41. Splay trees

41.1. Introduction

.intro: This document explains the design of impl.c.splay, an implementation of Splay Trees, including its interface and implementation.

.readership: This document is intended for any MM developer.

.source: The primary sources for this design are paper.st85(0) and paper.sleator96(0). Also as CBS is a client, design.mps.cbs. As PoolMVFF is an indirect client, design.mps.poolmvff(1). Also, as PoolMV2 is an (obsolescent?) indirect client, design.mps.poolmv2.

.background: The following background documents influence the design: guide.impl.c.adt(0).

41.2. Overview

.overview: Splay trees are a form of binary tree where each access brings the accessed element (or the nearest element) to the root of the tree. The restructuring of the tree caused by the access gives excellent amortised performance, as the splay tree adapts its shape to usage patterns. Unused nodes have essentially no time overhead.

41.3. Definitions

.def.splay-tree: A “Splay Tree” is a self-adjusting binary tree as described in paper.st85(0), paper.sleator96(0).

.def.node: A “node” is used in the typical datastructure sense to mean an element of a tree (see also .type.splay.node).

.def.key: A “key” is a value associated with each node; the keys are totally ordered by a client provided comparator.

.def.comparator: A “comparator” is a function that compares keys to determine their ordering (see also .type.splay.compare.method).

.def.successor: Node N1 is the “successor” of node N2 if N1 and N2 are both in the same tree, and the key of N1 immediately follows the key of N2 in the ordering of all keys for the tree.

.def.left-child: Each node N contains a “left child”, which is a (possibly empty) sub-tree of nodes. The key of N is ordered after the keys of all nodes in this sub-tree.

.def.right-child: Each node N contains a “right child”, which is a (possibly empty) sub-tree of nodes. The key of N is ordered before the keys of all nodes in this sub-tree.

.def.neighbour: A node N which has key Kn is a “neighbour” of a key K if either Kn is the first key in the total order which compares greater than K or if Kn is the last key in the total order which compares less than K.

.def.first: A node is the “first” node in a set of nodes if its key compares less than the keys of all other nodes in the set.

.def.last: A node is the “last” node in a set of nodes if its key compares greater than the keys of all other nodes in the set.

.def.client-property: A “client property” is a value that the client may associate with each node in addition to the key (a block size, for example). This splay tree implementation provides support for efficiently finding the first or last nodes with suitably large client property values. See also .prop below.

41.4. Requirements

.req: These requirements are drawn from those implied by design.mps.poolmv2, design.mps.poolmvff(1), design.mps.cbs(2) and general inferred MPS requirements.

.req.order: Must maintain a set of abstract keys which is totally ordered for a comparator.

.req.tree: The keys must be associated with nodes arranged in a Splay Tree.

.req.splay: Common operations must balance the tree by splaying it, to achieve low amortized cost (see paper.st85(0)).

.req.add: Must be able to add new members. This is a common operation.

.req.remove: Must be able to remove members. This is a common operation.

.req.locate: Must be able to locate a member, given a key. This is a common operation.

.req.neighbours: Must be able to locate the neighbouring members (in order) of a non-member, given a key (see .def.neighbour). This is a common operation.

.req.iterate: Must be able to iterate over all members in order with reasonable efficiency.

.req.protocol: Must support detection of protocol violations.

.req.debug: Must support debugging of clients.

.req.stack: Must do all non-debugging operations with stack usage bounded by a constant size.

.req.adapt: Must adapt to regularities in usage pattern, for better performance.

.req.property: Must permit a client to associate a client property (such as a size) with each node in the tree.

.req.property.change: Must permit a client to dynamically reassign client properties to nodes in the tree. This is a common operation.

.req.property.find: Must support rapid finding of the first and last nodes which have a suitably large value for their client property. This is a common operation.

.req.root: Must be able to find the root of a splay tree (if one exists).

41.5. External types

struct SplayTreeStruct SplayTreeStruct

struct SplayTreeStruct *SplayTree

.type.splay.tree: SplayTree is the type of the main object at the root of the splay tree. It is intended that the SplayTreeStruct can be embedded in another structure (see .usage.client-tree for an example). No convenience functions are provided for allocation or deallocation.

struct SplayNodeStruct SplayNodeStruct

struct SplayNodeStruct *SplayNode

.type.splay.node: SplayNode is the type of a node of the splay tree. SplayNodeStruct contains no fields to store the key associated with the node, or the client property. Again, it is intended that the SplayNodeStruct can be embedded in another structure, and that this is how the association will be made (see .usage.client-node for an example). No convenience functions are provided for allocation or deallocation.

Compare (*SplayCompareMethod)(void *key, SplayNode node)

.type.splay.compare.method: A function of type SplayCompareMethod is required to compare key with the key the client associates with that splay tree node node, and return the appropriate Compare value (see .usage.compare for an example). The function compares a key with a node, rather than a pair of keys or nodes as might seem more obvious. This is because the details of the mapping between nodes and keys is left to the client (see .type.splay.node), and the splaying operations compare keys with nodes (see .impl.splay).

Res (*SplayNodeDescribeMethod)(SplayNode node, mps_lib_FILE *stream)

.type.splay.node.describe.method: A function of type SplayNodeDescribeMethod is required to write (via WriteF()) a client-oriented representation of the splay node. The output should be non-empty, short, and without return characters. This is provided for debugging purposes only.

Bool (*SplayTestNodeMethod)(SplayTree tree, SplayNode node, void *closureP, unsigned long closureS)

.type.splay.test.node.method: A function of type SplayTestNodeMethod required to determine whether the node itself meets some client determined property (see .prop and .usage.test.node for an example). Parameters closureP and closureS describe the environment for the function (see .function.splay.find.first and .function.splay.find.last).

Bool (*SplayTestTreeMethod)(SplayTree tree, SplayNode node, void *closureP, unsigned long closureS)

.type.splay.test.tree.method: A function of type SplayTestTreeMethod is required to determine whether any of the nodes in the sub-tree rooted at the given node meet some client determined property (see .prop and .usage.test.tree for an example). In particular, it must be a precise (not conservative) indication of whether there are any nodes in the sub-tree for which the testNode method (see .type.splay.test.node.method) would return TRUE. Parameters closureP and closureS describe the environment for the function (see .function.splay.find.first and .function.splay.find.last).

void (*SplayUpdateNodeMethod)(SplayTree tree, SplayNode node, SplayNode leftChild, SplayNode rightChild)

.type.splay.update.node.method: A function of type SplayUpdateNodeMethod is required to update any client datastructures associated with a node to maintain some client determined property (see .prop) given that the children of the node have changed. If the node does not have one or both children, then NULL will be passed as the relevant parameter. (See .usage.callback for an example)

41.6. External functions

.function.no-thread: The interface functions are not designed to be either thread-safe or re-entrant. Clients of the interface are responsible for synchronization, and for ensuring that client-provided methods invoked by the splay module (.type.splay.compare.method, .type.splay.test.node.method, .type.splay.test.tree.method, .type.splay.update.node.method) do not call functions of the splay module.

Bool SplayTreeCheck(SplayTree tree)

.function.splay.tree.check: This is a check function for the SplayTree type (see guide.impl.c.adt.method.check & design.mps.check(0)).

Bool SplayNodeCheck(SplayNode node)

.function.splay.node.check: This is a check function for the SplayNode type (see guide.impl.c.adt.method.check & design.mps.check(0)).

void SplayTreeInit(SplayTree tree, SplayCompareMethod compare, SplayUpdateNodeMethod updateNode)

.function.splay.tree.init: This function initialises a SplayTree (see guide.impl.c.adt.method.init). It requires a compare method that defines a total ordering on nodes (see .req.order); the effect of supplying a compare method that does not implement a total ordering is undefined. It also requires an updateNode method, which will be used to keep client properties up to date when the tree structure changes; the value SplayTrivUpdateNode may be used for this method if there is no need to maintain client properties. (See .usage.initialization for an example use).

void SplayTreeFinish(SplayTree tree)

.function.splay.tree.finish: This function clears the fields of a SplayTree (see guide.impl.c.adt.method.finish). Note that it does not attempt to finish or deallocate any associated SplayNode objects; clients wishing to destroy a non-empty SplayTree must first explicitly descend the tree and call SplayNodeFinish() on each node from the bottom up.

void SplayNodeInit(SplayNode node)

.function.splay.node.init: This function initialises a SplayNode (see guide.impl.c.adt.method.init).

void SplayNodeFinish(SplayNode node)

.function.splay.node.finish: This function clears the fields of a SplayNode (see guide.impl.c.adt.method.finish). Note that it does not attempt to finish or deallocate any referenced SplayNode objects (see.function.splay.tree.finish).

Bool SplayRoot(SplayNode *nodeReturn, SplayTree tree)

.function.splay.root: This function returns the root node of the tree, if any (see .req.root). If the tree is empty, FALSE is returned and *nodeReturn is not changed. Otherwise, TRUE is returned and *nodeReturn is set to the root.

Res SplayTreeInsert(SplayTree tree, SplayNode node, void *key)

.function.splay.tree.insert: This function is used to insert into a splay tree a new node which is associated with the supplied key (see .req.add). It first splays the tree at the key. If an attempt is made to insert a node that compares CompareEQUAL to an existing node in the tree, then ResFAIL will be returned and the node will not be inserted. (See .usage.insert for an example use).

Res SplayTreeDelete(SplayTree tree, SplayNode node, void *key)

.function.splay.tree.delete: This function is used to delete from a splay tree a node which is associated with the supplied key (see .req.remove). If the tree does not contain the given node, or the given node does not compare CompareEQUAL with the given key, then ResFAIL will be returned, and the node will not be deleted. The function first splays the tree at the given key. (See .usage.delete for an example use).

Res SplayTreeSearch(SplayNode *nodeReturn, SplayTree tree, void *key)

.function.splay.tree.search: This function searches the splay tree for a node that compares CompareEQUAL to the given key (see .req.locate). It splays the tree at the key. It returns ResFAIL if there is no such node in the tree, otherwise *nodeReturn will be set to the node.

Res SplayTreeNeighbours(SplayNode *leftReturn, SplayNode *rightReturn, SplayTree tree, void *key)

.function.splay.tree.neighbours: This function searches a splay tree for the two nodes that are the neighbours of the given key (see .req.neighbours). It splays the tree at the key. *leftReturn will be the neighbour which compares less than the key if such a neighbour exists; otherwise it will be NULL. *rightReturn will be the neighbour which compares greater than the key if such a neighbour exists; otherwise it will be NULL. The function returns ResFAIL if any node in the tree compares CompareEQUAL with the given key. (See .usage.insert for an example use).

SplayNode SplayTreeFirst(SplayTree tree, void *zeroKey)

.function.splay.tree.first: This function splays the tree at the first node, and returns that node (see .req.iterate). The supplied key should compare CompareLESS with all nodes in the tree. It will return NULL if the tree has no nodes.

SplayNode SplayTreeNext(SplayTree tree, SplayNode oldNode, void *oldKey)

.function.splay.tree.next: This function receives a node and key and returns the successor node to that node (see .req.iterate). This function is intended for use in iteration when the received node will be the current root of the tree, but is robust against being interspersed with other splay operations (provided the old node still exists). The supplied key must compare CompareEQUAL to the supplied node. Note that use of this function rebalances the tree for each node accessed. If many nodes are accessed as a result of multiple uses, the resultant tree will be generally well balanced. But if the tree was previously beneficially balanced for a small working set of accesses, then this local optimization will be lost. (see .future.parent).

Res SplayTreeDescribe(SplayTree tree, mps_lib_FILE *stream, SplayNodeDescribeMethod nodeDescribe)

.function.splay.tree.describe: This function prints (using WriteF) to the stream a textual representation of the given splay tree, using nodeDescribe to print client-oriented representations of the nodes (see .req.debug).

Bool SplayFindFirst(SplayNode *nodeReturn, SplayTree tree, SplayTestNodeMethod testNode, SplayTestTreeMethod testTree, void *closureP, unsigned long closureS)

.function.splay.find.first: SplayFindFirst() finds the first node in the tree that satisfies some client property (as determined by the testNode and testTree methods) (see .req.property.find). closureP and closureS are arbitrary values, and are passed to the testNode and testTree methods which may use the values as closure environments. If there is no satisfactory node, then FALSE is returned, otherwise *nodeReturn is set to the node. (See .usage.delete for an example use).

Bool SplayFindFirst(SplayNode *nodeReturn, SplayTree tree, SplayTestNodeMethod testNode, SplayTestTreeMethod testTree, void *closureP, unsigned long closureS)

.function.splay.find.last: SplayFindLast() finds the last node in the tree that satisfies some client property (as determined by the testNode and testTree methods) (see .req.property.find). closureP and closureS are arbitrary values, and are passed to the testNode and testTree methods which may use the values as closure environments. If there is no satisfactory node, then FALSE is returned, otherwise *nodeReturn is set to the node.

void SplayNodeRefresh(SplayTree tree, SplayNode node, void *key)

.function.splay.node.refresh: SplayNodeRefresh() must be called whenever the client property (see .prop) at a node changes (see .req.property.change). It will call the updateNode method on the given node, and any other nodes that may require update. The client key for the node must also be supplied; the function splays the tree at this key. (See .usage.insert for an example use).

41.7. Client-determined properties

.prop: To support .req.property.find, this splay tree implementation provides additional features to permit clients to cache maximum (or minimum) values of client properties for all the nodes in a subtree. The splay tree implementation uses the cached values as part of SplayFindFirst and SplayFindLast via the testNode and testTree methods. The client is free to choose how to represent the client property, and how to compute and store the cached value.

.prop.update: The cached values depend upon the topology of the tree, which may vary as a result of operations on the tree. The client is given the opportunity to compute new cache values whenever necessary, via the updateNode method (see .function.splay.tree.init). This happens whenever the tree is restructured. The client may use the SplayNodeRefresh method to indicate that the client attributes at a node have changed (see .req.property.change). A call to SplayNodeRefresh splays the tree at the specified node, which may provoke calls to the updateNode method as a result of the tree restructuring. The updateNode method will also be called whenever a new splay node is inserted into the tree.

.prop.example: For example, if implementing an address ordered tree of free blocks using a splay tree, a client might choose to use the base address of each block as the key for each node, and the size of each block as the client property. The client can then maintain as a cached value in each node the size of the largest block in the subtree rooted at that node. This will permit a fast search for the first or last block of at least a given size. See .usage.callback for an example updateNode method for such a client.

.prop.ops: The splay operations must cause client properties for nodes to be updated in the following circumstances:- (see .impl for details):

.prop.ops.rotate: rotate left, rotate right – We need to update the value at the original root, and the new root, in that order.

.prop.ops.link: link left, link right – We know that the line of right descent from the root of the left tree and the line of left descent from the root of the right tree will both need to be updated. This is performed at the assembly stage. (We could update these chains every time we do a link left or link right instead, but this would be less efficient)

.prop.ops.assemble: assemble – This operation also invalidates the lines of right and left descent of the left and right trees respectively which need to be updated (see below). It also invalidates the root which must be updated last.

.prop.ops.assemble.reverse: To correct the chains of the left and right trees without requiring stack or high complexity, we use a judicious amount of pointer reversal.

.prop.ops.assemble.traverse: During the assembly, after the root’s children have been transplanted, we correct the chains of the left and right trees. For the left tree, we traverse the right child line, reversing pointers, until we reach the node that was the last node prior to the transplantation of the root’s children. Then we update from that node back to the left tree’s root, restoring pointers. Updating the right tree is the same, mutatis mutandis. (See .future.reverse for an alternative approach).

41.8. Usage

.usage: Here’s a simple example of a client which uses a splay tree to implement an address ordered tree of free blocks. The significant client usages of the splay tree interface might look as follows:-

.usage.client-tree: Tree structure to embed a SplayTree (see .type.splay.tree):

typedef struct FreeTreeStruct {
  SplayTreeStruct splayTree;  /* Embedded splay tree */
  /* no obvious client fields for this simple example */
} FreeTreeStruct;

.usage.client-node: Node structure to embed a SplayNode (see .type.splay.node):

typedef struct FreeBlockStruct {
  SplayNodeStruct splayNode; /* embedded splay node */
  Addr base;                 /* base address of block is also the key */
  Size size;                 /* size of block is also the client property */
  Size maxSize;              /* cached value for maximum size in subtree */
} FreeBlockStruct;

.usage.callback: updateNode callback method (see .type.splay.update.node.method):

void FreeBlockUpdateNode(SplayTree tree, SplayNode node,
                         SplayNode leftChild, SplayNode rightChild)
{
  /* Compute the maximum size of any block in this subtree. */
  /* The value to cache is the maximum of the size of this block, */
  /* the cached value for the left subtree (if any) and the cached */
  /* value of the right subtree (if any) */

  FreeBlock freeNode = FreeBlockOfSplayNode(node);

  Size maxSize = freeNode.size;

  if(leftChild != NULL) {
    FreeBlock leftNode = FreeBlockOfSplayNode(leftChild);
    if(leftNode.maxSize > maxSize)
      maxSize = leftNode->maxSize;
  }

  if(rightChild != NULL) {
    FreeBlock rightNode = FreeBlockOfSplayNode(rightChild);
    if(rightNode.maxSize > maxSize)
      maxSize = rightNode->maxSize;
  }

  freeNode->maxSize = maxSize;
}

.usage.compare: Comparison function (see .type.splay.compare.method):

Compare FreeBlockCompare(void *key, SplayNode node) {
  Addr base1, base2, limit2;
  FreeBlock freeNode = FreeBlockOfSplayNode(node);

  base1 = (Addr *)key;
  base2 = freeNode->base;
  limit2 = AddrAdd(base2, freeNode->size);

  if (base1 < base2)
    return CompareLESS;
  else if (base1 >= limit2)
    return CompareGREATER;
  else
    return CompareEQUAL;
}

.usage.test.tree: Test tree function (see .type.splay.test.tree.method):

Bool FreeBlockTestTree(SplayTree tree, SplayNode node
                       void *closureP, unsigned long closureS) {
  /* Closure environment has wanted size as value of closureS. */
  /* Look at the cached value for the node to see if any */
  /* blocks in the subtree are big enough. */

  Size size = (Size)closureS;
  FreeBlock freeNode = FreeBlockOfSplayNode(node);
  return freeNode->maxSize >= size;
}

.usage.test.node: Test node function (see .type.splay.test.node.method):

Bool FreeBlockTestNode(SplayTree tree, SplayNode node
                       void *closureP, unsigned long closureS) {
  /* Closure environment has wanted size as value of closureS. */
  /* Look at the size of the node to see if is big enough. */

  Size size = (Size)closureS;
  FreeBlock freeNode = FreeBlockOfSplayNode(node);
  return freeNode->size >= size;
}

:mps:tag:`usage.initialization` Client's initialization function (see
:mps:ref:`.function.splay.tree.init`):
void FreeTreeInit(FreeTree tree) {
  /* Initialize the embedded splay tree. */
  SplayTreeInit(&tree->splayTree, FreeBlockCompare, FreeBlockUpdateNode);
}

.usage.insert: Client function to add a new free block into the tree, merging it with an existing block if possible:

void FreeTreeInsert(FreeTree tree, Addr base, Addr limit) {
  SplayTree splayTree = &tree->splayTree;
  SplayNode leftNeighbour, rightNeighbour;
  void *key = (void *)base;  /* use the base of the block as the key */
  Res res;

  /* Look for any neighbouring blocks. (.function.splay.tree.neighbours) */
  res = SplayTreeNeighbours(&leftNeighbour, &rightNeighbour,
                            splayTree, key);
  AVER(res == ResOK);  /* this client doesn't duplicate free blocks */

  /* Look to see if the neighbours are contiguous. */

  if (leftNeighbour != NULL &&
      FreeBlockLimitOfSplayNode(leftNeighbour) == base) {
    /* Inserted block is contiguous with left neighbour, so merge it. */
    /* The client housekeeping is left as an exercise to the reader. */
    /* This changes the size of a block, which is the client */
    /* property of the splay node. See :mps:ref:`.function.splay.node.refresh` */
    SplayNodeRefresh(tree, leftNeighbour, key);

  } else if (rightNeighbour != NULL &&
             FreeBlockBaseOfSplayNode(rightNeighbour) == limit) {
    /* Inserted block is contiguous with right neighbour, so merge it. */
    /* The client housekeeping is left as an exercise to the reader. */
    /* This changes the size of a block, which is the client */
    /* property of the splay node. See :mps:ref:`.function.splay.node.refresh` */
    SplayNodeRefresh(tree, rightNeighbour, key);

  } else {
    /* Not contiguous - so insert a new node */
    FreeBlock newBlock = (FreeBlock)allocate(sizeof(FreeBlockStruct));
    splayNode = &newBlock->splayNode;

    newBlock->base = base;
    newBlock->size = AddrOffset(base, limit);
    SplayNodeInit(splayNode);  /* :mps:ref:`.function.splay.node.init` */
    /* :mps:ref:`.function.splay.tree.insert` */
    res = SplayTreeInsert(splayTree, splayNode, key);
    AVER(res == ResOK);  /* this client doesn't duplicate free blocks */
  }
}

.usage.delete: Client function to allocate the first block of a given size in address order. For simplicity, this allocates the entire block:

Bool FreeTreeAllocate(Addr *baseReturn, Size *sizeReturn,
                      FreeTree tree, Size size) {
  SplayTree splayTree = &tree->splayTree;
  SplayNode splayNode;
  Bool found;

  /* look for the first node of at least the given size. */
  /* closureP parameter is not used. See `.function.splay.find.first.`_  */
  found = SplayFindFirst(&splayNode, splayTree,
                         FreeBlockTestNode, FreeBlockTestTree,
                         NULL, (unsigned long)size);

  if (found) {
    FreeBlock freeNode = FreeBlockOfSplayNode(splayNode);
    Void *key = (void *)freeNode->base;  /* use base of block as the key */
    Res res;

    /* allocate the block */
    *baseReturn = freeNode->base;
    *sizeReturn = freeNode->size;

    /* remove the node from the splay tree - :mps:ref:`.function.splay.tree.delete` */
    res = SplayTreeDelete(splayTree, splayNode, key);
    AVER(res == ResOK);  /* Must be possible to delete node */

    /* Delete the block */
    deallocate(freeNode, (sizeof(FreeBlockStruct));

    return TRUE;

  } else {
    /* No suitable block */
    return FALSE;
  }
}

41.9. Implementation

.impl: For more details of how splay trees work, see paper.st85(0). For more details of how to implement operations on splay trees, see paper.sleator96(0). Here we describe the operations involved.

41.9.1. Top-down splaying

.impl.top-down: The method chosen to implement the splaying operation is called “top-down splay”. This is described as “procedure top-down splay” in paper.st85(0) - although the implementation here additionally permits attempts to access items which are not known to be in the tree. Top-down splaying is particularly efficient for the common case where the location of the node in a tree is not known at the start of an operation. Tree restructuring happens as the tree is descended, whilst looking for the node.

.impl.splay: The key to the operation of the splay tree is the internal function SplaySplay(). It searches the tree for a node with a given key and returns whether it suceeded. In the process, it brings the found node, or an arbitrary neighbour if not found, to the root of the tree. This “bring-to-root” operation is performed top-down during the search, and it is not the simplest possible bring-to-root operation, but the resulting tree is well-balanced, and will give good amortised cost for future calls to SplaySplay(). (See paper.st85(0))

.impl.splay.how: To perform this top-down splay, the tree is broken into three parts, a left tree, a middle tree and a right tree. We store the left tree and right tree in the right and left children respectively of a “sides” node to eliminate some boundary conditions. The initial condition is that the middle tree is the entire splay tree, and the left and right trees are empty. We also keep pointers to the last node in the left tree, and the first node in the right tree. Note that, at all times, the three trees are each validly ordered, and they form a partition with the ordering left, middle, right. The splay is then performed by comparing the middle tree with the following six cases, and performing the indicated operations, until none apply.

.impl.splay.cases: Note that paper.st85(0)(Fig. 3) describes only 3 cases: zig, zig-zig and zig-zag. The additional cases described here are the symmetric variants which are respectively called zag, zag-zag and zag-zig. In the descriptions of these cases, root is the root of the middle tree; node->left is the left child of node; node->right is the right child of node. The comparison operators (<, >, ==) are defined to compare a key and a node in the obvious way by comparing the supplied key with the node’s associated key.

.impl.splay.zig: The “zig” case is where key < root, and either:

• key == root->left;

• key < root->left && root->left->left == NULL; or

• key > root->left && root->left->right == NULL.

The operation for the zig case is: link right (see .impl.link.right).

.impl.splay.zag: The “zag” case is where key > root, and either:

• key == root->right;

• key < root->right && root->right->left == NULL; or

• key > root->right && root->right->right == NULL.

The operation for the zag case is: link left (see .impl.link.left).

.impl.splay.zig.zig: The “zig-zig” case is where

• key < root && key < root->left && root->left->left != NULL.

The operation for the zig-zig case is: rotate right (see .impl.rotate.right) followed by link right (see .impl.link.right).

.impl.splay.zig.zag: The “zig-zag” case is where

• key < root && key > root->left && root->left->right != NULL.

The operation for the zig-zag case is: link right (see .impl.link.right) followed by link left (see .impl.link.left).

.impl.splay.zag.zig: The “zag-zig” case is where

• key > root && key < root->right && root->right->left != NULL.

The operation for the zag-zig case is: link left (see .impl.link.left) followed by link right (see .impl.link.right).

.impl.splay.zag.zag: The “zag-zag” case is where

• key > root && key > root->right && root->right->right != NULL.

The operation for the zag-zag case is: rotate left (see .impl.rotate.left) followed by link left (see .impl.link.left).

.impl.splay.terminal.null: A special terminal case is when

• root == NULL.

This can only happen at the beginning, and cannot arise from the operations above. In this case, the splay operation must return NULL, and “not found”.

.impl.splay.terminal.found: One typical terminal case is when

• key == root.

This case is tested for at the beginning, in which case “found” is returned immediately. If this case happens as a result of other operations, the splay operation is complete, the three trees are assembled (see .impl.assemble), and “found” is returned.

.impl.splay.terminal.not-found: The other typical terminal cases are:

• key < root && root->left == NULL; and

• key > root && root->right == NULL.

In these cases, the splay operation is complete, the three trees are assembled (see .impl.assemble), and “not found” is returned.

.impl.rotate.left: The “rotate left” operation (see paper.st85(0) Fig. 1) rearranges the middle tree as follows (where any of sub-trees A, B and C may be empty):

[missing diagram]

.impl.rotate.right: The “rotate right” operation (see paper.st85(0) Fig. 1) rearranges the middle tree as follows (where any of sub-trees A, B and C may be empty):

[missing diagram]

.impl.link.left: The “link left” operation (see paper.st85(0) Fig. 11a for symmetric variant) rearranges the left and middle trees as follows (where any of sub-trees A, B, L and R may be empty):

[missing diagram]

The last node of the left tree is now x.

.impl.link.right: The “link right” operation (see paper.st85(0) Fig. 11a) rearranges the middle and right trees as follows (where any of sub-trees A, B, L and R may be empty):

[missing diagram]

The first node of the right tree is now x.

.impl.assemble: The “assemble” operation (see paper.st85(0) Fig. 12) merges the left and right trees with the middle tree as follows (where any of sub-trees A, B, L and R may be empty):

[missing diagram]

41.9.2. Top-level operations

.impl.insert: SplayTreeInsert(): (See paper.sleator96(0), chapter 4, function insert). If the tree has no nodes, [how does it smell?] add the inserted node and we’re done; otherwise splay the tree around the supplied key. If the splay successfully found a matching node, return failure. Otherwise, add the inserted node as a new root, with the old (newly splayed, but non-matching) root as its left or right child as appropriate, and the opposite child of the old root as the other child of the new root.

.impl.delete: SplayTreeDelete(): (See paper.sleator96(0), chapter 4, function delete). Splay the tree around the supplied key. Check that the newly splayed root is the same node as given by the caller, and that it matches the key; return failure if not. If the given node (now at the root) has fewer than two children, replace it (as root), with the non-null child or null. Otherwise, set the root of the tree to be the left child (arbitrarily) of the node to be deleted, and splay around the same key. The new root will be the last node in the sub-tree and will have a null right child; this is set to be the right child of the node to be deleted.

.impl.search: SplayTreeSearch(): Splay the node around the supplied key. If the splay found a matching node, return it; otherwise return failure.

.impl.neighbours: SplayTreeNeighbours(): Splay the tree around the supplied key. If the splay found a matching node, return failure. Otherwise, determine whether the (non-matching) found node is the left or right neighbour of the key (by comparison with the key). Set the tree root to be the right or left child of that first neighbour respectively, and again splay the tree around the supplied key. The new root will be the second neighbour, and will have a null left or right child respectively. Set this null child to be the first neighbour. Return the two neighbours.

.impl.neighbours.note: Note that it would be possible to implement SplayTreeNeighbours() with only one splay, and then a normal binary tree search for the left or right neighbour of the root. This would be a cheaper operation, but would give poorer amortised cost if the call to SplayTreeNeighbours() typically precedes a call to SplayTreeInsert() (which is expected to be a common usage pattern - see .usage.insert). It’s also possible to implement SplayTreeNeighbours() by simply keeping track of both neighbours during a single splay. This has about the same cost as a single splay, and hence about the same amortised cost if the call to SplayTreeNeighbours() typically precedes a call to SplayTreeInsert().

.impl.next: SplayTreeNext(): Splay the tree around the supplied oldKey. During iteration the “old node” found is probably already at the root, in which case this will be a null operation with little cost. If this old node has no right child, return NULL. Otherwise, split the tree into a right tree (which contains just the right child of the old node) and a left tree (which contains the old node, its left child and no right child). The next node is the first node in the right tree. Find this by splaying the right tree around oldKey (which is known to compare CompareLESS than any keys in the right tree). Rejoin the full tree, using the right tree as the root and setting the left child of root to be the left tree. Return the root of this tree.

41.10. Testing

.test: There is no plan to test splay trees directly. It is believed that the testing described in design.mps.cbs.test will be sufficient to test this implementation.

41.11. Error Handling

.error: This module detects and reports most common classes of protocol error. The cases it doesn’t handle will result in undefined behaviour and probably cause an AVER to fire. These are:

.error.bad-pointer: Passing an invalid pointer in place of a SplayTree or SplayNode.

.error.bad-compare: Initialising a SplayTree with a compare function that is not a valid compare function, or which doesn’t implement a total ordering on splay nodes.

.error.bad-describe: Passing an invalid describe method to SplayTreeDescribe().

.error.out-of-stack: Stack exhaustion under SplayTreeDescribe().

41.12. Future

.future.tree: It would be possible to split the splay tree module into two: one that implements binary trees; and one that implements splay trees on top of a binary tree.

.future.parent: The iterator could be made more efficient (in an amortized sense) if it didn’t splay at each node. To implement this (whilst meeting .req.stack) we really need parent pointers from the nodes. We could use the (first-child, right-sibling/parent) trick described in paper.st85 to implement this, at a slight cost to all other tree operations, and an increase in code complexity. paper.st85 doesn’t describe how to distinguish the first-child between left-child and right-child, and the right-sibling/parent between right-sibling and parent. One could either use the comparator to make these distinctions, or steal some bits from the pointers.

.future.reverse: The assembly phase could be made more efficient if the link left and link right operations were modified to add to the left and right trees with pointers reversed. This would remove the need for the assembly phase to reverse them.