Sethi–Ullman algorithm: Difference between revisions

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{{Short description|Algorithm for minimising register usage}}

In [[computer science]], the '''Sethi–Ullman algorithm''' is an [[algorithm]] named after [[Ravi Sethi]] and [[Jeffrey D. Ullman]], its inventors, for translating [[abstract syntax tree]]s into [[machine code]] that uses as few instructions[[Processor register|registers]] as possible.

==Overview==

When [[Codecode generation (compiler)|generating code]] for arithmetic expressions, the [[compiler]] has to decide which is the best way to translate the expression in terms of number of instructions used as well as number of registers needed to evaluate a certain subtree. Especially in the case that free registers are scarce, the [[order of evaluation]] can be important to the length of the generated code, because different orderings may lead to larger or smaller numbers of intermediate values being [[register allocation|spilled]] to memory and then restored. The Sethi–Ullman algorithm (also known as '''Sethi–Ullman numbering''') fulfills the property of producingproduces code which needs the least number offewest instructions possible as well as the least number offewest storage references (under the assumption that at the most [[commutativity]] and [[associativity]] apply to the operators used, but distributive laws i.e. <math>a * b + a * c = a * (b + c)</math> do not hold). Please note that theThe algorithm succeeds as well if neither [[commutativity]] nor [[associativity]] hold for the expressions used, and therefore arithmetic transformations can not be applied. The algorithm also does not take advantage of common subexpressions or apply directly to expressions represented as general directed acyclic graphs rather than trees.

==Simple Sethi–Ullman algorithm==

The '''simple Sethi–Ullman algorithm''' works as follows (for a [[RISC|load-/store architecture]]):

# Traverse the [[abstract syntax tree]] in pre- or postorder

## For every leaf node, if it is a non-constant leaf nodeleft-child, assign a 1 (i.e. 1 register is needed to hold the variable/field/etc.)., Forotherwise everyassign a 0 (it is a non-constant right child or constant leaf node (RHS of an operation – literals, values), assign a 0).

## For every non-leaf node ''n'', assignif the numberleft ofand registers needed to evaluate the respectiveright subtrees ofrespectively ''n''.need If thedifferent numbernumbers of registers needed in the left subtree (''l'') areand not equal to the number of registers needed in the right subtree (''r''), thethen number of registers needed for the current nodeassign max(''nl'' is max(l,&nbsp;''r'')., Ifotherwise assign ''l == r'', then the number of registers needed for the current node is l&nbsp;+&nbsp;1.

# To emit code, if the subtrees need different numbers of registers, evaluate the subtree needing the most registers first (since the register needed to save the result of one subtree may make the other one [[Register spilling|spill]]), otherwise the order is irrelevant.

# Code emission

## If the number of registers needed to compute the left subtree of node ''n'' is bigger than the number of registers for the right subtree, then the left subtree is evaluated first (since it may be possible that the one more register needed by the right subtree to save the result makes the left subtree [[Register spilling|spill]]). If the right subtree needs more registers than the left subtree, the right subtree is evaluated first accordingly. If both subtrees need equal as much registers, then the order of evaluation is irrelevant.

===Example===

For an arithmetic expression <math>a = (b + c + f * g) * (d + 3)</math>, the [[abstract syntax tree]] looks like this:

=

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/ \ / \

b_'''1''' c_'''0'''f_'''1''' g_'''0'''

From this tree it can be seen that we need 2 registers to compute the left subtree of the '*', but only 1 register to compute the right subtree.The reason ofNodes 'c' and 'g' do not need registerregisters isfor the following reasons: If T is a tree leaf, then the number of registers to evaluate T is either 1 or 0 depending whether T is a left or a right subtree (since an operation such as add R1, A can handle the right component A directly without storing it into a register). Therefore we shall start to emit code for the left subtree first, because we might run into the situation that we only have 2 registers left to compute the whole expression. If we now computed the right subtree first (which needs only 1 register), we would then need a register to hold the result of the right subtree while computing the left subtree (which would still need 2 registers), therefore needing 3 registers concurrently. Computing the left subtree first needs 2 registers, but the result can be stored in 1, and since the right subtree needs only 1 register to compute, the evaluation of the expression can do with only 2 registers left.

==Advanced Sethi–Ullman algorithm==

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==See also==

*[[Strahler number]], the minimum number of registers needed to evaluate an expression without any external storage

*[[Ershov Number]], basically the same concept as Strahler number

==References==

*{{citation|title=The Generation of Optimal Code for Arithmetic Expressions|first1=Ravi|last1=Sethi|author1-link=Ravi Sethi|first2=Jeffrey D.|last2=Ullman|author2-link=Jeffrey D. Ullman|journal=[[Journal of the Association for Computing Machinery]]|volume=17|issue=4|year=1970|pages=715–728|doi=10.1145/321607.321620|hdl=10338.dmlcz/101207|hdl-access=free}}.

==External links==

*[httphttps://lambda.uta.edu/cse5317/fall02/notes/node40node43.html Code Generation for Trees]

{{DEFAULTSORT:Sethi-Ullman algorithm}}

[[Category:Compiler theoryconstruction]]

[[Category:Graph algorithms]]

[[ja:セシィ-ウルマン法]]

[[pl:Algorytm Sethi-Ullmana]]