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<title>Induction variable</title>

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Induction variable

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In <a href="Computer_science" title="Computer science">computer science</a>, an induction variable is a variable that gets <a href="Successor_function" title="Successor function">increased</a> or decreased by a fixed amount on every iteration of a loop or is a <a href="Linear_function" title="Linear function">linear function</a> of another induction variable.<a href="#cite_note-MuchnickAssociates1997-1">[1]</a>

For example, in the following loop, <code>i</code> and <code>j</code> are induction variables:

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>for (i = 0; i < 10; ++i) {

j = 17 * i;

}

</pre></div>

<div class="mw-heading mw-heading2"><h2 id="Application_to_strength_reduction">Application to strength reduction</h2></div>

A common <a href="Compiler_optimization" class="mw-redirect" title="Compiler optimization">compiler optimization</a> is to recognize the existence of induction variables and replace them with simpler computations; for example, the code above could be rewritten by the compiler as follows, on the assumption that the addition of a constant will be cheaper than a multiplication.

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>j = -17;

for (i = 0; i < 10; ++i) {

j = j + 17;

}

</pre></div>

This optimization is a special case of <a href="Strength_reduction" title="Strength reduction">strength reduction</a>.

<div class="mw-heading mw-heading2"><h2 id="Application_to_reduce_register_pressure">Application to reduce register pressure</h2></div>

In some cases, it is possible to reverse this optimization in order to remove an induction variable from the code entirely. For example:

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>extern int sum;

int foo(int n) {

int j = 5;

for (int i = 0; i < n; ++i) {

j += 2;

sum += j;

}

return sum;

}

</pre></div>

This function's loop has two induction variables: <code>i</code> and <code>j</code>. Either one can be rewritten as a linear function of the other; therefore, the compiler may optimize this code as if it had been written

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>extern int sum;

int foo(int n) {

for (int i = 0; i < n; ++i) {

sum += 5 + 2 * (i + 1);

}

return sum;

}

</pre></div>

<div class="mw-heading mw-heading2"><h2 id="Induction_variable_substitution">Induction variable substitution</h2></div>

Induction variable substitution is a compiler transformation to recognize variables which can be expressed as functions of the indices of enclosing loops and replace them with expressions involving loop indices.

This transformation makes the relationship between the variables and loop indices explicit, which helps other compiler analysis, such as <a href="Dependence_analysis" title="Dependence analysis">dependence analysis</a>.

Example:

Input code:

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>int c = 10;

for (int i = 0; i < 10; i++) {

c = c + 5; // c is incremented by 5 for each loop iteration

}

</pre></div>

Output code

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>int c = 10;

for (int i = 0; i < 10; i++) {

c = 10 + 5 * (i + 1); // c is explicitly expressed as a function of loop index

}

</pre></div>

<div class="mw-heading mw-heading2"><h2 id="Non-linear_induction_variables">Non-linear induction variables</h2></div>

The same optimizations can be applied to induction variables that are not necessarily linear functions of the loop counter; for example, the loop

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>j = 1;

for (i = 0; i < 10; ++i) {

j = j << 1;

}

</pre></div>

may be converted to

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>for (i = 0; i < 10; ++i) {

j = 1 << (i+1);

}

</pre></div>

<ul><li><a href="Mathematical_induction" title="Mathematical induction">Mathematical induction</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFSteven_MuchnickMuchnick_and_Associates1997" class="citation book cs1">Steven Muchnick; Muchnick and Associates (15 August 1997). <a rel="nofollow" class="external text" href="https://archive.org/details/advancedcompiler00much">Advanced Compiler Design Implementation</a>. Morgan Kaufmann. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-55860-320-2</bdi>. <q>induction variable.</q></cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite class="citation cs2">Aho, Alfred V.; Sethi, Ravi; Ullman, Jeffrey D. (1986), <a rel="nofollow" class="external text" href="https://archive.org/details/compilersprincip00ahoa">Compilers: Principles, Techniques, and Tools</a> (2nd ed.), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-10088-4</bdi></cite></li>

<li><cite class="citation cs2">Allen, Francis E.; <a href="John_Cocke_(computer_scientist)" title="John Cocke (computer scientist)">Cocke, John</a>; Kennedy, Ken (1981), "Reduction of Operator Strength", in Munchnik, Steven S.; Jones, Neil D. (eds.), Program Flow Analysis: Theory and Applications, Prentice-Hall, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-729681-1</bdi></cite></li>

<li><cite class="citation cs2"><a href="John_Cocke_(computer_scientist)" title="John Cocke (computer scientist)">Cocke, John</a>; Kennedy, Ken (November 1977), "An algorithm for reduction of operator strength", Communications of the ACM, 20 (11): 850–856, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F359863.359888">10.1145/359863.359888</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1092505">1092505</a></cite></li>

<li><cite class="citation cs2">Cooper, Keith; Simpson, Taylor; Vick, Christopher (1995), <a rel="nofollow" class="external text" href="http://softlib.rice.edu/pub/CRPC-TRs/reports/CRPC-TR95635-S.pdf">Operator Strength Reduction</a> (PDF), Rice University, retrieved April 22, 2010</cite></li></ul>

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</style><div id="Compiler_optimizations253" style="font-size:114%;margin:0 4em"><a href="Optimizing_compiler" title="Optimizing compiler">Compiler optimizations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic block</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Peephole_optimization" title="Peephole optimization">Peephole optimization</a></li>

<li><a href="Local_value_numbering" class="mw-redirect" title="Local value numbering">Local value numbering</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Loop_optimization" title="Loop optimization">Loop</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Automatic_parallelization" title="Automatic parallelization">Automatic parallelization</a></li>

<li><a href="Automatic_vectorization" title="Automatic vectorization">Automatic vectorization</a></li>

<li><a href="Loop_fusion" class="mw-redirect" title="Loop fusion">Loop fusion</a></li>

<li><a href="Loop-invariant_code_motion" title="Loop-invariant code motion">Loop-invariant code motion</a></li>

<li><a href="Loop_inversion" title="Loop inversion">Loop inversion</a></li>

<li><a href="Loop_interchange" title="Loop interchange">Loop interchange</a></li>

<li><a href="Loop_nest_optimization" title="Loop nest optimization">Loop nest optimization</a></li>

<li><a href="Loop_splitting" title="Loop splitting">Loop splitting</a></li>

<li><a href="Loop_unrolling" title="Loop unrolling">Loop unrolling</a></li>

<li><a href="Loop_unswitching" title="Loop unswitching">Loop unswitching</a></li>

<li><a href="Software_pipelining" title="Software pipelining">Software pipelining</a></li>

<li><a href="Strength_reduction" title="Strength reduction">Strength reduction</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data-flow_analysis" title="Data-flow analysis">Data-flow analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Available_expression" title="Available expression">Available expression</a></li>

<li><a href="Common_subexpression_elimination" title="Common subexpression elimination">Common subexpression elimination</a></li>

<li><a href="Constant_folding" title="Constant folding">Constant folding</a></li>

<li><a href="Dead_store" title="Dead store">Dead store</a> elimination</li>

<li><a href="Induction_variable_recognition_and_elimination" class="mw-redirect" title="Induction variable recognition and elimination">Induction variable recognition and elimination</a></li>

<li><a href="Live-variable_analysis" title="Live-variable analysis">Live-variable analysis</a></li>

<li><a href="Upwards_exposed_uses" title="Upwards exposed uses">Upwards exposed uses</a></li>

<li><a href="Use-define_chain" title="Use-define chain">Use-define chain</a></li>

<li><a href="Reaching_definition" title="Reaching definition">Reaching definitions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Static_single-assignment_form" title="Static single-assignment form">SSA</a>-based</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Global_value_numbering" class="mw-redirect" title="Global value numbering">Global value numbering</a></li>

<li><a href="Sparse_conditional_constant_propagation" title="Sparse conditional constant propagation">Sparse conditional constant propagation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Code_generation_(compiler)" title="Code generation (compiler)">Code generation</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Instruction_scheduling" title="Instruction scheduling">Instruction scheduling</a></li>

<li><a href="Instruction_selection" title="Instruction selection">Instruction selection</a></li>

<li><a href="Register_allocation" title="Register allocation">Register allocation</a></li>

<li><a href="Rematerialization" title="Rematerialization">Rematerialization</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Functional</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Deforestation_(computer_science)" title="Deforestation (computer science)">Deforestation</a></li>

<li><a href="Tail_call" title="Tail call">Tail-call elimination</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Global</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Interprocedural_optimization" title="Interprocedural optimization">Interprocedural optimization</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bounds-checking_elimination" title="Bounds-checking elimination">Bounds-checking elimination</a></li>

<li><a href="Compile-time_function_execution" title="Compile-time function execution">Compile-time function execution</a></li>

<li><a href="Dead-code_elimination" title="Dead-code elimination">Dead-code elimination</a></li>

<li><a href="Expression_templates" title="Expression templates">Expression templates</a></li>

<li><a href="Inline_expansion" title="Inline expansion">Inline expansion</a></li>

<li><a href="Jump_threading" title="Jump threading">Jump threading</a></li>

<li><a href="Partial_evaluation" title="Partial evaluation">Partial evaluation</a></li>

<li><a href="Profile-guided_optimization" title="Profile-guided optimization">Profile-guided optimization</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Static analysis</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alias_analysis" title="Alias analysis">Alias analysis</a></li>

<li><a href="Array-access_analysis" title="Array-access analysis">Array-access analysis</a></li>

<li><a href="Control-flow_analysis" title="Control-flow analysis">Control-flow analysis</a></li>

<li><a href="Data-flow_analysis" title="Data-flow analysis">Data-flow analysis</a></li>

<li><a href="Dependence_analysis" title="Dependence analysis">Dependence analysis</a></li>

<li><a href="Escape_analysis" title="Escape analysis">Escape analysis</a></li>

<li><a href="Pointer_analysis" title="Pointer analysis">Pointer analysis</a></li>

<li><a href="Shape_analysis_(program_analysis)" title="Shape analysis (program analysis)">Shape analysis</a></li>

<li><a href="Value_range_analysis" title="Value range analysis">Value range analysis</a></li></ul>

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