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Incoming links: Doubly linked list
Deque endpoint invariant Created 2026-10-24 Updated 2026-11-05
In an endpoint-based double-ended queue represented by a doubly linked list, the first and last references are both null exactly when the double-ended queue is empty. Otherwise the first node has null predecessor, the last has null successor, and the finite successor chain is acyclic.
The two directions must agree: if node has successor , then has predecessor , and conversely for a nonnull predecessor. Thus following successors and following predecessors visit the same linked list nodes in reverse order.
Removing the only linked list node must clear both endpoints. Clearing only the endpoint used by that operation would leave one traversal apparently empty and the reverse traversal still reaching the removed node. The representation invariant makes that error visible before it affects later operations.
Destructive binary search tree flattening 2026-10-31
Turn a binary search tree into a sorted doubly linked list by reusing each linked list node's left and right child fields as predecessor and successor fields. An inorder traversal determines the required order. A recursive implementation first saves both child references, flattens the disjoint child subtrees, and splices their endpoints around the root. Structural induction proves that it preserves every original entry exactly once and produces reciprocal links with null outside endpoints. An explicit stack implements the same traversal without recursive calls; each original right child must be saved before its field is overwritten. Both versions take linear time complexity and auxiliary space for original height . The old binary search tree representation is consumed, so outside aliases must not continue to interpret the changed links as child pointers.
Lazy reversal of a doubly linked list 2026-10-31
Keep the physical doubly linked list unchanged and represent the logical sequence in either forward or backward order according to a Boolean flag. Toggling the flag reverses the logical contents in constant time complexity. Endpoint insertion and indexed access must interpret their positions using the flag, and the abstraction function explicitly accounts for this choice.
A list header is a stable entry record for a linked list. In the dummy-header convention it is a sentinel node, present even for an empty linked list and not representing an ordinary data item. It may also carry information such as the length.
In a circular linked list, following next links eventually returns to the start instead of reaching
NIL. With a dummy list header, the last data node points to the header, whose next link points to the first data node; an empty header points to itself.A singly linked list stores a next pointer in every linked list node. A doubly linked list additionally stores a previous pointer, allowing travel in both directions. For adjacent nodes, the consistency law iswith suitable endpoint conventions in a noncircular representation.
A dummy list header makes insertion or deletion at the front resemble insertion or deletion in the middle: there is always a predecessor record to update. In a circular doubly linked list, it also supplies a stable end position, handles the empty case uniformly, and avoids changing an external head pointer every time the first data node changes.
Its disadvantage is the extra record and associated storage and initialization work; traversal must also distinguish the sentinel node from real data. Thus
A doubly linked list supports backward traversal and removal of a known linked list node in constant time, because that node supplies its own predecessor. In a singly linked list, removing a known node generally requires finding its predecessor by a forward scan unless the caller already supplies it. Inserting before a known node has the same advantage.
The cost is another pointer per node, more link assignments on every update, and a stronger consistency invariant to maintain. HenceThis does not make locating an arbitrary data value faster: a search still takes linear time in the worst case.
Choose a circular doubly linked list with a permanent dummy list header. For a concrete Oberon definition take
data = INTEGER; the same link operations work with any chosen data data type:TYPE data = INTEGER;
pointerToNode = POINTER TO node;
node = RECORD
value : data;
prev, next : pointerToNode
END;
PROCEDURE NewList() : pointerToNode;
VAR h : pointerToNode;
BEGIN
NEW(h);
h.prev := h;
h.next := h;
RETURN h
END NewList;
PROCEDURE Insert(v : data; n : pointerToNode);
VAR t, p : pointerToNode;
BEGIN
p := n.prev;
NEW(t);
t.value := v;
t.prev := p;
t.next := n;
p.next := t;
n.prev := t
END Insert;
PROCEDURE Delete(n : pointerToNode);
VAR p, q : pointerToNode;
BEGIN
p := n.prev;
q := n.next;
p.next := q;
q.prev := p;
n.prev := NIL;
n.next := NIL
END Delete;The representation invariant is that the live nodes and header form one finite circular chain, with every neighboring next/previous pair mutually consistent.
Insert replaces the link by . Delete replaces by . All other links remain unchanged. These arguments also cover insertion into an empty linked list and deletion of its sole data node, where the neighboring positions can refer to the same header.Insert requires a live node n belonging to this representation and successful allocation. It may be called with n equal to the header, meaning append just before the end. Delete requires a live data node and must never be called on the header, on NIL, or on a node already deleted or belonging to an incompatible representation. The signature alone cannot enforce those preconditions. The detached node is left for garbage collection; external references must no longer treat it as a member of the linked list. Neither update contains a loop. Apart from allocation and memory management, the link work isThe predecessor disk block number turns the forward chain into a doubly linked list. Starting from the header's last block, one can traverse backwards; removing or detaching the current block can find its predecessor directly rather than scanning from the beginning. When modifying a chain, the affected predecessor and successor links must both be updated consistently.
The file number identifies the owner of each file data block. A file-system consistency checker can scan the medium, group blocks by owner, detect a cross-link between different files, and help reconstruct chains or headers after damage. It also lets the system check that a block reached through a chain belongs to the expected file.
ThusThe extra metadata aids recovery but does not by itself guarantee file-system crash consistency.
Use a doubly linked list, store both physical endpoints and its length, and maintain a flag for lazy reversal of a doubly linked list. This makes endpoint insertion and reversal constant time. Finding an arbitrary indexed element takes a traversal from whichever physical endpoint is nearer.Define to be the sequence of values obtained by starting at
class LLBuffer extends ReversableBuffer {
private class Node(val value: Int) {
var prev: Node = null
var next: Node = null
}
private var first: Node = null
private var last: Node = null
private var size = 0
private var backwards = false
// Methods from (ii) go here.
}first and repeatedly following next until null. The representation invariant says that size is nonnegative; size == 0 holds exactly when both endpoints are null; otherwise there are exactly size distinct reachable linked list nodes, the final one is last, first.prev == null, last.next == null, and all neighboring links are reciprocal. These conditions make finite and the physical linked list acyclic. The abstraction function isThe flag does not change the representation invariant. An empty physical linked list represents the empty sequence for either flag value. No client receives a linked list node reference, so clients cannot invalidate its links.
Physical doubly linked list order and logical sequence order under lazy reversal of a doubly linked list
. Details. 
A list header is a permanent dummy linked list node, distinct from the ordinary elements. It gives a stable reference to the linked list and can store metadata such as its length. In a circular linked list, following successor links eventually returns to the starting point; with a list header, the last ordinary element links to the header, and the empty structure has the header linked to itself.
A singly linked list stores a successor reference per linked list node. A doubly linked list additionally stores a predecessor reference. The latter supports traversal in both directions and deletion of a known linked list node in constant time, without first searching for its predecessor. It costs another reference per linked list node and additional updates whose mutual consistency must be maintained. A list header makes insertion and deletion at an endpoint use the same local splicing rules as an interior change; the empty case does not require replacing an external head reference.
Replace suspended recursive calls by an explicit stack for an inorder traversal. Save each original right child before overwriting that link:The method belongs to the concrete class owning
def flattenIterative(): (Tree, Tree) = {
val pending = new scala.collection.mutable.Stack[Tree]
var current = root
var start: Tree = null
var end: Tree = null
while (current != null || pending.nonEmpty) {
while (current != null) {
pending.push(current)
current = current.left
}
val t = pending.pop()
val nextSubtree = t.right
t.left = end
if (end == null) start = t else end.right = t
t.right = null
end = t
current = nextSubtree
}
root = null
(start, end)
}root. Clearing root records that ownership has transferred to the returned doubly linked list; callers retain both returned endpoints. The inner loop invariant says that the current subtree lies before all its pending ancestors in inorder traversal; pushing and descending left preserves this fact. Its loop variant is the number of entries along the remaining left-child path, a finite decreasing quantity.The outer loop invariant says that
start through end is precisely the already emitted sorted prefix, with reciprocal links and null outside endpoints. current and the pending ancestors describe exactly the remaining traversal in order; original links that are still needed have been retained. A pending ancestor has not yet had its right child overwritten. The next popped entry is therefore the next sorted entry. Saving nextSubtree preserves its future work before assigning t.right = null; linking it to end extends the prefix by exactly one entry. The number of original entries not yet emitted is an outer loop variant, since each full outer iteration emits one. At exit nothing remains, proving the same preservation and ordering postconditions as the recursive version.Each entry is pushed and popped once, giving time complexity and auxiliary space. No new
Tree entries are allocated. The historical mutable stack interface provides the push and pop operations used here; its later deprecation does not change their meaning.[ref]Perform a destructive binary search tree flattening by converting both child subtrees and splicing their endpoints around the original root:Save the child references before changing their interpretation. The input is a finite binary search tree satisfying part (b), and no external client may simultaneously rely on the old tree links. By structural induction, the two recursive calls return disjoint, ordered doubly linked lists containing exactly their original entries. The binary search tree bounds place every left word before
def flatten(t: Tree): (Tree, Tree) = {
if (t == null) (null, null)
else {
val oldLeft = t.left
val oldRight = t.right
val (ls, le) = flatten(oldLeft)
val (rs, re) = flatten(oldRight)
t.left = le
if (le != null) le.right = t
t.right = rs
if (rs != null) rs.left = t
val start = if (ls == null) t else ls
val end = if (re == null) t else re
(start, end)
}
}t.word and every right word after it. Splicing the left end, t, and the right start therefore preserves strict order and reciprocal links. A missing left list makes t.left null and t the start; a missing right list similarly makes t the end. The resulting outside links are null as required by part (i).The recursive loop variant analogue is the strictly smaller number of entries in each child subtree. Every entry is processed once and incurs a constant number of link writes, so time complexity is and recursion uses auxiliary space. No new
Tree entry is created; returning endpoint tuples and using a call stack are permitted auxiliary storage. The old binary search tree representation is consumed by this operation.start == null if and only if end == null. Otherwise there is a finite list of distinct software objects , with start eq v0, end eq v(m-1), v0.left == null, and v(m-1).right == null. For every valid adjacent pair, v(i).right eq v(i+1) and v(i+1).left eq v(i). These are exactly the entries reachable through the list links, so there is no extra cycle or branch. All counts are positive, all words are non-null, and words strictly increase from start to end.This is the representation invariant for a null-terminated doubly linked list without a list header. The abstraction function reads the ordered word-count pairs from left to right. Conversion additionally requires the output entries, counts and words to be precisely those of the input binary search tree; that preservation property is an operation postcondition, rather than an invariant of an arbitrary list by itself.
A list header is a permanent dummy node that represents the list boundary rather than a data entry. A circular linked list links the last node back to the first or to the header. A singly linked list stores only a successor; a doubly linked list stores both successor and predecessor references.
A header provides a stable object even for an empty list, makes insertions and deletions at the ends resemble interior operations, and can hold metadata such as the vector length in this question. In a circular header-based representation, the empty list is the header linked to itself.
A doubly linked list permits backward traversal and constant-time unlinking of a known node without searching for its predecessor. It uses an extra reference per node and requires more pointer updates and a stronger consistency invariant. Finding a node by an arbitrary index can still require a linear scan: bidirectional links alone do not provide random access.
Sparse linked vector representation 2026-10-31
Represent a finite sparse vector by a sorted doubly linked list of its nonzero coordinates. A list header stores the dimension; every ordinary linked list node stores a distinct valid coordinate index, its nonzero value, and predecessor and successor references. In a circular linked list, the endpoints connect to the header, and an all-zero vector has the header linked to itself in both directions. The representation invariant requires strictly increasing ordinary indices, valid bounds and mutually consistent forward and backward links. The abstraction function assigns zero to every coordinate absent from the list. The header is metadata, never an ordinary coordinate.
For example, has dimension nine and only three stored coordinates. The header's index field can store nine even though nine is not a valid coordinate index. Keeping the header distinct is therefore essential when searching for or removing entries.
A circular sparse linked vector representation stores only nonzero coordinates
. Details. 