Invent "rainbow" arcs within the regex engine.

Some regular expression constructs, most notably the "." match-anything
metacharacter, produce a sheaf of parallel NFA arcs covering all
possible colors (that is, character equivalence classes).  We can make
a noticeable improvement in the space and time needed to process large
regexes by replacing such cases with a single arc bearing the special
color code "RAINBOW".  This requires only minor additional complication
in places such as pull() and push().

Callers of pg_reg_getoutarcs() must now be prepared for the possibility
of seeing a RAINBOW arc.  For the one known user, contrib/pg_trgm,
that's a net benefit since it cuts the number of arcs to be dealt with,
and the handling isn't any different than for other colors that contain
too many characters to be dealt with individually.

This is part of a patch series that in total reduces the regex engine's
runtime by about a factor of four on a large corpus of real-world regexes.

Patch by me, reviewed by Joel Jacobson

Discussion: https://postgr.es/m/1340281.1613018383@sss.pgh.pa.us

contrib/pg_trgm/trgm_regexp.c

@@ -282,8 +282,8 @@ typedef struct
 typedef int TrgmColor;
 
 /* We assume that colors returned by the regexp engine cannot be these: */
-#define COLOR_UNKNOWN	(-1)
-#define COLOR_BLANK		(-2)
+#define COLOR_UNKNOWN	(-3)
+#define COLOR_BLANK		(-4)
 
 typedef struct
 {
@@ -780,7 +780,8 @@ getColorInfo(regex_t *regex, TrgmNFA *trgmNFA)
 		palloc0(colorsCount * sizeof(TrgmColorInfo));
 
 	/*
-	 * Loop over colors, filling TrgmColorInfo about each.
+	 * Loop over colors, filling TrgmColorInfo about each.  Note we include
+	 * WHITE (0) even though we know it'll be reported as non-expandable.
 	 */
 	for (i = 0; i < colorsCount; i++)
 	{
@@ -1098,9 +1099,9 @@ addKey(TrgmNFA *trgmNFA, TrgmState *state, TrgmStateKey *key)
 			/* Add enter key to this state */
 			addKeyToQueue(trgmNFA, &destKey);
 		}
-		else
+		else if (arc->co >= 0)
 		{
-			/* Regular color */
+			/* Regular color (including WHITE) */
 			TrgmColorInfo *colorInfo = &trgmNFA->colorInfo[arc->co];
 
 			if (colorInfo->expandable)
@@ -1156,6 +1157,14 @@ addKey(TrgmNFA *trgmNFA, TrgmState *state, TrgmStateKey *key)
 				addKeyToQueue(trgmNFA, &destKey);
 			}
 		}
+		else
+		{
+			/* RAINBOW: treat as unexpandable color */
+			destKey.prefix.colors[0] = COLOR_UNKNOWN;
+			destKey.prefix.colors[1] = COLOR_UNKNOWN;
+			destKey.nstate = arc->to;
+			addKeyToQueue(trgmNFA, &destKey);
+		}
 	}
 
 	pfree(arcs);
@@ -1216,10 +1225,10 @@ addArcs(TrgmNFA *trgmNFA, TrgmState *state)
 			/*
 			 * Ignore non-expandable colors; addKey already handled the case.
 			 *
-			 * We need no special check for begin/end pseudocolors here.  We
-			 * don't need to do any processing for them, and they will be
-			 * marked non-expandable since the regex engine will have reported
-			 * them that way.
+			 * We need no special check for WHITE or begin/end pseudocolors
+			 * here.  We don't need to do any processing for them, and they
+			 * will be marked non-expandable since the regex engine will have
+			 * reported them that way.
 			 */
 			if (!colorInfo->expandable)
 				continue;

src/backend/regex/README

@@ -261,6 +261,18 @@ and the NFA has these arcs:
 	states 4 -> 5 on color 2 ("x" only)
 which can be seen to be a correct representation of the regex.
 
+There is one more complexity, which is how to handle ".", that is a
+match-anything atom.  We used to do that by generating a "rainbow"
+of arcs of all live colors between the two NFA states before and after
+the dot.  That's expensive in itself when there are lots of colors,
+and it also typically adds lots of follow-on arc-splitting work for the
+color splitting logic.  Now we handle this case by generating a single arc
+labeled with the special color RAINBOW, meaning all colors.  Such arcs
+never need to be split, so they help keep NFAs small in this common case.
+(Note: this optimization doesn't help in REG_NLSTOP mode, where "." is
+not supposed to match newline.  In that case we still handle "." by
+generating an almost-rainbow of all colors except newline's color.)
+
 Given this summary, we can see we need the following operations for
 colors:
 
@@ -349,6 +361,8 @@ The possible arc types are:
 
     PLAIN arcs, which specify matching of any character of a given "color"
     (see above).  These are dumped as "[color_number]->to_state".
+    In addition there can be "rainbow" PLAIN arcs, which are dumped as
+    "[*]->to_state".
 
     EMPTY arcs, which specify a no-op transition to another state.  These
     are dumped as "->to_state".
@@ -356,11 +370,11 @@ The possible arc types are:
     AHEAD constraints, which represent a "next character must be of this
     color" constraint.  AHEAD differs from a PLAIN arc in that the input
     character is not consumed when crossing the arc.  These are dumped as
-    ">color_number>->to_state".
+    ">color_number>->to_state", or possibly ">*>->to_state".
 
     BEHIND constraints, which represent a "previous character must be of
     this color" constraint, which likewise consumes no input.  These are
-    dumped as "<color_number<->to_state".
+    dumped as "<color_number<->to_state", or possibly "<*<->to_state".
 
     '^' arcs, which specify a beginning-of-input constraint.  These are
     dumped as "^0->to_state" or "^1->to_state" for beginning-of-string and
@@ -396,14 +410,20 @@ substring, or an imaginary following EOS character if the substring is at
 the end of the input.
 3. If the NFA is (or can be) in the goal state at this point, it matches.
 
+This definition is necessary to support regexes that begin or end with
+constraints such as \m and \M, which imply requirements on the adjacent
+character if any.  The executor implements that by checking if the
+adjacent character (or BOS/BOL/EOS/EOL pseudo-character) is of the
+right color, and it does that in the same loop that checks characters
+within the match.
+
 So one can mentally execute an untransformed NFA by taking ^ and $ as
 ordinary constraints that match at start and end of input; but plain
 arcs out of the start state should be taken as matches for the character
 before the target substring, and similarly, plain arcs leading to the
 post state are matches for the character after the target substring.
-This definition is necessary to support regexes that begin or end with
-constraints such as \m and \M, which imply requirements on the adjacent
-character if any.  NFAs for simple unanchored patterns will usually have
-pre-state outarcs for all possible character colors as well as BOS and
-BOL, and post-state inarcs for all possible character colors as well as
-EOS and EOL, so that the executor's behavior will work.
+After the optimize() transformation, there are explicit arcs mentioning
+BOS/BOL/EOS/EOL adjacent to the pre-state and post-state.  So a finished
+NFA for a pattern without anchors or adjacent-character constraints will
+have pre-state outarcs for RAINBOW (all possible character colors) as well
+as BOS and BOL, and likewise post-state inarcs for RAINBOW, EOS, and EOL.

src/backend/regex/regc_color.c

@@ -977,6 +977,7 @@ colorchain(struct colormap *cm,
 {
 	struct colordesc *cd = &cm->cd[a->co];
 
+	assert(a->co >= 0);
 	if (cd->arcs != NULL)
 		cd->arcs->colorchainRev = a;
 	a->colorchain = cd->arcs;
@@ -994,6 +995,7 @@ uncolorchain(struct colormap *cm,
 	struct colordesc *cd = &cm->cd[a->co];
 	struct arc *aa = a->colorchainRev;
 
+	assert(a->co >= 0);
 	if (aa == NULL)
 	{
 		assert(cd->arcs == a);
@@ -1012,6 +1014,9 @@ uncolorchain(struct colormap *cm,
 
 /*
  * rainbow - add arcs of all full colors (but one) between specified states
+ *
+ * If there isn't an exception color, we now generate just a single arc
+ * labeled RAINBOW, saving lots of arc-munging later on.
  */
 static void
 rainbow(struct nfa *nfa,
@@ -1025,6 +1030,13 @@ rainbow(struct nfa *nfa,
 	struct colordesc *end = CDEND(cm);
 	color		co;
 
+	if (but == COLORLESS)
+	{
+		newarc(nfa, type, RAINBOW, from, to);
+		return;
+	}
+
+	/* Gotta do it the hard way.  Skip subcolors, pseudocolors, and "but" */
 	for (cd = cm->cd, co = 0; cd < end && !CISERR(); cd++, co++)
 		if (!UNUSEDCOLOR(cd) && cd->sub != co && co != but &&
 			!(cd->flags & PSEUDO))
@@ -1034,13 +1046,16 @@ rainbow(struct nfa *nfa,
 /*
  * colorcomplement - add arcs of complementary colors
  *
+ * We add arcs of all colors that are not pseudocolors and do not match
+ * any of the "of" state's PLAIN outarcs.
+ *
  * The calling sequence ought to be reconciled with cloneouts().
  */
 static void
 colorcomplement(struct nfa *nfa,
 				struct colormap *cm,
 				int type,
-				struct state *of,	/* complements of this guy's PLAIN outarcs */
+				struct state *of,
 				struct state *from,
 				struct state *to)
 {
@@ -1049,6 +1064,11 @@ colorcomplement(struct nfa *nfa,
 	color		co;
 
 	assert(of != from);
+
+	/* A RAINBOW arc matches all colors, making the complement empty */
+	if (findarc(of, PLAIN, RAINBOW) != NULL)
+		return;
+
 	for (cd = cm->cd, co = 0; cd < end && !CISERR(); cd++, co++)
 		if (!UNUSEDCOLOR(cd) && !(cd->flags & PSEUDO))
 			if (findarc(of, PLAIN, co) == NULL)

src/backend/regex/regc_nfa.c

@@ -271,6 +271,11 @@ destroystate(struct nfa *nfa,
  *
  * This function checks to make sure that no duplicate arcs are created.
  * In general we never want duplicates.
+ *
+ * However: in principle, a RAINBOW arc is redundant with any plain arc
+ * (unless that arc is for a pseudocolor).  But we don't try to recognize
+ * that redundancy, either here or in allied operations such as moveins().
+ * The pseudocolor consideration makes that more costly than it seems worth.
  */
 static void
 newarc(struct nfa *nfa,
@@ -1170,6 +1175,9 @@ copyouts(struct nfa *nfa,
 
 /*
  * cloneouts - copy out arcs of a state to another state pair, modifying type
+ *
+ * This is only used to convert PLAIN arcs to AHEAD/BEHIND arcs, which share
+ * the same interpretation of "co".  It wouldn't be sensible with LACONs.
  */
 static void
 cloneouts(struct nfa *nfa,
@@ -1181,9 +1189,13 @@ cloneouts(struct nfa *nfa,
 	struct arc *a;
 
 	assert(old != from);
+	assert(type == AHEAD || type == BEHIND);
 
 	for (a = old->outs; a != NULL; a = a->outchain)
+	{
+		assert(a->type == PLAIN);
 		newarc(nfa, type, a->co, from, to);
+	}
 }
 
 /*
@@ -1597,7 +1609,7 @@ pull(struct nfa *nfa,
 	for (a = from->ins; a != NULL && !NISERR(); a = nexta)
 	{
 		nexta = a->inchain;
-		switch (combine(con, a))
+		switch (combine(nfa, con, a))
 		{
 			case INCOMPATIBLE:	/* destroy the arc */
 				freearc(nfa, a);
@@ -1624,6 +1636,10 @@ pull(struct nfa *nfa,
 				cparc(nfa, a, s, to);
 				freearc(nfa, a);
 				break;
+			case REPLACEARC:	/* replace arc's color */
+				newarc(nfa, a->type, con->co, a->from, to);
+				freearc(nfa, a);
+				break;
 			default:
 				assert(NOTREACHED);
 				break;
@@ -1764,7 +1780,7 @@ push(struct nfa *nfa,
 	for (a = to->outs; a != NULL && !NISERR(); a = nexta)
 	{
 		nexta = a->outchain;
-		switch (combine(con, a))
+		switch (combine(nfa, con, a))
 		{
 			case INCOMPATIBLE:	/* destroy the arc */
 				freearc(nfa, a);
@@ -1791,6 +1807,10 @@ push(struct nfa *nfa,
 				cparc(nfa, a, from, s);
 				freearc(nfa, a);
 				break;
+			case REPLACEARC:	/* replace arc's color */
+				newarc(nfa, a->type, con->co, from, a->to);
+				freearc(nfa, a);
+				break;
 			default:
 				assert(NOTREACHED);
 				break;
@@ -1810,9 +1830,11 @@ push(struct nfa *nfa,
  * #def INCOMPATIBLE	1	// destroys arc
  * #def SATISFIED		2	// constraint satisfied
  * #def COMPATIBLE		3	// compatible but not satisfied yet
+ * #def REPLACEARC		4	// replace arc's color with constraint color
  */
 static int
-combine(struct arc *con,
+combine(struct nfa *nfa,
+		struct arc *con,
 		struct arc *a)
 {
 #define  CA(ct,at)	 (((ct)<<CHAR_BIT) | (at))
@@ -1827,14 +1849,46 @@ combine(struct arc *con,
 		case CA(BEHIND, PLAIN):
 			if (con->co == a->co)
 				return SATISFIED;
+			if (con->co == RAINBOW)
+			{
+				/* con is satisfied unless arc's color is a pseudocolor */
+				if (!(nfa->cm->cd[a->co].flags & PSEUDO))
+					return SATISFIED;
+			}
+			else if (a->co == RAINBOW)
+			{
+				/* con is incompatible if it's for a pseudocolor */
+				if (nfa->cm->cd[con->co].flags & PSEUDO)
+					return INCOMPATIBLE;
+				/* otherwise, constraint constrains arc to be only its color */
+				return REPLACEARC;
+			}
 			return INCOMPATIBLE;
 			break;
 		case CA('^', '^'):		/* collision, similar constraints */
 		case CA('$', '$'):
-		case CA(AHEAD, AHEAD):
+			if (con->co == a->co)	/* true duplication */
+				return SATISFIED;
+			return INCOMPATIBLE;
+			break;
+		case CA(AHEAD, AHEAD):	/* collision, similar constraints */
 		case CA(BEHIND, BEHIND):
 			if (con->co == a->co)	/* true duplication */
 				return SATISFIED;
+			if (con->co == RAINBOW)
+			{
+				/* con is satisfied unless arc's color is a pseudocolor */
+				if (!(nfa->cm->cd[a->co].flags & PSEUDO))
+					return SATISFIED;
+			}
+			else if (a->co == RAINBOW)
+			{
+				/* con is incompatible if it's for a pseudocolor */
+				if (nfa->cm->cd[con->co].flags & PSEUDO)
+					return INCOMPATIBLE;
+				/* otherwise, constraint constrains arc to be only its color */
+				return REPLACEARC;
+			}
 			return INCOMPATIBLE;
 			break;
 		case CA('^', BEHIND):	/* collision, dissimilar constraints */
@@ -2895,6 +2949,7 @@ compact(struct nfa *nfa,
 					break;
 				case LACON:
 					assert(s->no != cnfa->pre);
+					assert(a->co >= 0);
 					ca->co = (color) (cnfa->ncolors + a->co);
 					ca->to = a->to->no;
 					ca++;
@@ -3068,13 +3123,22 @@ dumparc(struct arc *a,
 	switch (a->type)
 	{
 		case PLAIN:
-			fprintf(f, "[%ld]", (long) a->co);
+			if (a->co == RAINBOW)
+				fprintf(f, "[*]");
+			else
+				fprintf(f, "[%ld]", (long) a->co);
 			break;
 		case AHEAD:
-			fprintf(f, ">%ld>", (long) a->co);
+			if (a->co == RAINBOW)
+				fprintf(f, ">*>");
+			else
+				fprintf(f, ">%ld>", (long) a->co);
 			break;
 		case BEHIND:
-			fprintf(f, "<%ld<", (long) a->co);
+			if (a->co == RAINBOW)
+				fprintf(f, "<*<");
+			else
+				fprintf(f, "<%ld<", (long) a->co);
 			break;
 		case LACON:
 			fprintf(f, ":%ld:", (long) a->co);
@@ -3161,7 +3225,9 @@ dumpcstate(int st,
 	pos = 1;
 	for (ca = cnfa->states[st]; ca->co != COLORLESS; ca++)
 	{
-		if (ca->co < cnfa->ncolors)
+		if (ca->co == RAINBOW)
+			fprintf(f, "\t[*]->%d", ca->to);
+		else if (ca->co < cnfa->ncolors)
 			fprintf(f, "\t[%ld]->%d", (long) ca->co, ca->to);
 		else
 			fprintf(f, "\t:%ld:->%d", (long) (ca->co - cnfa->ncolors), ca->to);

src/backend/regex/regcomp.c

@@ -158,7 +158,8 @@ static int	push(struct nfa *, struct arc *, struct state **);
 #define INCOMPATIBLE	1		/* destroys arc */
 #define SATISFIED	2			/* constraint satisfied */
 #define COMPATIBLE	3			/* compatible but not satisfied yet */
-static int	combine(struct arc *, struct arc *);
+#define REPLACEARC	4			/* replace arc's color with constraint color */
+static int	combine(struct nfa *nfa, struct arc *con, struct arc *a);
 static void fixempties(struct nfa *, FILE *);
 static struct state *emptyreachable(struct nfa *, struct state *,
 									struct state *, struct arc **);
@@ -289,9 +290,11 @@ struct vars
 #define SBEGIN	'A'				/* beginning of string (even if not BOL) */
 #define SEND	'Z'				/* end of string (even if not EOL) */
 
-/* is an arc colored, and hence on a color chain? */
+/* is an arc colored, and hence should belong to a color chain? */
+/* the test on "co" eliminates RAINBOW arcs, which we don't bother to chain */
 #define COLORED(a) \
-	((a)->type == PLAIN || (a)->type == AHEAD || (a)->type == BEHIND)
+	((a)->co >= 0 && \
+	 ((a)->type == PLAIN || (a)->type == AHEAD || (a)->type == BEHIND))
 
 
 /* static function list */
@@ -1393,7 +1396,8 @@ bracket(struct vars *v,
  * cbracket - handle complemented bracket expression
  * We do it by calling bracket() with dummy endpoints, and then complementing
  * the result.  The alternative would be to invoke rainbow(), and then delete
- * arcs as the b.e. is seen... but that gets messy.
+ * arcs as the b.e. is seen... but that gets messy, and is really quite
+ * infeasible now that rainbow() just puts out one RAINBOW arc.
  */
 static void
 cbracket(struct vars *v,

src/backend/regex/rege_dfa.c

@@ -612,6 +612,7 @@ miss(struct vars *v,
 	unsigned	h;
 	struct carc *ca;
 	struct sset *p;
+	int			ispseudocolor;
 	int			ispost;
 	int			noprogress;
 	int			gotstate;
@@ -643,13 +644,15 @@ miss(struct vars *v,
 	 */
 	for (i = 0; i < d->wordsper; i++)
 		d->work[i] = 0;			/* build new stateset bitmap in d->work */
+	ispseudocolor = d->cm->cd[co].flags & PSEUDO;
 	ispost = 0;
 	noprogress = 1;
 	gotstate = 0;
 	for (i = 0; i < d->nstates; i++)
 		if (ISBSET(css->states, i))
 			for (ca = cnfa->states[i]; ca->co != COLORLESS; ca++)
-				if (ca->co == co)
+				if (ca->co == co ||
+					(ca->co == RAINBOW && !ispseudocolor))
 				{
 					BSET(d->work, ca->to);
 					gotstate = 1;

src/backend/regex/regexport.c

@@ -222,7 +222,8 @@ pg_reg_colorisend(const regex_t *regex, int co)
  * Get number of member chrs of color number "co".
  *
  * Note: we return -1 if the color number is invalid, or if it is a special
- * color (WHITE or a pseudocolor), or if the number of members is uncertain.
+ * color (WHITE, RAINBOW, or a pseudocolor), or if the number of members is
+ * uncertain.
  * Callers should not try to extract the members if -1 is returned.
  */
 int
@@ -233,7 +234,7 @@ pg_reg_getnumcharacters(const regex_t *regex, int co)
 	assert(regex != NULL && regex->re_magic == REMAGIC);
 	cm = &((struct guts *) regex->re_guts)->cmap;
 
-	if (co <= 0 || co > cm->max)	/* we reject 0 which is WHITE */
+	if (co <= 0 || co > cm->max)	/* <= 0 rejects WHITE and RAINBOW */
 		return -1;
 	if (cm->cd[co].flags & PSEUDO)	/* also pseudocolors (BOS etc) */
 		return -1;
@@ -257,7 +258,7 @@ pg_reg_getnumcharacters(const regex_t *regex, int co)
  * whose length chars_len must be at least as long as indicated by
  * pg_reg_getnumcharacters(), else not all chars will be returned.
  *
- * Fetching the members of WHITE or a pseudocolor is not supported.
+ * Fetching the members of WHITE, RAINBOW, or a pseudocolor is not supported.
  *
  * Caution: this is a relatively expensive operation.
  */

src/backend/regex/regprefix.c

@@ -165,9 +165,13 @@ findprefix(struct cnfa *cnfa,
 			/* We can ignore BOS/BOL arcs */
 			if (ca->co == cnfa->bos[0] || ca->co == cnfa->bos[1])
 				continue;
-			/* ... but EOS/EOL arcs terminate the search, as do LACONs */
+
+			/*
+			 * ... but EOS/EOL arcs terminate the search, as do RAINBOW arcs
+			 * and LACONs
+			 */
 			if (ca->co == cnfa->eos[0] || ca->co == cnfa->eos[1] ||
-				ca->co >= cnfa->ncolors)
+				ca->co == RAINBOW || ca->co >= cnfa->ncolors)
 			{
 				thiscolor = COLORLESS;
 				break;

src/include/regex/regexport.h

@@ -30,6 +30,10 @@
 
 #include "regex/regex.h"
 
+/* These macros must match corresponding ones in regguts.h: */
+#define COLOR_WHITE		0		/* color for chars not appearing in regex */
+#define COLOR_RAINBOW	(-2)	/* represents all colors except pseudocolors */
+
 /* information about one arc of a regex's NFA */
 typedef struct
 {

src/include/regex/regguts.h

@@ -130,11 +130,16 @@
 /*
  * As soon as possible, we map chrs into equivalence classes -- "colors" --
  * which are of much more manageable number.
+ *
+ * To further reduce the number of arcs in NFAs and DFAs, we also have a
+ * special RAINBOW "color" that can be assigned to an arc.  This is not a
+ * real color, in that it has no entry in color maps.
  */
 typedef short color;			/* colors of characters */
 
 #define MAX_COLOR	32767		/* max color (must fit in 'color' datatype) */
 #define COLORLESS	(-1)		/* impossible color */
+#define RAINBOW		(-2)		/* represents all colors except pseudocolors */
 #define WHITE		0			/* default color, parent of all others */
 /* Note: various places in the code know that WHITE is zero */
 
@@ -276,7 +281,7 @@ struct state;
 struct arc
 {
 	int			type;			/* 0 if free, else an NFA arc type code */
-	color		co;
+	color		co;				/* color the arc matches (possibly RAINBOW) */
 	struct state *from;			/* where it's from (and contained within) */
 	struct state *to;			/* where it's to */
 	struct arc *outchain;		/* link in *from's outs chain or free chain */
@@ -284,6 +289,7 @@ struct arc
 #define  freechain	outchain	/* we do not maintain "freechainRev" */
 	struct arc *inchain;		/* link in *to's ins chain */
 	struct arc *inchainRev;		/* back-link in *to's ins chain */
+	/* these fields are not used when co == RAINBOW: */
 	struct arc *colorchain;		/* link in color's arc chain */
 	struct arc *colorchainRev;	/* back-link in color's arc chain */
 };
@@ -344,6 +350,9 @@ struct nfa
  * Plain arcs just store the transition color number as "co".  LACON arcs
  * store the lookaround constraint number plus cnfa.ncolors as "co".  LACON
  * arcs can be distinguished from plain by testing for co >= cnfa.ncolors.
+ *
+ * Note that in a plain arc, "co" can be RAINBOW; since that's negative,
+ * it doesn't break the rule about how to recognize LACON arcs.
  */
 struct carc
 {