NAME

SPLAY_PROTOTYPE, SPLAY_GENERATE, SPLAY_ENTRY, SPLAY_HEAD, SPLAY_INITIALIZER, SPLAY_ROOT, SPLAY_EMPTY, SPLAY_NEXT, SPLAY_MIN, SPLAY_MAX, SPLAY_FIND, SPLAY_LEFT, SPLAY_RIGHT, SPLAY_FOREACH, SPLAY_INIT, SPLAY_INSERT, SPLAY_REMOVE, RB_PROTOTYPE, RB_PROTOTYPE_STATIC, RB_GENERATE, RB_GENERATE_STATIC, RB_ENTRY, RB_HEAD, RB_INITIALIZER, RB_ROOT, RB_EMPTY, RB_NEXT, RB_PREV, RB_MIN, RB_MAX, RB_FIND, RB_NFIND, RB_LEFT, RB_RIGHT, RB_PARENT, RB_FOREACH, RB_FOREACH_SAFE, RB_FOREACH_REVERSE, RB_FOREACH_REVERSE_SAFE, RB_INIT, RB_INSERT, RB_REMOVE — implementations of splay and red-black trees

SYNOPSIS

#include <sys/tree.h>

SPLAY_PROTOTYPE(NAME, TYPE, FIELD, CMP);

SPLAY_GENERATE(NAME, TYPE, FIELD, CMP);

SPLAY_ENTRY(TYPE);

SPLAY_HEAD(HEADNAME, TYPE);

struct TYPE *
SPLAY_INITIALIZER(SPLAY_HEAD *head);

SPLAY_ROOT(SPLAY_HEAD *head);

int
SPLAY_EMPTY(SPLAY_HEAD *head);

struct TYPE *
SPLAY_NEXT(NAME, SPLAY_HEAD *head, struct TYPE *elm);

struct TYPE *
SPLAY_MIN(NAME, SPLAY_HEAD *head);

struct TYPE *
SPLAY_MAX(NAME, SPLAY_HEAD *head);

struct TYPE *
SPLAY_FIND(NAME, SPLAY_HEAD *head, struct TYPE *elm);

struct TYPE *
SPLAY_LEFT(struct TYPE *elm, SPLAY_ENTRY NAME);

struct TYPE *
SPLAY_RIGHT(struct TYPE *elm, SPLAY_ENTRY NAME);

SPLAY_FOREACH(VARNAME, NAME, SPLAY_HEAD *head);

void
SPLAY_INIT(SPLAY_HEAD *head);

struct TYPE *
SPLAY_INSERT(NAME, SPLAY_HEAD *head, struct TYPE *elm);

struct TYPE *
SPLAY_REMOVE(NAME, SPLAY_HEAD *head, struct TYPE *elm);

RB_PROTOTYPE(NAME, TYPE, FIELD, CMP);

RB_PROTOTYPE_STATIC(NAME, TYPE, FIELD, CMP);

RB_GENERATE(NAME, TYPE, FIELD, CMP);

RB_GENERATE_STATIC(NAME, TYPE, FIELD, CMP);

RB_ENTRY(TYPE);

RB_HEAD(HEADNAME, TYPE);

RB_INITIALIZER(RB_HEAD *head);

struct TYPE *
RB_ROOT(RB_HEAD *head);

int
RB_EMPTY(RB_HEAD *head);

struct TYPE *
RB_NEXT(NAME, RB_HEAD *head, struct TYPE *elm);

struct TYPE *
RB_PREV(NAME, RB_HEAD *head, struct TYPE *elm);

struct TYPE *
RB_MIN(NAME, RB_HEAD *head);

struct TYPE *
RB_MAX(NAME, RB_HEAD *head);

struct TYPE *
RB_FIND(NAME, RB_HEAD *head, struct TYPE *elm);

struct TYPE *
RB_NFIND(NAME, RB_HEAD *head, struct TYPE *elm);

struct TYPE *
RB_LEFT(struct TYPE *elm, RB_ENTRY NAME);

struct TYPE *
RB_RIGHT(struct TYPE *elm, RB_ENTRY NAME);

struct TYPE *
RB_PARENT(struct TYPE *elm, RB_ENTRY NAME);

RB_FOREACH(VARNAME, NAME, RB_HEAD *head);

RB_FOREACH_SAFE(VARNAME, NAME, RB_HEAD *head, TEMP_VARNAME);

RB_FOREACH_REVERSE(VARNAME, NAME, RB_HEAD *head);

RB_FOREACH_REVERSE_SAFE(VARNAME, NAME, RB_HEAD *head, TEMP_VARNAME);

void
RB_INIT(RB_HEAD *head);

struct TYPE *
RB_INSERT(NAME, RB_HEAD *head, struct TYPE *elm);

struct TYPE *
RB_REMOVE(NAME, RB_HEAD *head, struct TYPE *elm);

DESCRIPTION

These macros define data structures for different types of trees: splay trees and red-black trees.

In the macro definitions, TYPE is the name tag of a user defined structure that must contain a field named FIELD, of type SPLAY_ENTRY or RB_ENTRY. The argument HEADNAME is the name tag of a user defined structure that must be declared using the macros SPLAY_HEAD() or RB_HEAD(). The argument NAME has to be a unique name prefix for every tree that is defined.

The function prototypes are declared with SPLAY_PROTOTYPE, RB_PROTOTYPE, or RB_PROTOTYPE_STATIC. The function bodies are generated with SPLAY_GENERATE, RB_GENERATE, or RB_GENERATE_STATIC. See the examples below for further explanation of how these macros are used.

SPLAY TREES

A splay tree is a self-organizing data structure. Every operation on the tree causes a splay to happen. The splay moves the requested node to the root of the tree and partly rebalances it.

This has the benefit that request locality causes faster lookups as the requested nodes move to the top of the tree. On the other hand, every lookup causes memory writes.

The Balance Theorem bounds the total access time for m operations and n inserts on an initially empty tree as O((m + n)lg n). The amortized cost for a sequence of m accesses to a splay tree is O(lg n).

A splay tree is headed by a structure defined by the SPLAY_HEAD() macro. A SPLAY_HEAD structure is declared as follows:

SPLAY_HEAD(HEADNAME, TYPE) head;

where HEADNAME is the name of the structure to be defined, and struct TYPE is the type of the elements to be inserted into the tree.

The SPLAY_ENTRY() macro declares a structure that allows elements to be connected in the tree.

In order to use the functions that manipulate the tree structure, their prototypes need to be declared with the SPLAY_PROTOTYPE() macro, where NAME is a unique identifier for this particular tree. The TYPE argument is the type of the structure that is being managed by the tree. The FIELD argument is the name of the element defined by SPLAY_ENTRY().

The function bodies are generated with the SPLAY_GENERATE() macro. It takes the same arguments as the SPLAY_PROTOTYPE() macro, but should be used only once.

Finally, the CMP argument is the name of a function used to compare tree nodes with each other. The function takes two arguments of type struct TYPE *. If the first argument is smaller than the second, the function returns a value smaller than zero. If they are equal, the function returns zero. Otherwise, it should return a value greater than zero. The compare function defines the order of the tree elements.

The SPLAY_INIT() macro initializes the tree referenced by head.

The splay tree can also be initialized statically by using the SPLAY_INITIALIZER() macro like this:

SPLAY_HEAD(HEADNAME, TYPE) head = SPLAY_INITIALIZER(&head);

The SPLAY_INSERT() macro inserts the new element elm into the tree. Upon success, NULL is returned. If a matching element already exists in the tree, the insertion is aborted, and a pointer to the existing element is returned.

The SPLAY_REMOVE() macro removes the element elm from the tree pointed by head. Upon success, a pointer to the removed element is returned. NULL is returned if elm is not present in the tree.

The SPLAY_FIND() macro can be used to find a particular element in the tree.

struct TYPE find, *res; 
find.key = 30; 
res = SPLAY_FIND(NAME, &head, &find);

The SPLAY_ROOT(), SPLAY_MIN(), SPLAY_MAX(), and SPLAY_NEXT() macros can be used to traverse the tree:

for (np = SPLAY_MIN(NAME, &head); np != NULL; np = SPLAY_NEXT(NAME, &head, np))

Or, for simplicity, one can use the SPLAY_FOREACH() macro:

SPLAY_FOREACH(np, NAME, &head)

The SPLAY_EMPTY() macro should be used to check whether a splay tree is empty.

RED-BLACK TREES

A red-black tree is a binary search tree with the node color as an extra attribute. It fulfills a set of conditions:

1. every search path from the root to a leaf consists of the same number of black nodes,
2. each red node (except for the root) has a black parent,
3. each leaf node is black.

Every operation on a red-black tree is bounded as O(lg n). The maximum height of a red-black tree is 2lg (n+1).

A red-black tree is headed by a structure defined by the RB_HEAD() macro. A RB_HEAD structure is declared as follows:

RB_HEAD(HEADNAME, TYPE) head;

where HEADNAME is the name of the structure to be defined, and struct TYPE is the type of the elements to be inserted into the tree.

The RB_ENTRY() macro declares a structure that allows elements to be connected in the tree.

In order to use the functions that manipulate the tree structure, their prototypes need to be declared with the RB_PROTOTYPE() or RB_PROTOTYPE_STATIC() macros, where NAME is a unique identifier for this particular tree. The TYPE argument is the type of the structure that is being managed by the tree. The FIELD argument is the name of the element defined by RB_ENTRY().

The function bodies are generated with the RB_GENERATE() or RB_GENERATE_STATIC() macros. These macros take the same arguments as the RB_PROTOTYPE() and RB_PROTOTYPE_STATIC() macros, but should be used only once.

Finally, the CMP argument is the name of a function used to compare trees' nodes with each other. The function takes two arguments of type struct TYPE *. If the first argument is smaller than the second, the function returns a value smaller than zero. If they are equal, the function returns zero. Otherwise, it should return a value greater than zero. The compare function defines the order of the tree elements.

The RB_INIT() macro initializes the tree referenced by head.

The red-black tree can also be initialized statically by using the RB_INITIALIZER() macro like this:

RB_HEAD(HEADNAME, TYPE) head = RB_INITIALIZER(&head);

The RB_INSERT() macro inserts the new element elm into the tree. Upon success, NULL is returned. If a matching element already exists in the tree, the insertion is aborted, and a pointer to the existing element is returned.

The RB_REMOVE() macro removes the element elm from the tree pointed to by head. The element must be present in that tree. RB_REMOVE() returns elm.

The RB_FIND() macro can be used to find a particular element in the tree. and RB_NFIND() macros can be used to find a particular element in the tree. RB_FIND() finds the node with the same key as elm. RB_NFIND() finds the first node greater than or equal to the search key.

struct TYPE find, *res; 
find.key = 30; 
res = RB_FIND(NAME, &head, &find);

The RB_ROOT(), RB_MIN(), RB_MAX(), RB_NEXT(), and RB_PREV() macros can be used to traverse the tree:

for (np = RB_MIN(NAME, &head); np != NULL; np = RB_NEXT(NAME, &head, np))

Or, for simplicity, one can use the RB_FOREACH() or RB_FOREACH_REVERSE() macros:

RB_FOREACH(np, NAME, &head)

The macros RB_FOREACH_SAFE() and RB_FOREACH_REVERSE_SAFE() traverse the tree referenced by head in a forward or reverse direction respectively, assigning each element in turn to np. However, unlike their unsafe counterparts, they permit both the removal of np as well as freeing it from within the loop safely without interfering with the traversal.

The RB_EMPTY() macro should be used to check whether a red-black tree is empty.

EXAMPLES

The following example demonstrates how to declare a red-black tree holding integers. Values are inserted into it and the contents of the tree are printed in order. Lastly, the internal structure of the tree is printed.

#include <sys/tree.h> 
#include <err.h> 
#include <stdio.h> 
#include <stdlib.h> 
 
struct node { 
	RB_ENTRY(node) entry; 
	int i; 
}; 
 
int 
intcmp(struct node *e1, struct node *e2) 
{ 
	return (e1->i < e2->i ? -1 : e1->i > e2->i); 
} 
 
RB_HEAD(inttree, node) head = RB_INITIALIZER(&head); 
RB_GENERATE(inttree, node, entry, intcmp) 
 
int testdata[] = { 
	20, 16, 17, 13, 3, 6, 1, 8, 2, 4, 10, 19, 5, 9, 12, 15, 18, 
	7, 11, 14 
}; 
 
void 
print_tree(struct node *n) 
{ 
	struct node *left, *right; 
 
	if (n == NULL) { 
		printf("nil"); 
		return; 
	} 
	left = RB_LEFT(n, entry); 
	right = RB_RIGHT(n, entry); 
	if (left == NULL && right == NULL) 
		printf("%d", n->i); 
	else { 
		printf("%d(", n->i); 
		print_tree(left); 
		printf(","); 
		print_tree(right); 
		printf(")"); 
	} 
} 
 
int 
main() 
{ 
	int i; 
	struct node *n; 
 
	for (i = 0; i < sizeof(testdata) / sizeof(testdata[0]); i++) { 
		if ((n = malloc(sizeof(struct node))) == NULL) 
			err(1, NULL); 
		n->i = testdata[i]; 
		RB_INSERT(inttree, &head, n); 
	} 
 
	RB_FOREACH(n, inttree, &head) { 
		printf("%d\n", n->i); 
	} 
	print_tree(RB_ROOT(&head)); 
	printf("\n"); 
	return (0); 
}

NOTES

Some of these macros or functions perform no error checking, and invalid usage leads to undefined behaviour. In the case of macros or functions that expect their arguments to be elements that are present in the tree, passing an element that is not present in the tree is invalid.

Trying to free a tree in the following way is a common error:

SPLAY_FOREACH(var, NAME, &head) { 
	SPLAY_REMOVE(NAME, &head, var); 
	free(var); 
} 
free(head);

Since var is free'd, the FOREACH() macro refers to a pointer that may have been reallocated already. Proper code needs a second variable.

for (var = SPLAY_MIN(NAME, &head); var != NULL; var = nxt) { 
	nxt = SPLAY_NEXT(NAME, &head, var); 
	SPLAY_REMOVE(NAME, &head, var); 
	free(var); 
}

AUTHORS

The author of the tree macros is Niels Provos.