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  <title>P3159R0: C++ Range Adaptors and Parallel Algorithms</title>
<style data-fill-with="stylesheet">/******************************************************************************
 *                   Style sheet for the W3C specifications                   *
 *
 * Special classes handled by this style sheet include:
 *
 * Indices
 *   - .toc for the Table of Contents (<ol class="toc">)
 *     + <span class="secno"> for the section numbers
 *   - #toc for the Table of Contents (<nav id="toc">)
 *   - ul.index for Indices (<a href="#ref">term</a><span>, in § N.M</span>)
 *   - table.index for Index Tables (e.g. for properties or elements)
 *
 * Structural Markup
 *   - table.data for general data tables
 *     -> use 'scope' attribute, <colgroup>, <thead>, and <tbody> for best results !
 *     -> use <table class='complex data'> for extra-complex tables
 *     -> use <td class='long'> for paragraph-length cell content
 *     -> use <td class='pre'> when manual line breaks/indentation would help readability
 *   - dl.switch for switch statements
 *   - ol.algorithm for algorithms (helps to visualize nesting)
 *   - .figure and .caption (HTML4) and figure and figcaption (HTML5)
 *     -> .sidefigure for right-floated figures
 *   - ins/del
 *     -> ins/del.c### for candidate and proposed changes (amendments)
 *
 * Code
 *   - pre and code
 *
 * Special Sections
 *   - .note       for informative notes             (div, p, span, aside, details)
 *   - .example    for informative examples          (div, p, pre, span)
 *   - .issue      for issues                        (div, p, span)
 *   - .advisement for loud normative statements     (div, p, strong)
 *   - .annoying-warning for spec obsoletion notices (div, aside, details)
 *   - .correction for "candidate corrections"       (div, aside, details, section)
 *   - .addition   for "candidate additions"         (div, aside, details, section)
 *   - .correction.proposed for "proposed corrections" (div, aside, details, section)
 *   - .addition.proposed   for "proposed additions"   (div, aside, details, section)
 *
 * Definition Boxes
 *   - pre.def   for WebIDL definitions
 *   - table.def for tables that define other entities (e.g. CSS properties)
 *   - dl.def    for definition lists that define other entitles (e.g. HTML elements)
 *
 * Numbering
 *   - .secno for section numbers in .toc and headings (<span class='secno'>3.2</span>)
 *   - .marker for source-inserted example/figure/issue numbers (<span class='marker'>Issue 4</span>)
 *   - ::before styled for CSS-generated issue/example/figure numbers:
 *     -> Documents wishing to use this only need to add
 *        figcaption::before,
 *        .caption::before { content: "Figure "  counter(figure) " ";  }
 *        .example::before { content: "Example " counter(example) " "; }
 *        .issue::before   { content: "Issue "   counter(issue) " ";   }
 *
 * Header Stuff (ignore, just don't conflict with these classes)
 *   - .head for the header
 *   - .copyright for the copyright
 *
 * Outdated warning for old specs
 *
 * Miscellaneous
 *   - .overlarge for things that should be as wide as possible, even if
 *     that overflows the body text area. This can be used on an item or
 *     on its container, depending on the effect desired.
 *     Note that this styling basically doesn't help at all when printing,
 *     since A4 paper isn't much wider than the max-width here.
 *     It's better to design things to fit into a narrower measure if possible.
 *
 *   - js-added ToC jump links (see fixup.js)
 *
 ******************************************************************************/

/* color variables included separately for reliability */

/******************************************************************************/
/*                                    Body                                    */
/******************************************************************************/

	html {
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	body {
		counter-reset: example figure issue;

		/* Layout */
		max-width: 50em;			  /* limit line length to 50em for readability   */
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		padding: 1.6em 1.5em 2em calc(26px + 1.5em); /* leave space for status flag    */

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		color: var(--text);
		background: white top left fixed no-repeat;
		background: var(--bg) top left fixed no-repeat;
		background-size: 25px auto;
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/******************************************************************************/
/*                         Front Matter & Navigation                          */
/******************************************************************************/

/** Header ********************************************************************/

	div.head { margin-bottom: 1em; }
	div.head hr { border-style: solid; }

	div.head h1 {
		font-weight: bold;
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	div.head h2 { margin-bottom: 1.5em;}

/** W3C Logo ******************************************************************/

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	.head img[src*="logos/W3C"] {
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	.head a:hover > img[src*="logos/W3C"],
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	.head a:active > img[src*="logos/W3C"] {
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		border-color: var(--logo-active-bg);
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	/* see also additional rules in Link Styling section */

/** Copyright *****************************************************************/

	p.copyright,
	p.copyright small { font-size: small; }

/** Back to Top / ToC Toggle **************************************************/

	@media print {
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			display: none;
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	@media not print {
		#toc-nav {
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			height: 1.33em;
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			margin: 0;

			box-shadow: 0 0 2px;
			border: none;
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			background: var(--tocnav-hover-bg);
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			color: var(--tocnav-active-text);
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		#toc-nav > #toc-jump {
			padding-bottom: 2em;
			margin-bottom: -1.9em;
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		/* statusbar gets in the way on keyboard focus; remove once browsers fix */
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			padding-bottom: 1.5rem;
		}

		#toc-nav:not(:hover) > a:not(:focus) > span + span {
			/* Ideally this uses :focus-within on #toc-nav */
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		#toc-nav > a > span + span {
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/** ToC Sidebar ***************************************************************/

	/* Floating sidebar */
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			top: 0; bottom: 0;
			left: 0;
			width: 23.5em;
			max-width: 80%;
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			color: var(--tocsidebar-text);
			background: inherit;
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			box-shadow: -.1em 0 .25em rgba(0,0,0,.1) inset;
			box-shadow: -.1em 0 .25em var(--tocsidebar-shadow) inset;
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			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
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			width: 23.5em;
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			padding-left: calc(1em + 26px);
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			color: var(--tocsidebar-text);
			background: inherit;
			background-color: #f7f8f9;
			background-color: var(--tocsidebar-bg);
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			box-shadow: -.1em 0 .25em var(--tocsidebar-shadow) inset;
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		body:not(.toc-inline) #toc h2 {
			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
			font-weight: bold;
			color: gray;
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			color: var(--tocsidebar-heading-text);
		}

		body:not(.toc-inline) {
			padding-left: 29em;
		}
		/* See also Overflow section at the bottom */

		body:not(.toc-inline) #toc-jump:not(:focus) {
			width: 0;
			height: 0;
			padding: 0;
			position: absolute;
			overflow: hidden;
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	@media screen and (min-width: 90em) {
		body:not(.toc-inline) {
			margin: 0 4em;
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/******************************************************************************/
/*                                Sectioning                                  */
/******************************************************************************/

/** Headings ******************************************************************/

	h1, h2, h3, h4, h5, h6, dt {
		page-break-after: avoid;
		page-break-inside: avoid;
		font: 100% sans-serif;   /* Reset all font styling to clear out UA styles */
		font-family: inherit;	/* Inherit the font family. */
		line-height: 1.2;		/* Keep wrapped headings compact */
		hyphens: manual;		/* Hyphenated headings look weird */
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	h2, h3, h4, h5, h6 {
		margin-top: 3rem;
	}

	h1, h2, h3 {
		color: #005A9C;
		color: var(--heading-text);
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	h1 { font-size: 170%; }
	h2 { font-size: 140%; }
	h3 { font-size: 120%; }
	h4 { font-weight: bold; }
	h5 { font-style: italic; }
	h6 { font-variant: small-caps; }
	dt { font-weight: bold; }

/** Subheadings ***************************************************************/

	h1 + h2,
	#profile-and-date {
		/* #profile-and-date is a subtitle in an H2 under the H1 */
		margin-top: 0;
	}
	h2 + h3,
	h3 + h4,
	h4 + h5,
	h5 + h6 {
		margin-top: 1.2em; /* = 1 x line-height */
	}

/** Section divider ***********************************************************/

	:not(.head) > :not(.head) + hr {
		font-size: 1.5em;
		text-align: center;
		margin: 1em auto;
		height: auto;
		color: black;
		color: var(--hr-text);
		border: transparent solid 0;
		background: transparent;
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	:not(.head) > hr::before {
		content: "\2727\2003\2003\2727\2003\2003\2727";
	}

/******************************************************************************/
/*                            Paragraphs and Lists                            */
/******************************************************************************/

	p {
		margin: 1em 0;
	}

	dd > p:first-child,
	li > p:first-child {
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	ul, ol {
		margin-left: 0;
		padding-left: 2em;
	}

	li {
		margin: 0.25em 0 0.5em;
		padding: 0;
	}

	dl dd {
		margin: 0 0 .5em 2em;
	}

	.head dd + dd { /* compact for header */
		margin-top: -.5em;
	}

	/* Style for algorithms */
	ol.algorithm ol:not(.algorithm),
	.algorithm > ol ol:not(.algorithm) {
	border-left: 0.5em solid #DEF;
	border-left: 0.5em solid var(--algo-border);
	}

	/* Put nice boxes around each algorithm. */
	[data-algorithm]:not(.heading) {
	 padding: .5em;
	 border: thin solid #ddd;
	 border: thin solid var(--algo-border);
	 border-radius: .5em;
	 margin: .5em calc(-0.5em - 1px);
	}
	[data-algorithm]:not(.heading) > :first-child {
	 margin-top: 0;
	}
	[data-algorithm]:not(.heading) > :last-child {
	 margin-bottom: 0;
	}

	/* Style for switch/case <dl>s */
	dl.switch > dd > ol.only,
	dl.switch > dd > .only > ol {
	margin-left: 0;
	}
	dl.switch > dd > ol.algorithm,
	dl.switch > dd > .algorithm > ol {
	margin-left: -2em;
	}
	dl.switch {
	padding-left: 2em;
	}
	dl.switch > dt {
	text-indent: -1.5em;
	margin-top: 1em;
	}
	dl.switch > dt + dt {
	margin-top: 0;
	}
	dl.switch > dt::before {
	content: '\21AA';
	padding: 0 0.5em 0 0;
	display: inline-block;
	width: 1em;
	text-align: right;
	line-height: 0.5em;
	}

/** Terminology Markup ********************************************************/


/******************************************************************************/
/*                                 Inline Markup                              */
/******************************************************************************/

/** Terminology Markup ********************************************************/
	dfn   { /* Defining instance */
		font-weight: bolder;
	}
	a > i { /* Instance of term */
		font-style: normal;
	}
	dt dfn code, code.idl {
		font-size: inherit;
	}
	dfn var {
		font-style: normal;
	}

/** Change Marking ************************************************************/

	del {
		color: #aa0000;
		color: var(--del-text);
		background: transparent;
		background: var(--del-bg);
		text-decoration: line-through;
	}
	ins {
		color: #006100;
		color: var(--ins-text);
		background: transparent;
		background: var(--ins-bg);
		text-decoration: underline;
	}

	/* for amendments (candidate/proposed changes) */

	.amendment ins, .correction ins, .addition ins,
	ins[class^=c] {
		text-decoration-style: dotted;
	}
	.amendment del, .correction del, .addition del,
	del[class^=c] {
		text-decoration-style: dotted;
	}
	.amendment.proposed ins, .correction.proposed ins, .addition.proposed ins,
	ins[class^=c].proposed {
		text-decoration-style: double;
	}
	.amendment.proposed del, .correction.proposed del, .addition.proposed del,
	del[class^=c].proposed {
		text-decoration-style: double;
	}

/** Miscellaneous improvements to inline formatting ***************************/

	sup {
		vertical-align: super;
		font-size: 80%
	}

/******************************************************************************/
/*                                    Code                                    */
/******************************************************************************/

/** General monospace/pre rules ***********************************************/

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		font-family: Menlo, Consolas, "DejaVu Sans Mono", Monaco, monospace;
		font-size: .9em;
		hyphens: none;
		text-transform: none;
		text-align: left;
		text-align: start;
		font-variant: normal;
		orphans: 3;
		widows: 3;
		page-break-before: avoid;
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	pre code,
	code code {
		font-size: 100%;
	}

	pre {
		margin-top: 1em;
		margin-bottom: 1em;
		overflow: auto;
	}

/** Inline Code fragments *****************************************************/

	/* Do something nice. */

/******************************************************************************/
/*                                    Links                                   */
/******************************************************************************/

/** General Hyperlinks ********************************************************/

	/* We hyperlink a lot, so make it less intrusive */
	a[href] {
		color: #034575;
		color: var(--a-normal-text);
		text-decoration: underline #707070;
		text-decoration: underline var(--a-normal-underline);
		text-decoration-skip-ink: none;
	}
	a:visited {
		color: #034575;
		color: var(--a-visited-text);
		text-decoration-color: #bbb;
		text-decoration-color: var(--a-visited-underline);
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	/* Indicate interaction with the link */
	a[href]:focus,
	a[href]:hover {
		text-decoration-thickness: 2px;
	}
	a[href]:active {
		color: #c00;
		color: var(--a-active-text);
		text-decoration-color: #c00;
		text-decoration-color: var(--a-active-underline);
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	/* Backout above styling for W3C logo */
	.head .logo,
	.head .logo a {
		border: none;
		text-decoration: none;
		background: transparent;
	}

/******************************************************************************/
/*                                    Images                                  */
/******************************************************************************/

	img {
		border-style: none;
	}

	img, svg {
		/* Intentionally not color-scheme aware. */
		background: white;
	}

	/* For autogen numbers, add
	  .caption::before, figcaption::before { content: "Figure " counter(figure) ". "; }
	*/

	figure, .figure, .sidefigure {
		page-break-inside: avoid;
		text-align: center;
		margin: 2.5em 0;
	}
	.figure img,	.sidefigure img,	figure img,
	.figure object, .sidefigure object, figure object {
		max-width: 100%;
		margin: auto;
		height: auto;
	}
	.figure pre, .sidefigure pre, figure pre {
		text-align: left;
		display: table;
		margin: 1em auto;
	}
	.figure table, figure table {
		margin: auto;
	}
	@media screen and (min-width: 20em) {
		.sidefigure {
			float: right;
			width: 50%;
			margin: 0 0 0.5em 0.5em;
		}
	}
	.caption, figcaption, caption {
		font-style: italic;
		font-size: 90%;
	}
	.caption::before, figcaption::before, figcaption > .marker {
		font-weight: bold;
	}
	.caption, figcaption {
		counter-increment: figure;
	}

	/* DL list is indented 2em, but figure inside it is not */
	dd > .figure, dd > figure { margin-left: -2em; }

/******************************************************************************/
/*                             Colored Boxes                                  */
/******************************************************************************/

	.issue, .note, .example, .assertion, .advisement, blockquote,
	.amendment, .correction, .addition {
		margin: 1em auto;
		padding: .5em;
		border: .5em;
		border-left-style: solid;
		page-break-inside: avoid;
	}
	span.issue, span.note {
		padding: .1em .5em .15em;
		border-right-style: solid;
	}

	blockquote > :first-child,
	.note  > p:first-child,
	.issue > p:first-child,
	.amendment > p:first-child,
	.correction > p:first-child,
	.addition > p:first-child {
		margin-top: 0;
	}
	blockquote > :last-child,
	.note  > p:last-child,
	.issue > p:last-child,
	.amendment > p:last-child,
	.correction > p:last-child,
	.addition > p:last-child {
		margin-bottom: 0;
	}


	.issue::before, .issue > .marker,
	.example::before, .example > .marker,
	.note::before, .note > .marker,
	details.note > summary > .marker,
	.amendment::before, .amendment > .marker,
	details.amendment > summary > .marker,
	.addition::before, .addition > .marker,
	addition.amendment > summary > .marker,
	.correction::before, .correction > .marker,
	correction.amendment > summary > .marker
	{
		text-transform: uppercase;
		padding-right: 1em;
	}

	.example::before, .example > .marker {
		display: block;
		padding-right: 0em;
	}

/** Blockquotes ***************************************************************/

	blockquote {
		border-color: silver;
		border-color: var(--blockquote-border);
		background: transparent;
		background: var(--blockquote-bg);
		color: currentcolor;
		color: var(--blockquote-text);
	}

/** Open issue ****************************************************************/

	.issue {
		border-color: #e05252;
		border-color: var(--issue-border);
		background: #fbe9e9;
		background: var(--issue-bg);
		color: black;
		color: var(--issue-text);
		counter-increment: issue;
		overflow: auto;
	}
	.issue::before, .issue > .marker {
		color: #831616;
		color: var(--issueheading-text);
	}
	/* Add .issue::before { content: "Issue " counter(issue) " "; } for autogen numbers,
	  or use class="marker" to mark up the issue number in source. */

/** Example *******************************************************************/

	.example {
		border-color: #e0cb52;
		border-color: var(--example-border);
		background: #fcfaee;
		background: var(--example-bg);
		color: black;
		color: var(--example-text);
		counter-increment: example;
		overflow: auto;
		clear: both;
	}
	.example::before, .example > .marker {
		color: #574b0f;
		color: var(--exampleheading-text);
	}
	/* Add .example::before { content: "Example " counter(example) " "; } for autogen numbers,
	  or use class="marker" to mark up the example number in source. */

/** Non-normative Note ********************************************************/

	.note {
		border-color: #52e052;
		border-color: var(--note-border);
		background: #e9fbe9;
		background: var(--note-bg);
		color: black;
		color: var(--note-text);
		overflow: auto;
	}

	.note::before, .note > .marker,
	details.note > summary {
		color: hsl(120, 70%, 30%);
		color: var(--noteheading-text);
	}
	/* Add .note::before { content: "Note "; } for autogen label,
	  or use class="marker" to mark up the label in source. */

	details.note[open] > summary {
		border-bottom: 1px silver solid;
		border-bottom: 1px var(--notesummary-underline) solid;
	}

/** Assertion Box *************************************************************/
	/*  for assertions in algorithms */

	.assertion {
		border-color: #AAA;
		border-color: var(--assertion-border);
		background: #EEE;
		background: var(--assertion-bg);
		color: black;
		color: var(--assertion-text);
	}

/** Advisement Box ************************************************************/
	/*  for attention-grabbing normative statements */

	.advisement {
		border-color: orange;
		border-color: var(--advisement-border);
		border-style: none solid;
		background: #fec;
		background: var(--advisement-bg);
		color: black;
		color: var(--advisement-text);
	}
	strong.advisement {
		display: block;
		text-align: center;
	}
	.advisement::before, .advisement > .marker {
		color: #b35f00;
		color: var(--advisementheading-text);
	}

/** Amendment Box *************************************************************/

	.amendment, .correction, .addition {
		border-color: #330099;
		border-color: var(--amendment-border);
		background: #F5F0FF;
		background: var(--amendment-bg);
		color: black;
		color: var(--amendment-text);
	}
	.amendment.proposed, .correction.proposed, .addition.proposed {
		border-style: solid;
		border-block-width: 0.25em;
	}
	.amendment::before, .amendment > .marker,
	details.amendment > summary::before, details.amendment > summary > .marker,
	.correction::before, .correction > .marker,
	details.correction > summary::before, details.correction > summary > .marker,
	.addition::before, .addition > .marker,
	details.addition > summary::before, details.addition > summary > .marker {
		color: #220066;
		color: var(--amendmentheading-text);
	}
	.amendment.proposed::before, .amendment.proposed > .marker,
	details.amendment.proposed > summary::before, details.amendment.proposed > summary > .marker,
	.correction.proposed::before, .correction.proposed > .marker,
	details.correction.proposed > summary::before, details.correction.proposed > summary > .marker,
	.addition.proposed::before, .addition.proposed > .marker,
	details.addition.proposed > summary::before, details.addition.proposed > summary > .marker {
		font-weight: bold;
	}

/** Spec Obsoletion Notice ****************************************************/
	/* obnoxious obsoletion notice for older/abandoned specs. */

	details {
		display: block;
	}
	summary {
		font-weight: bolder;
	}

	.annoying-warning:not(details),
	details.annoying-warning:not([open]) > summary,
	details.annoying-warning[open] {
		background: hsla(40,100%,50%,0.95);
		background: var(--warning-bg);
		color: black;
		color: var(--warning-text);
		padding: .75em 1em;
		border: red;
		border: var(--warning-border);
		border-style: solid none;
		box-shadow: 0 2px 8px black;
		text-align: center;
	}
	.annoying-warning :last-child {
		margin-bottom: 0;
	}

@media not print {
	details.annoying-warning[open] {
		position: fixed;
		left: 0;
		right: 0;
		bottom: 2em;
		z-index: 1000;
	}
}

	details.annoying-warning:not([open]) > summary {
		text-align: center;
	}

/** Entity Definition Boxes ***************************************************/

	.def {
		padding: .5em 1em;
		background: #def;
		background: var(--def-bg);
		margin: 1.2em 0;
		border-left: 0.5em solid #8ccbf2;
		border-left: 0.5em solid var(--def-border);
		color: black;
		color: var(--def-text);
	}

/******************************************************************************/
/*                                    Tables                                  */
/******************************************************************************/

	th, td {
		text-align: left;
		text-align: start;
	}

/** Property/Descriptor Definition Tables *************************************/

	table.def {
		/* inherits .def box styling, see above */
		width: 100%;
		border-spacing: 0;
	}

	table.def td,
	table.def th {
		padding: 0.5em;
		vertical-align: baseline;
		border-bottom: 1px solid #bbd7e9;
		border-bottom: 1px solid var(--defrow-border);
	}

	table.def > tbody > tr:last-child th,
	table.def > tbody > tr:last-child td {
		border-bottom: 0;
	}

	table.def th {
		font-style: italic;
		font-weight: normal;
		padding-left: 1em;
		width: 3em;
	}

	/* For when values are extra-complex and need formatting for readability */
	table td.pre {
		white-space: pre-wrap;
	}

	/* A footnote at the bottom of a def table */
	table.def td.footnote {
		padding-top: 0.6em;
	}
	table.def td.footnote::before {
		content: " ";
		display: block;
		height: 0.6em;
		width: 4em;
		border-top: thin solid;
	}

/** Data tables (and properly marked-up index tables) *************************/
	/*
		<table class="data"> highlights structural relationships in a table
		when correct markup is used (e.g. thead/tbody, th vs. td, scope attribute)

		Use class="complex data" for particularly complicated tables --
		(This will draw more lines: busier, but clearer.)

		Use class="long" on table cells with paragraph-like contents
		(This will adjust text alignment accordingly.)
		Alternately use class="longlastcol" on tables, to have the last column assume "long".
	*/

	table {
		word-wrap: normal;
		overflow-wrap: normal;
		hyphens: manual;
	}

	table.data,
	table.index {
		margin: 1em auto;
		border-collapse: collapse;
		border: hidden;
		width: 100%;
	}
	table.data caption,
	table.index caption {
		max-width: 50em;
		margin: 0 auto 1em;
	}

	table.data td,  table.data th,
	table.index td, table.index th {
		padding: 0.5em 1em;
		border-width: 1px;
		border-color: silver;
		border-color: var(--datacell-border);
		border-top-style: solid;
	}

	table.data thead td:empty {
		padding: 0;
		border: 0;
	}

	table.data  thead,
	table.index thead,
	table.data  tbody,
	table.index tbody {
		border-bottom: 2px solid;
	}

	table.data colgroup,
	table.index colgroup {
		border-left: 2px solid;
	}

	table.data  tbody th:first-child,
	table.index tbody th:first-child  {
		border-right: 2px solid;
		border-top: 1px solid silver;
		border-top: 1px solid var(--datacell-border);
		padding-right: 1em;
	}

	table.data th[colspan],
	table.data td[colspan] {
		text-align: center;
	}

	table.complex.data th,
	table.complex.data td {
		border: 1px solid silver;
		border: 1px solid var(--datacell-border);
		text-align: center;
	}

	table.data.longlastcol td:last-child,
	table.data td.long {
		vertical-align: baseline;
		text-align: left;
	}

	table.data img {
		vertical-align: middle;
	}


/*
Alternate table alignment rules

	table.data,
	table.index {
		text-align: center;
	}

	table.data  thead th[scope="row"],
	table.index thead th[scope="row"] {
		text-align: right;
	}

	table.data  tbody th:first-child,
	table.index tbody th:first-child  {
		text-align: right;
	}

Possible extra rowspan handling

	table.data  tbody th[rowspan]:not([rowspan='1']),
	table.index tbody th[rowspan]:not([rowspan='1']),
	table.data  tbody td[rowspan]:not([rowspan='1']),
	table.index tbody td[rowspan]:not([rowspan='1']) {
		border-left: 1px solid silver;
	}

	table.data  tbody th[rowspan]:first-child,
	table.index tbody th[rowspan]:first-child,
	table.data  tbody td[rowspan]:first-child,
	table.index tbody td[rowspan]:first-child{
		border-left: 0;
		border-right: 1px solid silver;
	}
*/

/******************************************************************************/
/*                                  Indices                                   */
/******************************************************************************/


/** Table of Contents *********************************************************/

	.toc a {
		/* More spacing; use padding to make it part of the click target. */
		padding: 0.1rem 1px 0;
		/* Larger, more consistently-sized click target */
		display: block;
		/* Switch to using border-bottom for underlines */
		text-decoration: none;
		border-bottom: 1px solid;
		/* Reverse color scheme */
		color: black;
		color: var(--toclink-text);
		border-color: #3980b5;
		border-color: var(--toclink-underline);
	}
	.toc a:visited {
		color: black;
		color: var(--toclink-visited-text);
		border-color: #054572;
		border-color: var(--toclink-visited-underline);
	}
	.toc a:focus,
	.toc a:hover {
		background: rgba(75%, 75%, 75%, .25);
		background: var(--a-hover-bg);
		border-bottom-width: 3px;
		margin-bottom: -2px;
	}
	.toc a:not(:focus):not(:hover) {
		/* Allow colors to cascade through from link styling */
		border-bottom-color: transparent;
	}

	.toc, .toc ol, .toc ul, .toc li {
		list-style: none; /* Numbers must be inlined into source */
		/* because generated content isn't search/selectable and markers can't do multilevel yet */
		margin:  0;
		padding: 0;
	}
	.toc {
		line-height: 1.1em;
	}

	/* ToC not indented until third level, but font style & margins show hierarchy */
	.toc > li			{ font-weight: bold;   }
	.toc > li li		 { font-weight: normal; }
	.toc > li li li	  { font-size:   95%;	}
	.toc > li li li li	{ font-size:   90%;	}
	.toc > li li li li li { font-size:   85%;	}

	/* @supports not (display:grid) { */
		.toc > li			{ margin: 1.5rem 0;	}
		.toc > li li		 { margin: 0.3rem 0;	}
		.toc > li li li	  { margin-left: 2rem;   }

		/* Section numbers in a column of their own */
		.toc .secno {
			float: left;
			width: 4rem;
			white-space: nowrap;
		}
		.toc > li li li li .secno { font-size: 85%; }
		.toc > li li li li li .secno { font-size: 100%; }

		.toc li {
			clear: both;
		}

		:not(li) > .toc			 { margin-left:  5rem; }
		.toc .secno				 { margin-left: -5rem; }
		.toc > li li li .secno	  { margin-left: -7rem; }
		.toc > li li li li .secno	{ margin-left: -9rem; }
		.toc > li li li li li .secno { margin-left: -11rem; }

		/* Tighten up indentation in narrow ToCs */
		@media (max-width: 30em) {
			:not(li) > .toc			 { margin-left:  4rem; }
			.toc .secno				 { margin-left: -4rem; }
			.toc > li li li			 { margin-left:  1rem; }
			.toc > li li li .secno	  { margin-left: -5rem; }
			.toc > li li li li .secno	{ margin-left: -6rem; }
			.toc > li li li li li .secno { margin-left: -7rem; }
		}
		/* Loosen it on wide screens */
		@media screen and (min-width: 78em) {
			body:not(.toc-inline) :not(li) > .toc			 { margin-left:  4rem; }
			body:not(.toc-inline) .toc .secno				 { margin-left: -4rem; }
			body:not(.toc-inline) .toc > li li li			 { margin-left:  1rem; }
			body:not(.toc-inline) .toc > li li li .secno	  { margin-left: -5rem; }
			body:not(.toc-inline) .toc > li li li li .secno	{ margin-left: -6rem; }
			body:not(.toc-inline) .toc > li li li li li .secno { margin-left: -7rem; }
	}
	/* } */

	@supports (display:grid) and (display:contents) {
		/* Use #toc over .toc to override non-@supports rules. */
		#toc {
			display: grid;
			align-content: start;
			grid-template-columns: auto 1fr;
			grid-column-gap: 1rem;
			column-gap: 1rem;
			grid-row-gap: .6rem;
			row-gap: .6rem;
		}
		#toc h2 {
			grid-column: 1 / -1;
			margin-bottom: 0;
		}
		#toc ol,
		#toc li,
		#toc a {
			display: contents;
			/* Switch <a> to subgrid when supported */
		}
		#toc span {
			margin: 0;
		}
		#toc > .toc > li > a > span {
			/* The spans of the top-level list,
			  comprising the first items of each top-level section. */
			margin-top: 1.1rem;
		}
		#toc#toc .secno { /* Ugh, need more specificity to override base.css */
			grid-column: 1;
			width: auto;
			margin-left: 0;
		}
		#toc .content {
			grid-column: 2;
			width: auto;
			margin-right: 1rem;
			border-bottom: 3px solid transparent;
			margin-bottom: -3px;
		}
		#toc .content:hover,
		#toc .content:focus {
			background: rgba(75%, 75%, 75%, .25);
			background: var(--a-hover-bg);
			border-bottom-color: #054572;
			border-bottom-color: var(--toclink-underline);
		}
		#toc li li li .content {
			margin-left: 1rem;
		}
		#toc li li li li .content {
			margin-left: 2rem;
		}
	}


/** Index *********************************************************************/

	/* Index Lists: Layout */
	ul.index	  { margin-left: 0; columns: 15em; text-indent: 1em hanging; }
	ul.index li	{ margin-left: 0; list-style: none; break-inside: avoid; }
	ul.index li li { margin-left: 1em; }
	ul.index dl	{ margin-top: 0; }
	ul.index dt	{ margin: .2em 0 .2em 20px;}
	ul.index dd	{ margin: .2em 0 .2em 40px;}
	/* Index Lists: Typography */
	ul.index ul,
	ul.index dl { font-size: smaller; }
	@media not print {
		ul.index li a + span {
			white-space: nowrap;
			color: transparent; }
		ul.index li a:hover + span,
		ul.index li a:focus + span {
			color: #707070;
			color: var(--indexinfo-text);
		}
	}

/** Index Tables *****************************************************/
	/* See also the data table styling section, which this effectively subclasses */

	table.index {
		font-size: small;
		border-collapse: collapse;
		border-spacing: 0;
		text-align: left;
		margin: 1em 0;
	}

	table.index td,
	table.index th {
		padding: 0.4em;
	}

	table.index tr:hover td:not([rowspan]),
	table.index tr:hover th:not([rowspan]) {
		color: black;
		color: var(--indextable-hover-text);
		background: #f7f8f9;
		background: var(--indextable-hover-bg);
	}

	/* The link in the first column in the property table (formerly a TD) */
	table.index th:first-child a {
		font-weight: bold;
	}

/** Outdated warning **********************************************************/

.outdated-spec {
	color: black;
	color: var(--outdatedspec-text);
	background-color: rgba(0,0,0,0.5);
	background-color: var(--outdatedspec-bg);
}

.outdated-warning {
	position: fixed;
	bottom: 50%;
	left: 0;
	right: 0;
	margin: 0 auto;
	width: 50%;
	background: maroon;
	background: var(--outdated-bg);
	color: white;
	color: var(--outdated-text);
	border-radius: 1em;
	box-shadow: 0 0 1em red;
	box-shadow: 0 0 1em var(--outdated-shadow);
	padding: 2em;
	text-align: center;
	z-index: 2;
}

.outdated-warning a {
	color: currentcolor;
	background: transparent;
}

.edited-rec-warning {
	background: darkorange;
	background: var(--editedrec-bg);
	box-shadow: 0 0 1em;
}

.outdated-warning button {
	color: var(--outdated-text);
	border-radius: 1em;
	box-shadow: 0 0 1em red;
	box-shadow: 0 0 1em var(--outdated-shadow);
	padding: 2em;
	text-align: center;
	z-index: 2;
}

.outdated-warning a {
	color: currentcolor;
	background: transparent;
}

.edited-rec-warning {
	background: darkorange;
	background: var(--editedrec-bg);
	box-shadow: 0 0 1em;
}

.outdated-warning button {
	position: absolute;
	top: 0;
	right:0;
	margin: 0;
	border: 0;
	padding: 0.25em 0.5em;
	background: transparent;
	color: white;
	color: var(--outdated-text);
	font:1em sans-serif;
	text-align:center;
}

.outdated-warning span {
	display: block;
}

.outdated-collapsed {
	bottom: 0;
	border-radius: 0;
	width: 100%;
	padding: 0;
}

/******************************************************************************/
/*                                    Print                                   */
/******************************************************************************/

	@media print {
		/* Pages have their own margins. */
		html {
			margin: 0;
		}
		/* Serif for print. */
		body {
			font-family: serif;
		}

		.outdated-warning {
			position: absolute;
			border-style: solid;
			border-color: red;
		}

		.outdated-warning input {
			display: none;
		}
	}
	@page {
		margin: 1.5cm 1.1cm;
	}



/******************************************************************************/
/*                             Overflow Control                               */
/******************************************************************************/

	.figure .caption, .sidefigure .caption, figcaption {
		/* in case figure is overlarge, limit caption to 50em */
		max-width: 50rem;
		margin-left: auto;
		margin-right: auto;
	}
	.overlarge {
		/* Magic to create good item positioning:
		  "content column" is 50ems wide at max; less on smaller screens.
		  Extra space (after ToC + content) is empty on the right.

		  1. When item < content column, centers item in column.
		  2. When content < item < available, left-aligns.
		  3. When item > available, fills available + scroll bar.
		*/
		display: grid;
		grid-template-columns: minmax(0, 50em);
	}
	.overlarge > table {
		/* limit preferred width of table */
		max-width: 50em;
		margin-left: auto;
		margin-right: auto;
	}

	@media (min-width: 55em) {
		.overlarge {
			margin-right: calc(13px + 26.5rem - 50vw);
			max-width: none;
		}
	}
	@media screen and (min-width: 78em) {
		body:not(.toc-inline) .overlarge {
			/* 30.5em body padding 50em content area */
			margin-right: calc(40em - 50vw) !important;
		}
	}
	@media screen and (min-width: 90em) {
		body:not(.toc-inline) .overlarge {
			/* 4em html margin 30.5em body padding 50em content area */
			margin-right: calc(84.5em - 100vw) !important;
		}
	}

	@media not print {
		.overlarge {
			overflow-x: auto;
			/* See Lea Verou's explanation background-attachment:
			* http://lea.verou.me/2012/04/background-attachment-local/
			*
			background: top left  / 4em 100% linear-gradient(to right,  #ffffff, rgba(255, 255, 255, 0)) local,
						top right / 4em 100% linear-gradient(to left, #ffffff, rgba(255, 255, 255, 0)) local,
						top left  / 1em 100% linear-gradient(to right,  #c3c3c5, rgba(195, 195, 197, 0)) scroll,
						top right / 1em 100% linear-gradient(to left, #c3c3c5, rgba(195, 195, 197, 0)) scroll,
						white;
			background-repeat: no-repeat;
			*/
		}
	}
</style>
<style>
    table, th, td {
      border: 1px solid black;
      border-collapse: collapse;
      vertical-align: top;
    }
    th, td {
      border-left: none;
      border-right: none;
      padding: 0px 10px;
    }
    th {
      text-align: center;
    }

    del { background: #fcc; color: #000; text-decoration: line-through; }
    ins { background: #cfc; color: #000; }
    blockquote .highlight:not(.idl) { background: initial; margin: initial; padding: 0.5em }
    blockquote ul { background: inherit; }
    blockquote code.highlight:not(.idl) { padding: initial; }
    blockquote c-[a] { color: inherit; } /* Keyword.Declaration */
    blockquote c-[b] { color: inherit; } /* Keyword.Type */
    blockquote c-[c] { color: inherit; } /* Comment */
    blockquote c-[d] { color: inherit; } /* Comment.Multiline */
    blockquote c-[e] { color: inherit; } /* Name.Attribute */
    blockquote c-[f] { color: inherit; } /* Name.Tag */
    blockquote c-[g] { color: inherit; } /* Name.Variable */
    blockquote c-[k] { color: inherit; } /* Keyword */
    blockquote c-[l] { color: inherit; } /* Literal */
    blockquote c-[m] { color: inherit; } /* Literal.Number */
    blockquote c-[n] { color: inherit; } /* Name */
    blockquote c-[o] { color: inherit; } /* Operator */
    blockquote c-[p] { color: inherit; } /* Punctuation */
    blockquote c-[s] { color: inherit; } /* Literal.String */
    blockquote c-[t] { color: inherit; } /* Literal.String.Single */
    blockquote c-[u] { color: inherit; } /* Literal.String.Double */
    blockquote c-[cp] { color: inherit; } /* Comment.Preproc */
    blockquote c-[c1] { color: inherit; } /* Comment.Single */
    blockquote c-[cs] { color: inherit; } /* Comment.Special */
    blockquote c-[kc] { color: inherit; } /* Keyword.Constant */
    blockquote c-[kn] { color: inherit; } /* Keyword.Namespace */
    blockquote c-[kp] { color: inherit; } /* Keyword.Pseudo */
    blockquote c-[kr] { color: inherit; } /* Keyword.Reserved */
    blockquote c-[ld] { color: inherit; } /* Literal.Date */
    blockquote c-[nc] { color: inherit; } /* Name.Class */
    blockquote c-[no] { color: inherit; } /* Name.Constant */
    blockquote c-[nd] { color: inherit; } /* Name.Decorator */
    blockquote c-[ni] { color: inherit; } /* Name.Entity */
    blockquote c-[ne] { color: inherit; } /* Name.Exception */
    blockquote c-[nf] { color: inherit; } /* Name.Function */
    blockquote c-[nl] { color: inherit; } /* Name.Label */
    blockquote c-[nn] { color: inherit; } /* Name.Namespace */
    blockquote c-[py] { color: inherit; } /* Name.Property */
    blockquote c-[ow] { color: inherit; } /* Operator.Word */
    blockquote c-[mb] { color: inherit; } /* Literal.Number.Bin */
    blockquote c-[mf] { color: inherit; } /* Literal.Number.Float */
    blockquote c-[mh] { color: inherit; } /* Literal.Number.Hex */
    blockquote c-[mi] { color: inherit; } /* Literal.Number.Integer */
    blockquote c-[mo] { color: inherit; } /* Literal.Number.Oct */
    blockquote c-[sb] { color: inherit; } /* Literal.String.Backtick */
    blockquote c-[sc] { color: inherit; } /* Literal.String.Char */
    blockquote c-[sd] { color: inherit; } /* Literal.String.Doc */
    blockquote c-[se] { color: inherit; } /* Literal.String.Escape */
    blockquote c-[sh] { color: inherit; } /* Literal.String.Heredoc */
    blockquote c-[si] { color: inherit; } /* Literal.String.Interpol */
    blockquote c-[sx] { color: inherit; } /* Literal.String.Other */
    blockquote c-[sr] { color: inherit; } /* Literal.String.Regex */
    blockquote c-[ss] { color: inherit; } /* Literal.String.Symbol */
    blockquote c-[vc] { color: inherit; } /* Name.Variable.Class */
    blockquote c-[vg] { color: inherit; } /* Name.Variable.Global */
    blockquote c-[vi] { color: inherit; } /* Name.Variable.Instance */
    blockquote c-[il] { color: inherit; } /* Literal.Number.Integer.Long */
  </style>
  <meta content="Bikeshed version d5d58a306, updated Fri Jan 26 16:12:28 2024 -0800" name="generator">
  <link href="https://wg21.link/P3159" rel="canonical">
  <link href="https://isocpp.org/favicon.ico" rel="icon">
  <meta content="03b31276f7f69f6c75101d8fc5bea4f2eb8591c3" name="revision">
<style>
table, th, tr, td {
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 <body class="h-entry">
  <div class="head">
   <p data-fill-with="logo"></p>
   <h1 class="p-name no-ref" id="title">P3159R0<br>C++ Range Adaptors and Parallel Algorithms</h1>
   <h2 class="no-num no-toc no-ref heading settled" id="profile-and-date"><span class="content">Published Proposal, <time class="dt-updated" datetime="2024-03-17">2024-03-17</time></span></h2>
   <div data-fill-with="spec-metadata">
    <dl>
     <dt class="editor">Author:
     <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:brycelelbach@gmail.com">Bryce Adelstein Lelbach (he/him/his)</a> (<span class="p-org org">NVIDIA</span>)
     <dt>Source:
     <dd><a href="https://github.com/brycelelbach/wg21_p3159_cpp_range_adaptors_and_parallel_algorihtms/blob/main/cpp_asynchronous_parallel_algorithms.bs">GitHub</a>
     <dt>Issue Tracking:
     <dd><a href="https://github.com/brycelelbach/wg21_p3159_cpp_range_adaptors_and_parallel_algorihtms/issues">GitHub</a>
     <dt>Project:
     <dd>ISO/IEC 14882 Programming Languages — C++, ISO/IEC JTC1/SC22/WG21
     <dt>Audience:
     <dd>WG21
    </dl>
   </div>
   <div data-fill-with="warning"></div>
   <hr title="Separator for header">
  </div>
  <nav data-fill-with="table-of-contents" id="toc">
   <h2 class="no-num no-toc no-ref" id="contents">Table of Contents</h2>
   <ol class="toc" role="directory">
    <li><a href="#introduction"><span class="secno">1</span> <span class="content">Introduction</span></a>
    <li>
     <a href="#optimizations"><span class="secno">2</span> <span class="content">Optimizations</span></a>
     <ol class="toc">
      <li><a href="#non-trivial-removal"><span class="secno">2.1</span> <span class="content">Non-Trivial Removal</span></a>
      <li><a href="#non-trivial-grouping"><span class="secno">2.2</span> <span class="content">Non-Trivial Grouping</span></a>
      <li><a href="#trivial-removal-and-grouping"><span class="secno">2.3</span> <span class="content">Trivial Removal and Grouping</span></a>
     </ol>
    <li><a href="#cheatsheet"><span class="secno">3</span> <span class="content">Cheatsheet</span></a>
   </ol>
  </nav>
  <main>
   <h2 class="heading settled" data-level="1" id="introduction"><span class="secno">1. </span><span class="content">Introduction</span><a class="self-link" href="#introduction"></a></h2>
   <p>Range factories and adaptors are a powerful compositional tool that allow users to elegantly express non-modifying transformation, filtering, grouping, and reshaping of a range without materializing the changes in global memory. Adaptors and factories produce range views, which are non-owning and lazily-evaluated ranges. Due to their lazy nature, parallelizing views presents challenges. Views apply their transformative logic one element at a time on-demand. But, for parallelization, we need to extract that transformative logic and apply it in bulk across the entire underlying range.</p>
   <p>To implement parallel algorithms that consume ranges, we must build a library-only optimizing kernel builder that recursively decomposes arbitrary ranges (and iterators to them) into the pipeline of adaptors and factories that constructed them. This kernel builder may then substitute each adaptor and factory for a parallel-friendly alternative and/or insert in-place preprocessing passes as needed.</p>
<pre class="highlight"><c- k>auto</c-> <c- n>optimize_range</c-><c- p>(</c-><c- n>range</c-> <c- k>auto</c-><c- o>&amp;&amp;</c-> <c- n>rng</c-><c- p>)</c-> <c- p>{</c->
  <c- k>if</c-> <c- k>constexpr</c-><c- p>(</c-><c- n>has_base</c-><c- p>(</c-><c- n>rng</c-><c- p>))</c-> <c- p>{</c->
    <c- c1>// Dispatch to logic that optimizes this adaptor.</c->
    <c- k>return</c-> <c- n>optimize_range</c-><c- p>(</c-><c- n>rng</c-><c- p>.</c-><c- n>base</c-><c- p>());</c->
  <c- p>}</c-> <c- k>else</c-> <c- p>{</c->
    <c- k>return</c-> <c- n>rng</c-><c- p>;</c->
  <c- p>}</c->
<c- p>}</c->
</pre>
   <p>If we restrict ourselves to the closed set of adaptors and factories in the C++ standard library, then for any given range:</p>
   <ul>
    <li data-md>
     <p>We can determine if it came from an adaptor or factory.</p>
    <li data-md>
     <p>We can determine what adaptor or factory it came from.</p>
    <li data-md>
     <p>We can get the base ranges (if any) from it.</p>
    <li data-md>
     <p>We can get any of the user-provided parameters to the adaptor or factory that we may need.</p>
   </ul>
   <h2 class="heading settled" data-level="2" id="optimizations"><span class="secno">2. </span><span class="content">Optimizations</span><a class="self-link" href="#optimizations"></a></h2>
   <h3 class="heading settled" data-level="2.1" id="non-trivial-removal"><span class="secno">2.1. </span><span class="content">Non-Trivial Removal</span><a class="self-link" href="#non-trivial-removal"></a></h3>
   <p>Certain range adaptors remove elements in a way that cannot be trivially computed a priori (ex: <code class="highlight"><c- n>filter</c-></code>, <code class="highlight"><c- n>take_while</c-></code>).</p>
   <p>Parallel algorithm implementations may distribute the input among threads a priori, creating N execution agents, each of which will process M initial elements. Since we cannot compute which elements will be removed during this distribution, each execution agent will need tostart with M initial elements and later discard any of those elements that meet the removal criterion.</p>
   <p>When consuming these removing adaptors in parallel, they need to be substituted into range adaptors that instead replace those elements with tombstones. A tombstone is an element of a range that represents non-existent data that needs to be removed from the range. A tombstone is represented with an <code class="highlight"><c- n>optional</c-></code>-like object.</p>
   <p>The tombstones must be removed when the range is consumed by a parallel algorithm. The simplest approach is to insert a <code class="highlight"><c- n>copy_if</c-></code> preprocessing pass. This must be done in-place to avoid materializing the adapted range in memory. Such a pass is known as stream compaction</p>
<pre class="highlight"><c- n>for_each</c-><c- p>(</c-><c- n>rng</c-> <c- o>|</c-> <c- n>filter</c-><c- p>(</c-><c- n>f</c-><c- p>),</c-> <c- n>g</c-><c- p>);</c->
</pre>
<pre class="highlight"><c- b>void</c-> <c- nf>kernel</c-><c- p>(</c-><c- n>range</c-> <c- k>auto</c-><c- o>&amp;&amp;</c-> <c- n>rng0</c-><c- p>)</c-> <c- p>{</c->
  <c- n>rng1</c-> <c- o>=</c-> <c- n>compact</c-><c- p>(</c-><c- n>rng0</c-> <c- o>|</c-> <c- n>filter_tombstone</c-><c- p>(</c-><c- n>f</c-><c- p>));</c->
  <c- n>for_each_collective</c-><c- p>(</c-><c- n>rng1</c-><c- p>);</c->
<c- p>}</c->
</pre>
   <p>Stream compaction is often implemented with a scan.</p>
<pre class="highlight"><c- k>auto</c-> <c- n>copy_if</c-><c- p>(</c-><c- n>range</c-> <c- k>auto</c-><c- o>&amp;&amp;</c-> <c- n>in</c-><c- p>,</c-> <c- n>output_iterator</c-> <c- k>auto</c-> <c- n>out</c-><c- p>,</c-> <c- k>auto</c-> <c- n>pred</c-><c- p>)</c-> <c- p>{</c->
  <c- n>vector</c-><c- o>&lt;</c-><c- b>uint8_t</c-><c- o>></c-> <c- n>flags</c-><c- p>(</c-><c- n>size</c-><c- p>(</c-><c- n>in</c-><c- p>));</c->

  <c- n>transform</c-><c- p>(</c-><c- n>par</c-><c- p>,</c-> <c- n>in</c-><c- p>,</c-> <c- n>begin</c-><c- p>(</c-><c- n>flags</c-><c- p>),</c-> <c- n>pred</c-><c- p>);</c->

  <c- n>vector</c-><c- o>&lt;</c-><c- b>size_t</c-><c- o>></c-> <c- n>indices</c-><c- p>(</c-><c- n>size</c-><c- p>(</c-><c- n>in</c-><c- p>));</c->

  <c- n>exclusive_scan</c-><c- p>(</c-><c- n>par</c-><c- p>,</c-> <c- n>flags</c-><c- p>,</c-> <c- n>begin</c-><c- p>(</c-><c- n>indices</c-><c- p>),</c-> <c- mi>0</c-><c- p>);</c->

  <c- n>for_each</c-><c- p>(</c-><c- n>par</c-><c- p>,</c-> <c- n>zip</c-><c- p>(</c-><c- n>in</c-><c- p>,</c-> <c- n>flags</c-><c- p>,</c-> <c- n>indices</c-><c- p>),</c->
    <c- n>apply</c-><c- p>([</c-><c- o>&amp;</c-><c- n>in</c-><c- p>]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>e</c-><c- p>,</c-> <c- k>auto</c-> <c- n>flag</c-><c- p>,</c-> <c- k>auto</c-> <c- n>index</c-><c- p>])</c-> <c- p>{</c->
      <c- k>if</c-> <c- p>(</c-><c- n>flag</c-><c- p>)</c-> <c- n>out</c-><c- p>[</c-><c- n>index</c-><c- p>]</c-> <c- o>=</c-> <c- n>e</c-><c- p>;</c->
    <c- p>}));</c->

  <c- k>return</c-> <c- n>subrange</c-><c- p>(</c-><c- n>begin</c-><c- p>(</c-><c- n>out</c-><c- p>),</c-> <c- n>next</c-><c- p>(</c-><c- n>out</c-><c- p>,</c-> <c- n>indices</c-><c- p>.</c-><c- n>back</c-><c- p>()));</c->
<c- p>}</c->
</pre>
   <p>An alternative approach is to defer removal of the tombstones, instead wrapping all subsequent adaptors, user-provided operations, and the parallel algorithm implementation to ignore the tombstones. In some cases, it will never be necessary to insert a stream compaction pass, because everything can simply be wrapped:</p>
<pre class="highlight"><c- n>for_each</c-><c- p>(</c-><c- n>rng</c-> <c- o>|</c-> <c- n>filter</c-><c- p>(</c-><c- n>f</c-><c- p>),</c-> <c- n>g</c-><c- p>);</c->
</pre>
<pre class="highlight"><c- n>for_each</c-><c- p>(</c-><c- n>rng</c-> <c- o>|</c-> <c- n>transform</c-><c- p>([</c-><c- n>f</c-><c- p>]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>x</c-><c- p>)</c-> <c- p>{</c->
                  <c- k>if</c-> <c- p>(</c-><c- o>!</c-><c- n>f</c-><c- p>(</c-><c- n>x</c-><c- p>))</c-> <c- k>return</c-> <c- n>tombstone</c-><c- p>(</c-><c- n>x</c-><c- p>);</c-> <c- k>else</c-> <c- k>return</c-> <c- n>nullopt</c-><c- p>;</c->
               <c- p>}),</c->
         <c- p>[</c-><c- n>g</c-><c- p>]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>x</c-><c- p>)</c-> <c- p>{</c-> <c- k>if</c-> <c- p>(</c-><c- n>x</c-><c- p>)</c-> <c- n>g</c-><c- p>(</c-><c- o>*</c-><c- n>x</c-><c- p>);</c-> <c- p>});</c->
</pre>
<pre class="highlight"><c- n>reduce</c-><c- p>(</c-><c- n>rng</c-> <c- o>|</c-> <c- n>filter</c-><c- p>(</c-><c- n>f</c-><c- p>),</c-> <c- n>g</c-><c- p>);</c->
</pre>
<pre class="highlight"><c- n>reduce</c-><c- p>(</c-><c- n>rng</c-> <c- o>|</c-> <c- n>transform</c-><c- p>([</c-><c- n>f</c-><c- p>]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>x</c-><c- p>)</c-> <c- p>{</c->
               <c- k>if</c-> <c- p>(</c-><c- o>!</c-><c- n>f</c-><c- p>(</c-><c- n>x</c-><c- p>))</c-> <c- k>return</c-> <c- n>tombstone</c-><c- p>(</c-><c- n>x</c-><c- p>);</c-> <c- k>else</c-> <c- k>return</c-> <c- n>nullopt</c-><c- p>;</c->
             <c- p>}),</c->
       <c- p>[</c-><c- n>g</c-><c- p>]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>l</c-><c- p>,</c-> <c- k>auto</c-> <c- n>r</c-><c- p>)</c-> <c- p>{</c->
         <c- k>if</c-> <c- p>(</c-><c- n>l</c-> <c- o>&amp;&amp;</c-> <c- n>r</c-><c- p>)</c-> <c- k>return</c-> <c- n>g</c-><c- p>(</c-><c- n>l</c-><c- p>,</c-> <c- n>r</c-><c- p>);</c->
         <c- k>else</c-> <c- k>if</c-> <c- p>(</c-><c- n>l</c-><c- p>)</c-> <c- k>return</c-> <c- n>l</c-><c- p>;</c->
         <c- k>else</c-> <c- k>if</c-> <c- p>(</c-><c- n>r</c-><c- p>)</c-> <c- k>return</c-> <c- n>r</c-><c- p>;</c->
         <c- k>else</c->        <c- k>return</c-> <c- p>{};</c->
       <c- p>});</c->
</pre>
   <p>However, not all adaptors and parallel algorithms can be wrapped to ignore tombstones. For example, some adaptors like <code class="highlight"><c- n>adjacent</c-></code> and <code class="highlight"><c- n>enumerate</c-></code> are position-aware, meaning they either access neighboring elements or care about the position of the element in the range. If tombstones are present, they will throw off the position logic. <code class="highlight"><c- n>adjacent</c-></code> needs to give you adjacent non-tombstone elements and <code class="highlight"><c- n>enumerate</c-></code> needs to not count the tombstones, which would require the sort of lazy filtering logic that we seek to avoid.</p>
   <p>To defer removal of tombstones, the optimizing kernel builder must keep track of whether the reconstructed range is tombstoned as it recurses through it. If it encounters an operation that requires tombstoning, it must mark the reconstructed range as tombstoned. If it encounters an operation that cannot be wrapped to ignore tombstones, then it must insert a stream compaction pass and mark the reconstructed range as untombstoned. Maintaining this state adds complexity. A pipeline may enter and exit the tombstone state multiple times.</p>
<pre class="highlight"><c- n>sort</c-><c- p>(</c-><c- n>rng</c-> <c- o>|</c-> <c- n>filter</c-><c- p>(</c-><c- n>f</c-><c- p>)</c-> <c- o>|</c-> <c- n>transform</c-><c- p>(</c-><c- n>g</c-><c- p>)</c-> <c- o>|</c-> <c- n>adjacent</c-><c- o>&lt;</c-><c- mi>2</c-><c- o>></c-> <c- o>|</c-> <c- n>filter</c-><c- p>(</c-><c- n>h</c-><c- p>)</c-> <c- o>|</c-> <c- n>transform</c-><c- p>(</c-><c- n>i</c-><c- p>));</c->
  <c- c1>// ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^   XXXXXXXXXXX   ^^^^^^^^^^^^^^^^^^^^^^^^</c->
  <c- c1>//           Tombstoned             Untombstoned         Tombstoned</c->
</pre>
<pre class="highlight"><c- b>void</c-> <c- nf>kernel</c-><c- p>(</c-><c- n>range</c-> <c- k>auto</c-><c- o>&amp;&amp;</c-> <c- n>rng0</c-><c- p>)</c-> <c- p>{</c->
  <c- n>rng1</c-> <c- o>=</c-> <c- n>compact</c-><c- p>(</c-><c- n>rng0</c-> <c- o>|</c-> <c- n>filter_tombstone</c-><c- p>(</c-><c- n>f</c-><c- p>)</c-> <c- o>|</c-> <c- n>transform_tombstone</c-><c- p>(</c-><c- n>g</c-><c- p>));</c->
  <c- n>rng2</c-> <c- o>=</c-> <c- n>compact</c-><c- p>(</c-><c- n>rng1</c-> <c- o>|</c-> <c- n>adjacent</c-><c- o>&lt;</c-><c- mi>2</c-><c- o>></c-> <c- o>|</c-> <c- n>filter_tombstone</c-><c- p>(</c-><c- n>h</c-><c- p>)</c-> <c- o>|</c-> <c- n>transform_tombstone</c-><c- p>(</c-><c- n>i</c-><c- p>));</c->
  <c- n>sort_collective</c-><c- p>(</c-><c- n>rng2</c-><c- p>);</c->
<c- p>}</c->
</pre>
   <p>Deferred removal of tombstones offers potential performance benefits by avoiding unnecessary or redundant stream compaction passes, at the cost of additional complexity. For the sake of simplicity, we propose to not implement deferred removal at this time, and instead immediately inserting the stream compaction preprocessing pass when encountering range adaptors that perform non-trivial removal.</p>
   <h3 class="heading settled" data-level="2.2" id="non-trivial-grouping"><span class="secno">2.2. </span><span class="content">Non-Trivial Grouping</span><a class="self-link" href="#non-trivial-grouping"></a></h3>
   <p>Certain range adaptors group or combine elements in a way that cannot be trivially computed a priori (ex: <code class="highlight"><c- n>chunk_by</c-></code>, <code class="highlight"><c- n>split</c-></code>). These adaptors reduce the size of the sequence. When considering how to parallelize them, we should think of them as both transforming and removing elements. A parallel implementation of these adaptors must do two things:</p>
   <ul>
    <li data-md>
     <p>Compute the groupings, which requires some form of summation.</p>
    <li data-md>
     <p>Discard whichever of the M initial elements per execution agent do not represent one of the groupings.</p>
   </ul>
   <p>The computation of the groupings can be implemented by inserting an in-place scan preprocessing pass.</p>
<pre class="highlight"><c- n>for_each</c-><c- p>(</c-><c- n>rng</c-> <c- o>|</c-> <c- n>chunk_by</c-><c- p>(</c-><c- n>f</c-><c- p>)</c-> <c- o>|</c-> <c- n>adjacent</c-><c- o>&lt;</c-><c- mi>2</c-><c- o>></c-><c- p>,</c-> <c- n>g</c-><c- p>);</c->
</pre>
<pre class="highlight"><c- b>void</c-> <c- nf>kernel</c-><c- p>(</c-><c- n>range</c-> <c- k>auto</c-><c- o>&amp;&amp;</c-> <c- n>rng0</c-><c- p>)</c-> <c- p>{</c->
  <c- n>rng1</c-> <c- o>=</c-> <c- n>grouping</c-><c- p>(</c-><c- n>rng0</c-><c- p>);</c->
  <c- n>for_each_collective</c-><c- p>(</c-><c- n>rng1</c-> <c- o>|</c-> <c- n>adjacent</c-><c- o>&lt;</c-><c- mi>2</c-><c- o>></c-><c- p>,</c-> <c- n>g</c-><c- p>);</c->
<c- p>}</c->
</pre>
   <p>As discussed in the Non-Trivial Removal section, the removal of elements can be done by inserting an in-place <code class="highlight"><c- n>copy_if</c-></code> preprocessing pass, and <code class="highlight"><c- n>copy_if</c-></code> can be implemented with a scan. These two operations can be combined into a single scan-based preprocessing pass that both computes the groupings and removes all but one element per grouping.</p>
<pre class="highlight"><c- k>auto</c-> <c- n>chunk_by</c-><c- p>(</c-><c- n>range</c-> <c- k>auto</c-><c- o>&amp;&amp;</c-> <c- n>in</c-><c- p>,</c-> <c- k>auto</c-> <c- n>out</c-><c- p>,</c-> <c- k>auto</c-> <c- n>pred</c-><c- p>)</c-> <c- p>{</c->
  <c- n>vector</c-><c- o>&lt;</c-><c- b>uint8_t</c-><c- o>></c-> <c- n>flags</c-><c- p>(</c-><c- n>size</c-><c- p>(</c-><c- n>in</c-><c- p>));</c->

  <c- n>transform</c-><c- p>(</c-><c- n>in</c-> <c- o>|</c-> <c- n>adjacent</c-><c- o>&lt;</c-><c- mi>2</c-><c- o>></c-><c- p>,</c-> <c- n>begin</c-><c- p>(</c-><c- n>flags</c-><c- p>),</c->
    <c- n>apply</c-><c- p>([</c-><c- o>&amp;</c-><c- p>]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>l</c-><c- p>,</c-> <c- k>auto</c-> <c- n>r</c-><c- p>)</c-> <c- p>{</c-> <c- k>return</c-> <c- n>pred</c-><c- p>(</c-><c- n>l</c-><c- p>,</c-> <c- n>r</c-><c- p>);</c-> <c- p>});</c->

  <c- k>struct</c-> <c- nc>interval</c-> <c- p>{</c->
    <c- b>bool</c-> <c- n>flag</c-><c- p>;</c->
    <c- b>size_t</c-> <c- n>index</c-><c- p>;</c->
    <c- b>size_t</c-> <c- n>start</c-><c- p>;</c->
    <c- b>size_t</c-> <c- n>end</c-><c- p>;</c->
  <c- p>};</c->

  <c- n>vector</c-><c- o>&lt;</c-><c- n>interval</c-><c- o>></c-> <c- n>intervals</c-><c- p>(</c-><c- n>size</c-><c- p>(</c-><c- n>in</c-><c- p>));</c->

  <c- n>exclusive_scan</c-><c- p>(</c-><c- n>par</c-><c- p>,</c->
    <c- n>flags</c-> <c- o>|</c-> <c- n>transform</c-><c- p>([]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>b</c-><c- p>)</c-> <c- p>{</c-> <c- k>return</c-> <c- n>interval</c-><c- p>{</c-><c- n>b</c-><c- p>,</c-> <c- mi>0</c-><c- p>,</c-> <c- mi>0</c-><c- p>,</c-> <c- mi>1</c-><c- p>};</c-> <c- p>}),</c->
    <c- n>begin</c-><c- p>(</c-><c- n>intervals</c-><c- p>),</c->
    <c- n>interval</c-><c- p>{</c->true<c- p>,</c-> <c- mi>0</c-><c- p>,</c-> <c- mi>0</c-><c- p>,</c-> <c- mi>1</c-><c- p>},</c->
    <c- p>[]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>l</c-><c- p>,</c-> <c- k>auto</c-> <c- n>r</c-><c- p>)</c-> <c- p>{</c->
      <c- k>return</c-> <c- n>interval</c-><c- p>{</c-><c- n>r</c-><c- p>.</c-><c- n>flag</c-><c- p>,</c->
                      <c- n>r</c-><c- p>.</c-><c- n>flag</c-> <c- o>?</c-> <c- n>l</c-><c- p>.</c-><c- n>index</c-> <c- o>+</c-> <c- n>r</c-><c- p>.</c-><c- n>index</c-> <c- o>:</c-> <c- n>l</c-><c- p>.</c-><c- n>index</c-> <c- o>+</c-> <c- n>r</c-><c- p>.</c-><c- n>index</c-> <c- o>+</c-> <c- mi>1</c-><c- p>,</c->
                      <c- n>r</c-><c- p>.</c-><c- n>flag</c-> <c- o>?</c-> <c- n>l</c-><c- p>.</c-><c- n>start</c-> <c- o>+</c-> <c- n>r</c-><c- p>.</c-><c- n>start</c-> <c- o>:</c-> <c- n>l</c-><c- p>.</c-><c- n>end</c-><c- p>,</c->
                      <c- n>l</c-><c- p>.</c-><c- n>end</c-> <c- o>+</c-> <c- n>r</c-><c- p>.</c-><c- n>end</c-><c- p>};</c->
    <c- p>});</c->

  <c- n>for_each</c-><c- p>(</c-><c- n>par</c-><c- p>,</c-> <c- n>zip</c-><c- p>(</c-><c- n>flags</c-><c- p>,</c-> <c- n>intervals</c-><c- p>),</c->
    <c- n>apply</c-><c- p>([</c-><c- o>&amp;</c-><c- p>]</c-> <c- p>(</c-><c- k>auto</c-> <c- n>flag</c-><c- p>,</c-> <c- k>auto</c-> <c- n>i</c-><c- p>)</c-> <c- p>{</c->
      <c- k>if</c-> <c- p>(</c-><c- o>!</c-><c- n>flag</c-><c- p>)</c->
        <c- n>out</c-><c- p>[</c-><c- n>i</c-><c- p>.</c-><c- n>index</c-><c- p>]</c-> <c- o>=</c-> <c- n>subrange</c-><c- p>(</c-><c- n>next</c-><c- p>(</c-><c- n>begin</c-><c- p>(</c-><c- n>in</c-><c- p>),</c-> <c- n>i</c-><c- p>.</c-><c- n>start</c-><c- p>),</c->
                                <c- n>next</c-><c- p>(</c-><c- n>begin</c-><c- p>(</c-><c- n>in</c-><c- p>),</c-> <c- n>i</c-><c- p>.</c-><c- n>end</c-><c- p>));</c->
    <c- p>}));</c->

  <c- k>return</c-> <c- n>subrange</c-><c- p>(</c-><c- n>out</c-><c- p>,</c-> <c- n>next</c-><c- p>(</c-><c- n>out</c-><c- p>,</c-> <c- n>intervals</c-><c- p>.</c-><c- n>back</c-><c- p>().</c-><c- n>index</c-><c- p>));</c->
<c- p>}</c->
</pre>
   <h3 class="heading settled" data-level="2.3" id="trivial-removal-and-grouping"><span class="secno">2.3. </span><span class="content">Trivial Removal and Grouping</span><a class="self-link" href="#trivial-removal-and-grouping"></a></h3>
   <p>Certain range adaptors remove elements in a way that can be trivially computed a priori (ex: <code class="highlight"><c- n>drop</c-></code>, <code class="highlight"><c- n>stride</c-></code>) or group or combine elements in a way that can be trivially computed a priori (ex: <code class="highlight"><c- n>chunk</c-></code>, <code class="highlight"><c- n>adjacent</c-></code>). Thus, when consuming these adaptors in parallel, we can account for the removal or grouping while doing work distribution.</p>
<pre class="highlight"><c- n>for_each</c-><c- p>(</c-><c- n>rng</c-> <c- o>|</c-> <c- n>drop</c-><c- p>(</c-><c- n>X</c-><c- p>),</c-> <c- n>f</c-><c- p>);</c->
</pre>
<pre class="highlight"><c- k>auto</c-> <c- n>start</c-> <c- o>=</c-> <c- n>begin</c-><c- p>(</c-><c- n>rng</c-><c- p>)</c-> <c- o>+</c-> <c- n>X</c-><c- p>;</c->
<c- k>auto</c-> <c- n>end</c-> <c- o>=</c-> <c- n>end</c-><c- p>(</c-><c- n>rng</c-><c- p>);</c->
<c- n>for_each</c-><c- p>(</c-><c- n>start</c-><c- p>,</c-> <c- n>end</c-><c- p>,</c-> <c- n>f</c-><c- p>);</c->
</pre>
   <p>However, if these range adaptors adapt a range that contained a non-trivial removal or grouping adaptor, then they become non-trivial as well, and need to be handled with an in-place stream compaction pass.</p>
   <p>There’s two approaches to handling trivial removals and groupings:
0) Always insert an in-place scan pass and don’t handle them during work distribution. This has the advantage of simplicity.
1) Only insert in-place scan pass if there is a prior non-trivial removal or grouping, and otherwise handle it during work distribution. This requires maintaining some state when reconstructing the range, but is more efficient.</p>
   <p>We propose to do (1).</p>
   <h2 class="heading settled" data-level="3" id="cheatsheet"><span class="secno">3. </span><span class="content">Cheatsheet</span><a class="self-link" href="#cheatsheet"></a></h2>
   <table>
    <tbody>
     <tr>
      <th>Range Adaptor or Factory 
      <th>Iterator Category 
      <th>Internal Iteration? 
      <th>Output Size Unknown? 
      <th>Position Aware? 
      <th>Reshapes? 
      <th>Optimization 
     <tr>
      <td><code class="highlight"><c- n>ref_view</c-></code> 
      <td>Contig 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>owning_view</c-></code> 
      <td>Contig 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>all</c-></code> 
      <td>Contig 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>common</c-></code> 
      <td>Contig 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>as_const</c-></code> <br><code class="highlight"><c- n>as_rvalue</c-></code> 
      <td>Contig 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>iota</c-></code> 
      <td>RA 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>transform</c-></code> 
      <td>RA 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>elements</c-></code> 
      <td>RA 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>keys</c-></code> <br><code class="highlight"><c- n>values</c-></code> 
      <td>RA 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>enumerate</c-></code> 
      <td>RA 
      <td>
      <td>
      <td>Yes 
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>reverse</c-></code> 
      <td>RA 
      <td>
      <td>
      <td>
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>zip</c-></code> <br><code class="highlight"><c- n>zip_transform</c-></code> 
      <td>RA 
      <td>
      <td>
      <td>Yes 
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>cartesian_product</c-></code> 
      <td>RA 
      <td>
      <td>
      <td>Yes 
      <td>
      <td>
     <tr>
      <td><code class="highlight"><c- n>filter</c-></code> 
      <td>Forward 
      <td>Yes 
      <td>Yes 
      <td>
      <td>
      <td>Non-Trivial Removal 
     <tr>
      <td><code class="highlight"><c- n>take</c-></code> 
      <td>Contig 
      <td>
      <td>
      <td>Yes 
      <td>
      <td>Trivial Removal 
     <tr>
      <td><code class="highlight"><c- n>take_while</c-></code> 
      <td>Contig 
      <td>Yes 
      <td>Yes 
      <td>Yes 
      <td>
      <td>Non-Trivial Removal 
     <tr>
      <td><code class="highlight"><c- n>drop</c-></code> 
      <td>Contig 
      <td>
      <td>
      <td>Yes 
      <td>
      <td>Trivial Removal 
     <tr>
      <td><code class="highlight"><c- n>drop_while</c-></code> 
      <td>Contig 
      <td>Yes 
      <td>Yes 
      <td>Yes 
      <td>
      <td>Non-Trivial Removal 
     <tr>
      <td><code class="highlight"><c- n>join</c-></code> <br><code class="highlight"><c- n>join_with</c-></code> 
      <td>BiDi 
      <td>Yes 
      <td>
      <td>
      <td>Yes 
      <td>? 
     <tr>
      <td><code class="highlight"><c- n>split</c-></code> <br><code class="highlight"><c- n>lazy_split</c-></code> 
      <td>Forward 
      <td>Yes 
      <td>Yes 
      <td>Yes 
      <td>Yes 
      <td>Non-Trivial Grouping 
     <tr>
      <td><code class="highlight"><c- n>chunk</c-></code> 
      <td>Forward 
      <td>Yes 
      <td>
      <td>Yes 
      <td>Yes 
      <td>Trivial Grouping 
     <tr>
      <td><code class="highlight"><c- n>chunk_by</c-></code> 
      <td>Forward 
      <td>Yes 
      <td>Yes 
      <td>Yes 
      <td>Yes 
      <td>Non-Trivial Grouping 
     <tr>
      <td><code class="highlight"><c- n>adjacent</c-></code> <br><code class="highlight"><c- n>adjacent_transform</c-></code> 
      <td>RA 
      <td>Yes 
      <td>
      <td>Yes 
      <td>Yes 
      <td>Trivial Grouping 
     <tr>
      <td><code class="highlight"><c- n>slide</c-></code> 
      <td>RA 
      <td>Yes 
      <td>
      <td>Yes 
      <td>Yes 
      <td>Trivial Grouping 
     <tr>
      <td><code class="highlight"><c- n>stride</c-></code> 
      <td>RA 
      <td>Yes 
      <td>
      <td>Yes 
      <td>
      <td>Trivial Removal 
   </table>
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