chaitin and his colleagues in the PL.8 compile=-=r [8, 6]-=-. Chow and Hennessy later described a priority-based scheme for allocation based on a coloring paradigm [10, 11]. Almost all subsequent work on coloring-based allocation has followed from one of these...

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...d values and propagate this information throughout the SSA graph. 4.1.2 Propagating Rematerialization Tags. To propagate tags, we use an analog of Wegman and Zadeck’s sparse simple constant algorith=-=m [33]. -=-2 We modify their lattice slightly to represent the necessary rematerialization information. The new lattice elements may have one of three types: ⊤ Top means that no information is known. A value d...

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...es where it produced obviously flawed allocations. 3.1 A Motivating Problem A particularly interesting case arose in the code generated for the singular value decomposition (SVD) of Golub and Reinsch =-=[22]. -=-The actual code was from the software library distributed with Forsythe, Malcolm, and Moler’s book on numerical methods [18]. It has two hundred fourteen lines of code, excluding comments. The code ...

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...d L. Torczon ping from values in the procedure to registers in the target machine; the mapping must observe the constraints. The first graph coloring allocator was built by Chaitin and his colleagues =-=[8]-=-. Another approach, called priority-based coloring, was described by Chow and Hennessy [10, 11]. These two techniques have formed the core around which a rich literature has emerged (see Section 7). T...

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...ping must observe the constraints. The first graph coloring allocator was built by Chaitin and his colleagues [8]. Another approach, called priority-based coloring, was described by Chow and Hennessy =-=[10, 11]-=-. These two techniques have formed the core around which a rich literature has emerged (see Section 7). The techniques used in building graph coloring allocators can be improved. In recent years, seve...

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...a “splitting” allocator [11, 27, 23, 5]; the actual code might be worse. In fact, our work on rematerialization was motivated by problems observed during our own experiments with live range splitt=-=ing [3]-=-. Unfortunately, examples of this sort are not discussed in the literature on splitting allocators and it is unclear how best to extend these techniques to achieve the Ideal solution. This section div...

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...generated for the singular value decomposition (SVD) of Golub and Reinsch [22]. The actual code was from the software library distributed with Forsythe, Malcolm, and Moler’s book on numerical method=-=s [18]-=-. It has two hundred fourteen lines of code, excluding comments. The code contains thirty-seven DO-loops organized into five different loop nests. The first loop nest is a simple array copy; four larg...

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...osts encode simple information about the control-flow graph. Any improvements to Chaitin’s work should retain these properties. 3. OPTIMISTIC COLORING As part of the ParaScope programming environmen=-=t [12], -=-we built an optimizing compiler for FORTRAN running on uniprocessors. The initial implementation included a register allocator that used Chaitin’s technique as described in Section 2. The allocator ...

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...ction technique inserts special definitions called φ-nodes at those points where control-flow paths join and different values merge. We actually use the pruned SSA graph, with dead φ-nodes eliminate=-=d [9]. A-=-CM Transactions on Programming Languages and Systems, Vol. 16, No. 3, May 1994, pp. 428–455. ?sImprovements to Graph Coloring Register Allocation · 11 A natural way to view the SSA graph for a proc...

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...ping must observe the constraints. The first graph coloring allocator was built by Chaitin and his colleagues [8]. Another approach, called priority-based coloring, was described by Chow and Hennessy =-=[10, 11]-=-. These two techniques have formed the core around which a rich literature has emerged (see Section 7). The techniques used in building graph coloring allocators can be improved. In recent years, seve...

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... live range of p has been spilled to memory, with loads inserted before uses and stores inserted after definitions. Splitting. The final column shows code we would expect from a “splitting” alloca=-=tor [11, 27, 23, 5]-=-; the actual code might be worse. In fact, our work on rematerialization was motivated by problems observed during our own experiments with live range splitting [3]. Unfortunately, examples of this so...

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...p. Similarly, the allocator cannot retract the decisions later. Once spilled, a live range stays spilled. The coloring heuristic explored by Matula and Beck finds a two-coloring for the diamond graph =-=[29]. Th-=-eir algorithm differs only slightly from Chaitin’s approach. To simplify the graph, they repeatedly remove the node of smallest current degree, versus Chaitin’s approach of removing any node n whe...

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...ature has emerged (see Section 7). The techniques used in building graph coloring allocators can be improved. In recent years, several important extensions to Chaitin's basic techniques have appeared =-=[2, 28]-=-. Nevertheless, problems remain. In practice, most of these problems appear as either over-spilling or a poor spill choice. In the former case, the allocator fails to keep some value in a register, ev...

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...s appeared much slower on very small routines, but much faster on very large routines [3]. Of course, these speeds are not competitive with the fast, local techniques used in non-optimizing compilers =-=[19, 20]-=-; however, we believe that global optimizations require global register allocation. 6. IMPLEMENTATION INSIGHT To perform the experiments described in Section 5, we implemented several versions of our ...

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...and try to color them independently. Fabri, in the context of her work on minimizing memory storage requirements, introduces a renaming transformation that is analogous to Chow’s live range splittin=-=g [16]. Sh-=-e notes that, in some cases, splitting improved the resulting coloring. She mentions the idea’s applicability to register allocation problems. Several groups have refined Chow’s approach. Larus an...

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...locators are built on the graph coloring paradigm. Other approaches include the bin-packing allocators built by Digital Equipment Corporation and the probabilistic allocator of Fischer and Proebsting =-=[1, 17]-=-. It is difficult to compare these techniques because the implementations work on different intermediate representations, they follow different optimizers, and the compilers target different machines....

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... live range of p has been spilled to memory, with loads inserted before uses and stores inserted after definitions. Splitting. The final column shows code we would expect from a “splitting” alloca=-=tor [11, 27, 23, 5]-=-; the actual code might be worse. In fact, our work on rematerialization was motivated by problems observed during our own experiments with live range splitting [3]. Unfortunately, examples of this so...

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...an extra trip around the main loop for spill code insertion. In another paper, we have shown that optimistic allocators generate better allocations on machines that use register pairs for some values =-=[4]-=-. Any implementation of a Chaitin-style allocator should incorporate this improvement; it makes a significant difference in the allocation quality at little or no cost. Our method of rematerializing n...

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...e live range of p has been spilled to memory, with loads inserted before uses and stores inserted after definitions. Splitting The final column shows code we would expect from a "splitting" =-=allocator [11, 25, 21, 5]-=-; the actual code might be worse. In fact, our work on rematerialization was motivated by problems observed during our own experiments with live range splitting [3]. Unfortunately, examples of this so...

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...problem. (In the papers on the PL.8 compiler, this analysis is called “getting the right number of names”[8]. The Hewlett-Packard Precision Architecture compiler papers refer to this as “web ana=-=lysis”[25]-=-.) (2) Build constructs the interference graph. Our implementation closely follows the published descriptions of the PL.8 allocator [8, 6]. The interference graph is simultaneously represented as a bi...

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...ature has emerged (see Section 7). The techniques used in building graph coloring allocators can be improved. In recent years, several important extensions to Chaitin’s basic techniques have appeare=-=d [2, 30]-=-. Nevertheless, problems remain. In practice, most of these problems appear as either over-spilling or a poor spill choice. In the former case, the allocator fails to keep some value in a register, ev...

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... packing algorithm on it [14, 15]. By the late sixties, Cocke was clearly talking about applying these insights directly to register allocation; both Kennedy and Schwartz credit him with this insight =-=[26, 31]-=-. This early work on graph coloring register allocation emphasizes the coloring problem with little consideration for the questions of spill choice and placement. Algorithms by Cocke and Ershov (as re...

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...s appeared much slower on very small routines, but much faster on very large routines [3]. Of course, these speeds are not competitive with the fast, local techniques used in non-optimizing compilers =-=[19, 20]-=-; however, we believe that global optimizations require global register allocation. 6. IMPLEMENTATION INSIGHT To perform the experiments described in Section 5, we implemented several versions of our ...

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...or finding a coloring. Ershov and his colleagues built on this work in the ALPHA project. They solved storage allocation problems by building an interference graph and using a packing algorithm on it =-=[14, 15]-=-. By the late sixties, Cocke was clearly talking about applying these insights directly to register allocation; both Kennedy and Schwartz credit him with this insight [26, 31]. This early work on grap...

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...Finally, section 7 presents a discussion of related work. 2. REGISTER ALLOCATION VIA GRAPH COLORING The notion of abstracting storage allocation problems to graph coloring dates from the early 1960’=-=s [28]-=- (see Section 7). The first implementation of a graph coloring register allocator was done by Chaitin and his colleagues in the PL.8 compiler [8, 6]. Chow and Hennessy later described a priority-based...

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...or finding a coloring. Ershov and his colleagues built on this work in the ALPHA project. They solved storage allocation problems by building an interference graph and using a packing algorithm on it =-=[14, 15]-=-. By the late sixties, Cocke was clearly talking about applying these insights directly to register allocation; both Kennedy and Schwartz credit him with this insight [24, 29]. This early work on grap...

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...tion of this technique in the back-end of the IBM XL compiler family for the RS/6000 architecture resulted in a decrease of about twenty percent in estimated spill costs over the SPEC benchmark suite =-=[24]-=-. 4. REMATERIALIZATION Even with optimistic coloring, the allocator must spill some live ranges. When this happens, the allocator should choose the least expensive mechanism to accomplish the spill. I...

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... packing algorithm on it [14, 15]. By the late sixties, Cocke was clearly talking about applying these insights directly to register allocation; both Kennedy and Schwartz credit him with this insight =-=[26, 31]-=-. This early work on graph coloring register allocation emphasizes the coloring problem with little consideration for the questions of spill choice and placement. Algorithms by Cocke and Ershov (as re...

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...locators are built on the graph coloring paradigm. Other approaches include the bin-packing allocators built by Digital Equipment Corporation and the probabilistic allocator of Fischer and Proebsting =-=[1, 17]-=-. It is difficult to compare these techniques because the implementations work on different intermediate representations, they follow different optimizers, and the compilers target different machines....

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...ature has emerged (see Section 7). The techniques used in building graph coloring allocators can be improved. In recent years, several important extensions to Chaitin’s basic techniques have appeare=-=d [2, 30]-=-. Nevertheless, problems remain. In practice, most of these problems appear as either over-spilling or a poor spill choice. In the former case, the allocator fails to keep some value in a register, ev...

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...de is invoked if simplify decides to spill a node. Each spilled live range is converted into a collection of tiny live ranges by inserting loads before uses and stores after definitions, as required. =-=(7)-=- Select assigns colors to the nodes of the graph in the order determined by simplify. It repeats the following steps until the stack is empty: (a) pop a live range from the stack, (b) insert its corre...

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...or finding a coloring. Ershov and his colleagues built on this work in the ALPHA project. They solved storage allocation problems by building an interference graph and using a packing algorithm on it =-=[14, 15]-=-. By the late sixties, Cocke was clearly talking about applying these insights directly to register allocation; both Kennedy and Schwartz credit him with this insight [26, 31]. This early work on grap...

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... live range of p has been spilled to memory, with loads inserted before uses and stores inserted after definitions. Splitting. The final column shows code we would expect from a “splitting” alloca=-=tor [11, 27, 23, 5]-=-; the actual code might be worse. In fact, our work on rematerialization was motivated by problems observed during our own experiments with live range splitting [3]. Unfortunately, examples of this so...

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...or finding a coloring. Ershov and his colleagues built on this work in the ALPHA project. They solved storage allocation problems by building an interference graph and using a packing algorithm on it =-=[14, 15]-=-. By the late sixties, Cocke was clearly talking about applying these insights directly to register allocation; both Kennedy and Schwartz credit him with this insight [24, 29]. This early work on grap...