February 14, 2009 TREE 3

### 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_GENERATE, RB_ENTRY, RB_HEAD, RB_INITIALIZER, RB_ROOT, RB_EMPTY, RB_NEXT, RB_MIN, RB_MAX, RB_FIND, RB_LEFT, RB_RIGHT, RB_PARENT, RB_FOREACH, RB_INIT, RB_INSERT, RB_REMOVE - implementations of splay and red-black trees

### SYNOPSIS

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`)
```bool` 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_GENERATE(```NAME``, TYPE``, FIELD``, CMP`) RB_ENTRY(```TYPE`) RB_HEAD(```HEADNAME``, TYPE`) RB_INITIALIZER(```RB_HEAD *head`)
```struct TYPE *` RB_ROOT(```RB_HEAD *head`)
```bool` RB_EMPTY(```RB_HEAD *head`)
```struct TYPE *` RB_NEXT(```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_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`)
```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 of type ```SPLAY_ENTRY`, or ```RB_ENTRY`, named ```ENTRYNAME`. 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 either ```SPLAY_PROTOTYPE` or ```RB_PROTOTYPE`. The function bodies are generated with either ```SPLAY_GENERATE` or ```RB_GENERATE`. 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 trees noded 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.

The SPLAY_REMOVE(``) macro removes the element ```elm` from the tree pointed by ```head`.

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(``) 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 RB_ENTRY(``).

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

Finally, the ```CMP` argument is the name of a function used to compare trees noded 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 redblack 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.

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

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

  `

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

  `

The RB_ROOT(``), RB_MIN(``), RB_MAX(``), and RB_NEXT(``) 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(``) macro:

  `

      RB_FOREACH(np, NAME, head)

  `

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

### 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);
      }

  `

Both RB_INSERT(``) and SPLAY_INSERT(``) return ```NULL` if the element was inserted in the tree successfully, otherwise they return a pointer to the element with the colliding key.

Accordingly, RB_REMOVE(``) and SPLAY_REMOVE(``) return the pointer to the removed element, otherwise they return ```NULL` to indicate an error.

### AUTHORS

The author of the tree macros is Niels Provos
.