Double-ended queue

Doubly linked list

While a simple list may implement a deque, a doubly-linked list is more adequate for its symmetry to achieve a fast access to both ends of the list (head and tail, hence the name head-tail linked list). The obvious solution is to manage two references; alternatively the deque can be built as a circular list.

In a doubly-linked list implementation, and assuming no allocation/deallocation overhead, the time complexity of all deque operations is O(1). Additionally, insertion or deletion in the middle, given an iterator, can also be achieved in constant time; however, the time taken for random access by index is O(n). Similarly, finding a specific element normally requires O(n) time. Linked data structures generally have poor locality of reference.

Dynamic array

In this case as well, while a dynamic array can be used to implement a deque, a variant that can grow from both ends is more appropriate. This is sometimes called array deques. This can be achieved in various ways, e.g.:

• by offsetting the position of the first element of the array in the reserved memory: the unused space is distributed on both side of the data;
• with a circular array.

The amortized time complexity of all deque operations with an array deque is O(1), thanks to the geometric expansion of the back-end buffer. Additionally, random access by index takes constant time ; but the average time taken for insertion or deletion in the middle are O(n). Thank to the fast random access, finding an element in an ordered array is time O(log n) (binary search). Each time the array is resized, the whole content is moved: the memory usage is then momentarily doubled (or more), and all direct (external) references to the content of the array are lost.

Functional and persistent implementations

Doubly linked lists cannot be used as immutable data structures. And an immutable array would be highly inefficient (an array is often simulated by a tree). A purely functional implementation of the deque can be based on stack, that is easily implemented with a singly linked list as an immutable and persistent structure.

There are several works in the literature dealing with this problem. All of these use two key ideas. The first is that a deque can be represented by a pair of stacks, one representing the front part of the deque and the other representing the reversal of the rear part. When one side becomes empty because of too many pop or eject operations, the deque, now all on one stack, is copied into two stacks each containing half of the deque elements. This fifty-fifty split guarantees that such copying, even though expensive, happens infrequently. A simple amortization argument shows that this gives a linear-time simulation of a deque by a constant number of stacks: k deque operations starting from an empty deque are simulated by O(k) stack operations.
[…]
The second idea is to use incremental copying to convert this linear-time simulation into a real-time simulation: as soon as the two stack become sufficiently unbalanced, recopying to create two balanced stacks begins.

This last process could be rather complicated as it needs to be executed concurrently with other operations and completed before the next one, to achieve an amortized real-time complexity. The next step is to support the operations in O(1) worst-case time. Another challenge is the real-time catenation of deques. Okasaki gives a simple solution that uses lazy lists combined with memoization. The stack to stack balancing is then partially automatic by means of a precise scheduling of incremental functions. ^{[17]: 52−59 : 115} However some authors deem such algorithm as not purely functional since memoization is considered as a side effect.^[18]: 581 Kaplan and Tarjan gives their own version of the purely functional deque (non-catenable), based on three ideas:^[18]

• data-structural bootstrapping, resulting in a recursive structure that foresees the finger tree: a deque is a triple consisting of a sub-deque flanked by two size-bounded buffers. Enqueue and dequeue basically operate on the buffers (in real-time because of the bounded size), and move forward one step in the balancing process;
• recursive slow down, inspired by Redundant binary representation (RBR), where an additional 2 digit stands for a suspended carry: the sub-deque contains pairs of elements from the parent deque, and the propagation of the balancing process to the sub-deque is delayed like is the carry propagation after the increment or decrement of a RBR number;^[17]: 105
• and a modification of the spine of the finger tree-like structure (a stack) into a stack of stacks that can be thought of as a 2-level skip list. This allows a real-time access to the unbalanced sub-deques. In analogy to a RBR, the sub-stacks stand for contiguous blocks of 1 digits, that can be skipped to access the next 2, i.e. a suspended carry.

In this paper Kaplan and Tarjan also present an even more complex version that achieves catenation in real-time. However this description is mostly textual. J. Viennot, A. Wendling, A. Guéneau, and F. Pottier publish a verified implementation of this data structure (in OCaml and Rocq), alongside a formal description and detailed analysis of the algorithm.^[19]

More generally, real-time catenation requires a deque is a tuple consisting mainly in two sub-structures, that themselves contain deques or compounds of deques. The linear spine of the non-catenable deque is then replaced by a binary skeleton.

Kaplan, Okasaki, and Tarjan produced a simpler, amortized version that can be implemented either using lazy evaluation or more efficiently using mutation in a broader but still restricted fashion.^[20] Mihaescu and Tarjan created a simpler (but still highly complex) strictly purely functional implementation of catenable deques, and also a much simpler implementation of strictly purely functional non catenable deques, both of which have optimal worst-case bounds (not officially published).^[21]

Monotonic queue

An input-restricted deque can be used to build a monotonic queue, i.e. a sub-sequence whom elements are in given order, either increasing or decreasing. Given a sequence, the algorithm only keeps elements in the desired order and discards the other ones. The order of the elements is preserved. To build an increasing (resp. decreasing) monotonic sequence, only stack operations are used :

• Start with an empty deque,
• For each element in the input sequence:
  • while the last element of the deque is greater (resp. smaller) than the current element, pop it out,
  • push the current element into the deque.

std::deque<int> increasing_monotonic_queue(std::vector<int> &seq[])
{
  std::deque<int> q;
  for (std::size_t i = 0; i < seq.size(); i++) {
    while (!q.empty() && q.back() > seq[i])
      q.pop_back();
    q.push_back(seq[i]);
  }
  return q;
}

The elements of the monotonic sequence can then be dequeued from the other side (hence the usage of a deque).

A monotonic queue can be used to find the minimum or maximum value in a sliding window over a sequence in a linear time complexity.^[23] The complexity of a naive solution is O(n.k) time and O(1) space, where n is the length of the input sequence and k the size of the window. A solution that uses some kind of search method is O(n.log n) time and O(n) space.

In the following code, the monotonic queue stores references to the elements of the sequence.

#include <vector>
#include <deque>

typedef std::vector<int>::const_iterator seq_iterator;
typedef std::deque<seq_iterator> monotonic_queue;

monotonic_queue & decreasing_monotonic_queue_push
( monotonic_queue &q, seq_iterator i )
{
  while (!q.empty() && *q.back() < *i)
    q.pop_back();
  q.push_back(i);
  return q;
}

std::vector<int> max_of_subarrays(std::vector<int> &seq, std::size_t win_sz )
{
  std::vector<int> max_of_sub;
  monotonic_queue decreasing;
  seq_iterator i = seq.begin();
  // scan the 1st window
  for ( size_t win_i = 0;
        i < seq.end() && win_i < win_sz; 
        i++, win_i++ )
    decreasing_monotonic_queue_push( decreasing, i );
  max_of_sub.push_back( *decreasing.front() );
  // scan the rest of the sequence
  for ( /*keep i*/; i < seq.end(); i++ )
    {
      if ( decreasing.front() <= i - win_sz )
        decreasing.pop_front(); // fall out of scope
      decreasing_monotonic_queue_push( decreasing, i );
      max_of_sub.push_back( *decreasing.front() );
    }
  return max_of_sub;
}

Each element of the input sequence is pushed and popped at most once resulting in 2.n operations. The time complexity is then O(n). The space complexity is O(k), the maximum size of the deque.

In a similar way a monotonic queue allows the optimization of some dynamic programming cases equivalent to the least-weight sub-sequence problem : shortest path problem for a weighted directed graph, paragraph breaking, etc. These problems are said to be convex or concave, and then monotonic. The time complexity is then reduced from O(n²) to O(n.log n), and O(n) in conducive situations. ^{[24] [25]}

An other direct application of the monotonic queue is the minimum queue or minqueue. In this case the count of items in the window varies. The minqueue is a data structure with a queue interface that additionally gives a direct access to the minimum stored item. The key operations are then enqueue, dequeue and find min. Despite the name alludes to the (min/max-) heap, the min/max-queue is not a priority queue: the order of the items is maintained from the enter to the exit (FIFO policy). A simple version of a min-queue with O(1) amortized time is a normal queue combined with an auxiliary increasing monotonic queue, that gives the minimum item on its front. The addition of an item applies to both queues, and when the minimum item is dequeued from the normal queue (i.e. same front item), it is also dequeued from the monotonic queue.^[9]

Convex hull of a simple polyline

Melkman's algorithm computes the convex hull of a simple polygonal chain (or a simple polygon) in linear time. The main difference with other similar algorithm is that Melkman requires vertices to be added on removed on both ends of the forming hull chain. Hence the use of a deque. The algorithm computes the position of each new vertex relative to the first and last segments (two vertices each) of the hull chain (stored in the deque). The vertex is either ignored or added (enqueued) to both sides of the deque (the hull is a loop), after removing (dequeueing) some previous vertices that are now on the inner side of the hull.^[26][27][28]

from collections import deque

def position(A,B,C):  
	det=(B.X-A.X)*(C.Y-A.Y)-(B.Y-A.Y)*(C.X-A.X)
	if det>0:
		return 1 # C is on the left of the line AB
	elif det<0 :
		return 0 # C is on the right of the line AB
	else:
		return -1 # A B and C are colinear

def melkman(path):
	if position(path[0], path[1], path[2]) == 1 :
		hull = deque([path[2], path[0], path[1], path[2]])
	else :
		hull = deque([path[2], path[1], path[0], path[2]])
	for v in path[3:] :
		if position(hull[0], hull[1], v) == 1 and position(hull[-2], hull[-1], v) == 1 : 
			continue
		while position(hull[0], hull[1], v) <= 0 :
			hull.popleft()
		hull.appendleft(0,v)
		while position(hull[-2], hull[-1], v) <=0 :
			hull.pop()
		hull.append(v)
    return hull

Simple priority queue

Stacks and queues can be seen as a particular kinds of priority queues, with the priority determined by the order in which the elements are inserted. Similarly a deque can implement a priority queue with two levels of priority: high priority elements are added to the front side of the deque while low priority ones are added to the rear.^[1] Unless an element could be canceled or stolen and then ejected from the bottom, an output-restricted deque is adequate.

Thus it's possible to modify the standard Dijkstra's algorithm to find single source shortest path in a graph with 0-cost and 1-cost edges. A deque substitutes the min-priority queue. 0-cost elements are enqueued in front of the deque (high-priority) and then are always processed before the higher cost elements (low-priority) that are enqueued at the rear end.

A deque is used in the work stealing algorithm.^[29] This algorithm implements task scheduling for several processors. A separate deque with threads to be executed is maintained for each processor. To execute the next thread, the processor gets the first element from the deque (using the "remove first element" deque operation). If the current thread forks, it is put back to the front of the deque ("insert element at front") and a new thread is executed. When one of the processors finishes execution of its own threads (i.e. its deque is empty), it can "steal" a thread from another processor: it gets the last element from the deque of another processor ("remove last element") and executes it. The work stealing algorithm is used by Intel's Threading Building Blocks (TBB) library for parallel programming.

Deque automaton

A Deque automaton (DA) is a finite-state machine equipped with a deque auxiliary memory. It generalizes Pushdown automaton (PDA) (stack automaton) and Queue automaton (Pull up automaton, PUA). As such it is equivalent to a Turing machine, and therefore it can process the same class of formal languages. But unlike the PDA and the PUA which impose serialization, a deque automaton permits parallel or interleaved execution of some operations.^[30]