Data Structures & AlgorithmsDouble-Ended Queue (Deque)

Double-Ended Queue (Deque) — Flexible Two-End Access

LevelIntermediate

Duration55 mins

TopicDouble-Ended Queue (Deque)

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Implementation Considerations

From Abstract to Concrete

Understanding the deque as an abstract data type is only half the story. The other half—equally important for a practicing engineer—is understanding how deques are implemented and what trade-offs different implementations make.

A poorly chosen implementation can turn your O(1) operations into O(n), waste memory, or cause cache misses that slow your program by orders of magnitude. By understanding implementation considerations, you'll make informed choices that keep your systems fast and efficient.

What You Will Learn

By the end of this page, you'll understand the two main implementation strategies (circular array and doubly-linked list), their respective trade-offs, when to choose each, and how standard library implementations in popular languages are designed.

The Implementation Landscape

There are two primary approaches to implementing a deque:

Array-Based (Circular Array/Ring Buffer) — Store elements in a contiguous array, using modular arithmetic to wrap around
Linked-List Based (Doubly-Linked List) — Store elements in nodes with prev/next pointers

A third hybrid approach exists in some standard libraries:

Block-Based (Array of Arrays) — An array of fixed-size blocks, combining benefits of both approaches

Each approach has distinct characteristics that make it suitable for different scenarios:

Deque Implementation Strategies Overview
Characteristic	Circular Array	Doubly-Linked List	Block-Based
Memory Layout	Contiguous	Scattered	Partially contiguous
Cache Performance	Excellent	Poor	Good
addFront/addRear	O(1) amortized	O(1) worst-case	O(1) amortized
removeFront/removeRear	O(1)	O(1)	O(1)
Random Access	O(1)	O(n)	O(1)
Memory Overhead	Low	High (2 pointers/element)	Medium
Resize Cost	O(n) occasionally	N/A (dynamic)	O(1) amortized
Insert in Middle	O(n)	O(1) if position known	O(n)

Standard Library Choices

Python's collections.deque uses a block-based approach (linked list of blocks). C++'s std::deque uses a block-based approach. Java's ArrayDeque uses a circular array. Understanding these choices helps you predict performance characteristics.

Circular Array Implementation

The circular array (or ring buffer) implementation is elegant and cache-friendly. It uses a fixed-size array with two indices—front and rear—that wrap around using modular arithmetic.

Key Concept: Wrapping Around

Imagine the array as a circle where position n-1 is adjacent to position 0. When front or rear moves past the array bounds, it wraps to the other side.

Linear View:    [_, A, B, C, D, _, _, _]
                    ↑           ↑
                  front        rear

Circular View:   7  0  1  2  3  4  5  6
                 ↻  A  B  C  D  ↻       
                    ↑           ↑
                  front        rear

After addFront(X):  [X, A, B, C, D, _, _, _]
                     ↑           ↑
                   front        rear
                   (wrapped from 0)

circular_array_deque
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/**
 * Circular Array-Based Deque Implementation
 * 
 * - O(1) amortized for all end operations
 * - O(1) random access (bonus!)
 * - Excellent cache performance
 * - Automatic resizing when full
 */
class CircularArrayDeque<T> {
    private array: (T | undefined)[];
    private frontIndex: number;
    private rearIndex: number;
    private count: number;
    private static readonly DEFAULT_CAPACITY = 16;
 
    constructor(initialCapacity: number = CircularArrayDeque.DEFAULT_CAPACITY) {
        this.array = new Array(initialCapacity);
        this.frontIndex = 0;
        this.rearIndex = 0;
        this.count = 0;
    }
 
    /**
     * Add element to front. O(1) amortized.
     */
    addFront(element: T): void {
        if (this.count === this.array.length) {
            this.resize(this.array.length * 2);
        }
 
        // Move front index backward (with wrap)
        this.frontIndex = this.decrementIndex(this.frontIndex);
        this.array[this.frontIndex] = element;
        this.count++;
    }
 
    /**
     * Add element to rear. O(1) amortized.
     */
    addRear(element: T): void {
        if (this.count === this.array.length) {
            this.resize(this.array.length * 2);
        }
 
        this.array[this.rearIndex] = element;
        // Move rear index forward (with wrap)
        this.rearIndex = this.incrementIndex(this.rearIndex);
        this.count++;
    }
 
    /**
     * Remove and return front element. O(1).
     */
    removeFront(): T {
        if (this.isEmpty()) {
            throw new Error("Deque is empty");
        }
 
        const element = this.array[this.frontIndex] as T;
        this.array[this.frontIndex] = undefined;  // Help GC
        this.frontIndex = this.incrementIndex(this.frontIndex);
        this.count--;
 
        return element;
    }
 
    /**
     * Remove and return rear element. O(1).
     */
    removeRear(): T {
        if (this.isEmpty()) {
            throw new Error("Deque is empty");
        }
 
        this.rearIndex = this.decrementIndex(this.rearIndex);
        const element = this.array[this.rearIndex] as T;
        this.array[this.rearIndex] = undefined;  // Help GC
        this.count--;
 
        return element;
    }
 
    peekFront(): T {
        if (this.isEmpty()) throw new Error("Deque is empty");
        return this.array[this.frontIndex] as T;
    }
 
    peekRear(): T {
        if (this.isEmpty()) throw new Error("Deque is empty");
        return this.array[this.decrementIndex(this.rearIndex)] as T;
    }
 
    isEmpty(): boolean {
        return this.count === 0;
    }
 
    size(): number {
        return this.count;
    }
 
    /**
     * BONUS: Random access in O(1) - not part of standard deque ADT
     */
    get(index: number): T {
        if (index < 0 || index >= this.count) {
            throw new Error("Index out of bounds");
        }
        const actualIndex = (this.frontIndex + index) % this.array.length;
        return this.array[actualIndex] as T;
    }
 
    // Helper: increment index with wrap-around
    private incrementIndex(index: number): number {
        return (index + 1) % this.array.length;
    }
 
    // Helper: decrement index with wrap-around
    private decrementIndex(index: number): number {
        return (index - 1 + this.array.length) % this.array.length;
    }
 
    // Resize array when full (doubles capacity)
    private resize(newCapacity: number): void {
        const newArray = new Array(newCapacity);
 
        // Copy elements in logical order
        for (let i = 0; i < this.count; i++) {
            newArray[i] = this.array[(this.frontIndex + i) % this.array.length];
        }
 
        this.array = newArray;
        this.frontIndex = 0;
        this.rearIndex = this.count;
    }
}

The Modular Arithmetic Trick

The key insight is using (index + array.length) % array.length for decrementing. This handles negative wraparound correctly. For incrementing, simple (index + 1) % array.length suffices.

Doubly-Linked List Implementation

The doubly-linked list implementation provides O(1) worst-case operations without any resizing costs. Each element is wrapped in a node that points to both its predecessor and successor.

Structure:

    null ← head ↔ A ↔ B ↔ C ↔ D ↔ tail → null
            ↑                       ↑
          front                   rear

    Node structure:
    ┌──────┬───────┬──────┐
    │ prev │ value │ next │
    └──────┴───────┴──────┘

Trade-off: Each node carries two pointer overheads (prev and next). For small elements, this can more than double memory usage. For large elements, the overhead is proportionally smaller.

linked_list_deque
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/**
 * Doubly-Linked List Deque Implementation
 * 
 * - O(1) worst-case for all end operations
 * - No resizing overhead
 * - Higher memory overhead per element
 * - Poor cache locality
 */
class DoublyLinkedDeque<T> {
    private head: Node<T> | null = null;
    private tail: Node<T> | null = null;
    private count: number = 0;
 
    /**
     * Add element to front. O(1) worst-case.
     */
    addFront(element: T): void {
        const newNode = new Node(element);
 
        if (this.isEmpty()) {
            this.head = this.tail = newNode;
        } else {
            newNode.next = this.head;
            this.head!.prev = newNode;
            this.head = newNode;
        }
 
        this.count++;
    }
 
    /**
     * Add element to rear. O(1) worst-case.
     */
    addRear(element: T): void {
        const newNode = new Node(element);
 
        if (this.isEmpty()) {
            this.head = this.tail = newNode;
        } else {
            newNode.prev = this.tail;
            this.tail!.next = newNode;
            this.tail = newNode;
        }
 
        this.count++;
    }
 
    /**
     * Remove and return front element. O(1) worst-case.
     */
    removeFront(): T {
        if (this.isEmpty()) {
            throw new Error("Deque is empty");
        }
 
        const element = this.head!.value;
        this.head = this.head!.next;
 
        if (this.head === null) {
            // Was the only element
            this.tail = null;
        } else {
            this.head.prev = null;
        }
 
        this.count--;
        return element;
    }
 
    /**
     * Remove and return rear element. O(1) worst-case.
     */
    removeRear(): T {
        if (this.isEmpty()) {
            throw new Error("Deque is empty");
        }
 
        const element = this.tail!.value;
        this.tail = this.tail!.prev;
 
        if (this.tail === null) {
            // Was the only element
            this.head = null;
        } else {
            this.tail.next = null;
        }
 
        this.count--;
        return element;
    }
 
    peekFront(): T {
        if (this.isEmpty()) throw new Error("Deque is empty");
        return this.head!.value;
    }
 
    peekRear(): T {
        if (this.isEmpty()) throw new Error("Deque is empty");
        return this.tail!.value;
    }
 
    isEmpty(): boolean {
        return this.count === 0;
    }
 
    size(): number {
        return this.count;
    }
}
 
class Node<T> {
    value: T;
    prev: Node<T> | null = null;
    next: Node<T> | null = null;
 
    constructor(value: T) {
        this.value = value;
    }
}

Watch for Single-Element Edge Cases

The trickiest bug in linked deque implementations is handling the single-element case. When removing the only element, both head AND tail must become null. Forgetting this creates dangling pointers and memory leaks.

Memory and Cache Analysis

Understanding memory usage patterns is crucial for performance-critical applications. Let's analyze both implementations:

Circular Array Memory:

Fixed overhead: 3 integers (front, rear, count) + array metadata
Per-element: Just the element itself (no pointer overhead)
Total for n elements: O(capacity) where capacity ≥ n
Wasted space: Up to 50% if recently shrunk or initially oversized

Doubly-Linked List Memory:

Fixed overhead: 2 pointers (head, tail) + count integer
Per-element: Element + 2 pointers (prev, next) = element + 16 bytes (64-bit)
Total for n elements: n × (element_size + 16 bytes)
No wasted space, but high overhead for small elements

Memory Comparison for 1000 Elements (64-bit system)
Element Size	Circular Array	Doubly-Linked List	Overhead Ratio
4 bytes (int)	~8 KB (2x capacity)	20 KB	2.5x worse for linked
8 bytes (long)	~16 KB	24 KB	1.5x worse for linked
64 bytes (struct)	~128 KB	80 KB	0.6x better for linked
1 KB (large object)	~2 MB	~1 MB	Similar, linked slightly better

Cache Locality Analysis:

Modern CPUs fetch memory in cache lines (typically 64 bytes). When you access an array element, neighboring elements come along "for free."

Circular Array:

Contiguous memory = excellent cache utilization
Traversing elements hits mostly cache (after first fetch)
Prefetcher can predict access patterns

Doubly-Linked List:

Nodes scattered across heap = poor cache utilization
Each node access may cause a cache miss
Prefetcher cannot predict next node location

In practice, this difference can make circular arrays 5-10x faster for sequential access patterns, even though both have O(1) complexity for individual operations.

The Big-O Lie

Two O(1) operations can differ by 100x in actual time due to cache effects. Big-O notation hides constant factors, but those constants include cache miss penalties (~100 CPU cycles) vs cache hits (~1 cycle). For hot paths, cache locality often matters more than algorithmic complexity.

Resizing Strategies for Array-Based Deques

Array-based deques must handle capacity limits. The resizing strategy significantly impacts amortized performance:

Strategy 1: Double on Full (Standard)

When capacity is reached, create 2x larger array
Copy all elements to new array
Amortized O(1): expensive operation is rare
Space efficiency: 25-100% utilization

Strategy 2: Grow by Fixed Amount

When full, add k slots (e.g., 100)
Simpler but leads to O(n) amortized for repeated adds
Not recommended for general use

Strategy 3: Shrink When Sparse

If utilization drops below 25%, halve capacity
Prevents memory waste after bulk removals
Risk: thrashing if size oscillates around threshold

resize_strategy
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/**
 * Amortized Analysis of Doubling Strategy
 * 
 * Consider adding n elements starting from capacity 1:
 * 
 * Resizes occur at sizes: 1, 2, 4, 8, 16, ..., 2^k (where 2^k < n ≤ 2^(k+1))
 * 
 * Total copy operations: 1 + 2 + 4 + 8 + ... + 2^k = 2^(k+1) - 1 < 2n
 * 
 * Total operations for n adds:
 *   - n adds (O(1) each) = n operations
 *   - Copy operations < 2n
 *   - Total < 3n
 * 
 * Amortized per operation: 3n/n = O(1)
 */
 
class AmortizedDeque<T> {
    private array: (T | undefined)[];
    private front: number = 0;
    private rear: number = 0;
    private count: number = 0;
    private static readonly MIN_CAPACITY = 8;
 
    constructor() {
        this.array = new Array(AmortizedDeque.MIN_CAPACITY);
    }
 
    addRear(element: T): void {
        this.ensureCapacity();
        this.array[this.rear] = element;
        this.rear = (this.rear + 1) % this.array.length;
        this.count++;
    }
 
    removeFront(): T {
        if (this.isEmpty()) throw new Error("Empty");
        
        const element = this.array[this.front] as T;
        this.array[this.front] = undefined;
        this.front = (this.front + 1) % this.array.length;
        this.count--;
 
        this.tryShrink();  // Prevent memory waste
        return element;
    }
 
    private ensureCapacity(): void {
        if (this.count === this.array.length) {
            this.resize(this.array.length * 2);
        }
    }
 
    private tryShrink(): void {
        // Shrink if less than 25% utilized and above minimum capacity
        if (this.count < this.array.length / 4 && 
            this.array.length > AmortizedDeque.MIN_CAPACITY * 2) {
            this.resize(this.array.length / 2);
        }
    }
 
    private resize(newCapacity: number): void {
        const newArray = new Array(Math.max(newCapacity, AmortizedDeque.MIN_CAPACITY));
        
        for (let i = 0; i < this.count; i++) {
            newArray[i] = this.array[(this.front + i) % this.array.length];
        }
 
        this.array = newArray;
        this.front = 0;
        this.rear = this.count;
    }
 
    isEmpty(): boolean { return this.count === 0; }
    size(): number { return this.count; }
}

Shrinking Carefully

Aggressive shrinking can cause 'thrashing': if size oscillates around the threshold, you repeatedly resize. Use hysteresis: shrink at 25% utilization, not 50%, to create a buffer zone. Some implementations skip shrinking entirely.

Standard Library Implementations

Let's examine how major languages implement deques in their standard libraries:

Python: collections.deque

Implementation: Block-based (linked list of 64-element blocks)
O(1) append/pop at both ends
O(n) random access (but still possible via indexing)
maxlen parameter for bounded deques with automatic overflow
Rotation support: rotate(n) in O(n)

Java: ArrayDeque

Implementation: Circular array
O(1) amortized for end operations
Does NOT support random access via index
Does NOT allow null elements
Generally faster than LinkedList for deque operations

C++: std::deque

Implementation: Block-based (array of arrays)
O(1) for end operations
O(1) random access!
Iterator stability: some insertions invalidate iterators
Memory: slightly more overhead than vector, less than list

Standard Library Deque Comparison
Feature	Python deque	Java ArrayDeque	C++ std::deque
Implementation	Block-based	Circular array	Block-based
Random access	O(1) but slow	Not available	O(1) fast
Insert middle	O(n)	Not available	O(n)
Memory efficiency	Good (blocks)	Excellent	Good (blocks)
Bounded option	Yes (maxlen)	No	No
Thread-safe	Atomic ops only	No	No
Null elements	Yes	No	Yes

stdlib_deque_usage
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from collections import deque
 
# Python deque is the recommended choice for queues and stacks
# per official Python documentation
 
# Basic usage
d = deque([1, 2, 3])
d.append(4)        # Right end
d.appendleft(0)    # Left end
d.pop()            # Right end: 4
d.popleft()        # Left end: 0
 
# Bounded deque (automatic overflow handling)
limited = deque(maxlen=3)
limited.append(1)  # [1]
limited.append(2)  # [1, 2]
limited.append(3)  # [1, 2, 3]
limited.append(4)  # [2, 3, 4]  -- 1 was dropped!
 
# Rotation (unique to Python deque)
d = deque([1, 2, 3, 4, 5])
d.rotate(2)        # [4, 5, 1, 2, 3]
d.rotate(-2)       # [1, 2, 3, 4, 5]
 
# Extend left (useful for prepending sequences)
d.extendleft([0, -1, -2])  # [-2, -1, 0, 1, 2, 3, 4, 5]
# Note: extendleft reverses the input order!
 
# Count occurrences
count = d.count(1)
 
# Reverse in-place
d.reverse()

Choose Your Language's Deque

In Python, always prefer collections.deque over lists for queue/stack behavior. In Java, prefer ArrayDeque over LinkedList unless you need null elements or random access. In C++, std::deque is suitable for most use cases, but consider std::vector if you primarily access by index.

Choosing the Right Implementation

Given the trade-offs we've analyzed, here's a decision framework for choosing a deque implementation:

When to Choose Circular Array

•Cache performance is critical — sequential access patterns, hot loops
•Memory is constrained — per-element overhead matters
•Elements are small — integers, short strings, small structs
•Random access is needed — indexed access beyond ends
•Approximate size is known — can pre-allocate to avoid resizing
•Latency variance is acceptable — occasional resize is OK

When to Choose Doubly-Linked List

•Worst-case latency matters — O(1) guaranteed, no resize spikes
•Elements are large — pointer overhead becomes negligible
•Middle insertions needed — can insert in O(1) with node reference
•Size is highly variable — grows and shrinks dramatically
•Real-time systems — predictable operation timing required
•Elements can't be moved — pointers to elements remain valid

Default Choice

For most applications, circular array (or standard library deque) is the right choice. Cache performance advantages usually outweigh worst-case latency concerns. Start here and switch only if profiling reveals issues.

Real-Time Exception

In hard real-time systems (robotics, trading, safety-critical), linked-list implementations may be necessary. A single resize causing a deadline miss can be catastrophic. Always profile with realistic workloads.

Thread-Safety Considerations

In concurrent applications, standard deque implementations are not thread-safe. Multiple threads accessing a deque simultaneously can cause data corruption, lost updates, or crashes.

Options for Concurrent Deques:

External Synchronization — Wrap operations in locks
Concurrent Deque Implementations — Use specialized thread-safe variants
Lock-Free Algorithms — Advanced techniques for contention-free access

Concurrent Deque Options by Language
Language	Class/Type	Characteristics
Java	ConcurrentLinkedDeque	Lock-free, unbounded, excellent for high contention
Java	LinkedBlockingDeque	Blocking, bounded, good for producer-consumer
Python	queue.deque + Lock	Manual locking (deque ops are atomic but not compound ops)
C++	tbb::concurrent_queue	Intel TBB library, high performance
Rust	crossbeam::deque	Work-stealing deque, excellent for task schedulers

concurrent_deque
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/**
 * Thread-safe deque wrapper (conceptual - TypeScript is single-threaded)
 * 
 * In multi-threaded environments (e.g., with Web Workers), 
 * you'd need synchronization primitives.
 */
class ThreadSafeDeque<T> {
    private deque = new Deque<T>();
    private lock = new Mutex();  // Conceptual
 
    async addFront(element: T): Promise<void> {
        await this.lock.acquire();
        try {
            this.deque.addFront(element);
        } finally {
            this.lock.release();
        }
    }
 
    async removeFront(): Promise<T> {
        await this.lock.acquire();
        try {
            return this.deque.removeFront();
        } finally {
            this.lock.release();
        }
    }
 
    // For high-contention scenarios, consider:
    // - Lock-free algorithms (compare-and-swap)
    // - Sharded deques (one per thread, steal from others)
    // - Read-write locks (multiple readers OR one writer)
}

Work-Stealing Deques

In parallel task schedulers, work-stealing deques are a key pattern: each thread has its own deque, pushing/popping at one end. When idle, threads 'steal' from other threads' opposite ends. This minimizes contention while balancing load. See Chase-Lev deque for a famous lock-free implementation.

Summary: Implementation Considerations

Let's consolidate the key implementation insights:

Key Takeaways

•Two main implementations: Circular array (cache-friendly, random access) and doubly-linked list (O(1) worst-case, stable pointers)
•Standard libraries use block-based: Python and C++ use hybrid approaches combining benefits of both
•Cache locality matters: Array-based can be 5-10x faster for sequential access despite same Big-O
•Resizing strategy affects amortized cost: Doubling ensures O(1) amortized; fixed increments give O(n)
•Memory overhead varies: Linked lists use 2 pointers per element; arrays have no per-element overhead
•Thread-safety requires explicit handling: Standard deques are not thread-safe; use concurrent variants or locks
•Default to standard library: Unless you have specific requirements, use your language's deque implementation
•Profile before optimizing: Big-O hides constants; measure with realistic workloads

Module Complete:

You've now achieved a comprehensive understanding of the Double-Ended Queue (Deque):

What a deque is and its conceptual foundation
How to use all deque operations effectively
Why deques generalize stacks and queues
How to implement deques with different trade-offs

This knowledge prepares you for advanced algorithms like sliding window maximum, work-stealing schedulers, and 0-1 BFS that leverage deque capabilities.

Module Complete

Congratulations! You've mastered the Double-Ended Queue. You understand it abstractly, operationally, theoretically (as a generalization), and practically (implementation trade-offs). Use this knowledge to choose the right data structure and implementation for your specific needs.

4 / 4

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Data Structures & AlgorithmsDouble-Ended Queue (Deque)

Double-Ended Queue (Deque) — Flexible Two-End Access

LevelIntermediate

Duration55 mins

TopicDouble-Ended Queue (Deque)

4 / 4

Implementation Considerations

From Abstract to Concrete

What You Will Learn

The Implementation Landscape

There are two primary approaches to implementing a deque:

Array-Based (Circular Array/Ring Buffer) — Store elements in a contiguous array, using modular arithmetic to wrap around
Linked-List Based (Doubly-Linked List) — Store elements in nodes with prev/next pointers

A third hybrid approach exists in some standard libraries:

Block-Based (Array of Arrays) — An array of fixed-size blocks, combining benefits of both approaches

Each approach has distinct characteristics that make it suitable for different scenarios:

Deque Implementation Strategies Overview
Characteristic	Circular Array	Doubly-Linked List	Block-Based
Memory Layout	Contiguous	Scattered	Partially contiguous
Cache Performance	Excellent	Poor	Good
addFront/addRear	O(1) amortized	O(1) worst-case	O(1) amortized
removeFront/removeRear	O(1)	O(1)	O(1)
Random Access	O(1)	O(n)	O(1)
Memory Overhead	Low	High (2 pointers/element)	Medium
Resize Cost	O(n) occasionally	N/A (dynamic)	O(1) amortized
Insert in Middle	O(n)	O(1) if position known	O(n)

Standard Library Choices

Circular Array Implementation

The circular array (or ring buffer) implementation is elegant and cache-friendly. It uses a fixed-size array with two indices—front and rear—that wrap around using modular arithmetic.

Key Concept: Wrapping Around

Imagine the array as a circle where position n-1 is adjacent to position 0. When front or rear moves past the array bounds, it wraps to the other side.

Linear View:    [_, A, B, C, D, _, _, _]
                    ↑           ↑
                  front        rear

Circular View:   7  0  1  2  3  4  5  6
                 ↻  A  B  C  D  ↻       
                    ↑           ↑
                  front        rear

After addFront(X):  [X, A, B, C, D, _, _, _]
                     ↑           ↑
                   front        rear
                   (wrapped from 0)

circular_array_deque
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/**
 * Circular Array-Based Deque Implementation
 * 
 * - O(1) amortized for all end operations
 * - O(1) random access (bonus!)
 * - Excellent cache performance
 * - Automatic resizing when full
 */
class CircularArrayDeque<T> {
    private array: (T | undefined)[];
    private frontIndex: number;
    private rearIndex: number;
    private count: number;
    private static readonly DEFAULT_CAPACITY = 16;
 
    constructor(initialCapacity: number = CircularArrayDeque.DEFAULT_CAPACITY) {
        this.array = new Array(initialCapacity);
        this.frontIndex = 0;
        this.rearIndex = 0;
        this.count = 0;
    }
 
    /**
     * Add element to front. O(1) amortized.
     */
    addFront(element: T): void {
        if (this.count === this.array.length) {
            this.resize(this.array.length * 2);
        }
 
        // Move front index backward (with wrap)
        this.frontIndex = this.decrementIndex(this.frontIndex);
        this.array[this.frontIndex] = element;
        this.count++;
    }
 
    /**
     * Add element to rear. O(1) amortized.
     */
    addRear(element: T): void {
        if (this.count === this.array.length) {
            this.resize(this.array.length * 2);
        }
 
        this.array[this.rearIndex] = element;
        // Move rear index forward (with wrap)
        this.rearIndex = this.incrementIndex(this.rearIndex);
        this.count++;
    }
 
    /**
     * Remove and return front element. O(1).
     */
    removeFront(): T {
        if (this.isEmpty()) {
            throw new Error("Deque is empty");
        }
 
        const element = this.array[this.frontIndex] as T;
        this.array[this.frontIndex] = undefined;  // Help GC
        this.frontIndex = this.incrementIndex(this.frontIndex);
        this.count--;
 
        return element;
    }
 
    /**
     * Remove and return rear element. O(1).
     */
    removeRear(): T {
        if (this.isEmpty()) {
            throw new Error("Deque is empty");
        }
 
        this.rearIndex = this.decrementIndex(this.rearIndex);
        const element = this.array[this.rearIndex] as T;
        this.array[this.rearIndex] = undefined;  // Help GC
        this.count--;
 
        return element;
    }
 
    peekFront(): T {
        if (this.isEmpty()) throw new Error("Deque is empty");
        return this.array[this.frontIndex] as T;
    }
 
    peekRear(): T {
        if (this.isEmpty()) throw new Error("Deque is empty");
        return this.array[this.decrementIndex(this.rearIndex)] as T;
    }
 
    isEmpty(): boolean {
        return this.count === 0;
    }
 
    size(): number {
        return this.count;
    }
 
    /**
     * BONUS: Random access in O(1) - not part of standard deque ADT
     */
    get(index: number): T {
        if (index < 0 || index >= this.count) {
            throw new Error("Index out of bounds");
        }
        const actualIndex = (this.frontIndex + index) % this.array.length;
        return this.array[actualIndex] as T;
    }
 
    // Helper: increment index with wrap-around
    private incrementIndex(index: number): number {
        return (index + 1) % this.array.length;
    }
 
    // Helper: decrement index with wrap-around
    private decrementIndex(index: number): number {
        return (index - 1 + this.array.length) % this.array.length;
    }
 
    // Resize array when full (doubles capacity)
    private resize(newCapacity: number): void {
        const newArray = new Array(newCapacity);
 
        // Copy elements in logical order
        for (let i = 0; i < this.count; i++) {
            newArray[i] = this.array[(this.frontIndex + i) % this.array.length];
        }
 
        this.array = newArray;
        this.frontIndex = 0;
        this.rearIndex = this.count;
    }
}

The Modular Arithmetic Trick

The key insight is using (index + array.length) % array.length for decrementing. This handles negative wraparound correctly. For incrementing, simple (index + 1) % array.length suffices.

Doubly-Linked List Implementation

The doubly-linked list implementation provides O(1) worst-case operations without any resizing costs. Each element is wrapped in a node that points to both its predecessor and successor.

Structure:

    null ← head ↔ A ↔ B ↔ C ↔ D ↔ tail → null
            ↑                       ↑
          front                   rear

    Node structure:
    ┌──────┬───────┬──────┐
    │ prev │ value │ next │
    └──────┴───────┴──────┘

Trade-off: Each node carries two pointer overheads (prev and next). For small elements, this can more than double memory usage. For large elements, the overhead is proportionally smaller.

linked_list_deque
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/**
 * Doubly-Linked List Deque Implementation
 * 
 * - O(1) worst-case for all end operations
 * - No resizing overhead
 * - Higher memory overhead per element
 * - Poor cache locality
 */
class DoublyLinkedDeque<T> {
    private head: Node<T> | null = null;
    private tail: Node<T> | null = null;
    private count: number = 0;
 
    /**
     * Add element to front. O(1) worst-case.
     */
    addFront(element: T): void {
        const newNode = new Node(element);
 
        if (this.isEmpty()) {
            this.head = this.tail = newNode;
        } else {
            newNode.next = this.head;
            this.head!.prev = newNode;
            this.head = newNode;
        }
 
        this.count++;
    }
 
    /**
     * Add element to rear. O(1) worst-case.
     */
    addRear(element: T): void {
        const newNode = new Node(element);
 
        if (this.isEmpty()) {
            this.head = this.tail = newNode;
        } else {
            newNode.prev = this.tail;
            this.tail!.next = newNode;
            this.tail = newNode;
        }
 
        this.count++;
    }
 
    /**
     * Remove and return front element. O(1) worst-case.
     */
    removeFront(): T {
        if (this.isEmpty()) {
            throw new Error("Deque is empty");
        }
 
        const element = this.head!.value;
        this.head = this.head!.next;
 
        if (this.head === null) {
            // Was the only element
            this.tail = null;
        } else {
            this.head.prev = null;
        }
 
        this.count--;
        return element;
    }
 
    /**
     * Remove and return rear element. O(1) worst-case.
     */
    removeRear(): T {
        if (this.isEmpty()) {
            throw new Error("Deque is empty");
        }
 
        const element = this.tail!.value;
        this.tail = this.tail!.prev;
 
        if (this.tail === null) {
            // Was the only element
            this.head = null;
        } else {
            this.tail.next = null;
        }
 
        this.count--;
        return element;
    }
 
    peekFront(): T {
        if (this.isEmpty()) throw new Error("Deque is empty");
        return this.head!.value;
    }
 
    peekRear(): T {
        if (this.isEmpty()) throw new Error("Deque is empty");
        return this.tail!.value;
    }
 
    isEmpty(): boolean {
        return this.count === 0;
    }
 
    size(): number {
        return this.count;
    }
}
 
class Node<T> {
    value: T;
    prev: Node<T> | null = null;
    next: Node<T> | null = null;
 
    constructor(value: T) {
        this.value = value;
    }
}

Watch for Single-Element Edge Cases

Memory and Cache Analysis

Understanding memory usage patterns is crucial for performance-critical applications. Let's analyze both implementations:

Circular Array Memory:

Fixed overhead: 3 integers (front, rear, count) + array metadata
Per-element: Just the element itself (no pointer overhead)
Total for n elements: O(capacity) where capacity ≥ n
Wasted space: Up to 50% if recently shrunk or initially oversized

Doubly-Linked List Memory:

Fixed overhead: 2 pointers (head, tail) + count integer
Per-element: Element + 2 pointers (prev, next) = element + 16 bytes (64-bit)
Total for n elements: n × (element_size + 16 bytes)
No wasted space, but high overhead for small elements

Memory Comparison for 1000 Elements (64-bit system)
Element Size	Circular Array	Doubly-Linked List	Overhead Ratio
4 bytes (int)	~8 KB (2x capacity)	20 KB	2.5x worse for linked
8 bytes (long)	~16 KB	24 KB	1.5x worse for linked
64 bytes (struct)	~128 KB	80 KB	0.6x better for linked
1 KB (large object)	~2 MB	~1 MB	Similar, linked slightly better

Cache Locality Analysis:

Modern CPUs fetch memory in cache lines (typically 64 bytes). When you access an array element, neighboring elements come along "for free."

Circular Array:

Contiguous memory = excellent cache utilization
Traversing elements hits mostly cache (after first fetch)
Prefetcher can predict access patterns

Doubly-Linked List:

Nodes scattered across heap = poor cache utilization
Each node access may cause a cache miss
Prefetcher cannot predict next node location

In practice, this difference can make circular arrays 5-10x faster for sequential access patterns, even though both have O(1) complexity for individual operations.

The Big-O Lie

Resizing Strategies for Array-Based Deques

Array-based deques must handle capacity limits. The resizing strategy significantly impacts amortized performance:

Strategy 1: Double on Full (Standard)

When capacity is reached, create 2x larger array
Copy all elements to new array
Amortized O(1): expensive operation is rare
Space efficiency: 25-100% utilization

Strategy 2: Grow by Fixed Amount

When full, add k slots (e.g., 100)
Simpler but leads to O(n) amortized for repeated adds
Not recommended for general use

Strategy 3: Shrink When Sparse

If utilization drops below 25%, halve capacity
Prevents memory waste after bulk removals
Risk: thrashing if size oscillates around threshold

resize_strategy
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/**
 * Amortized Analysis of Doubling Strategy
 * 
 * Consider adding n elements starting from capacity 1:
 * 
 * Resizes occur at sizes: 1, 2, 4, 8, 16, ..., 2^k (where 2^k < n ≤ 2^(k+1))
 * 
 * Total copy operations: 1 + 2 + 4 + 8 + ... + 2^k = 2^(k+1) - 1 < 2n
 * 
 * Total operations for n adds:
 *   - n adds (O(1) each) = n operations
 *   - Copy operations < 2n
 *   - Total < 3n
 * 
 * Amortized per operation: 3n/n = O(1)
 */
 
class AmortizedDeque<T> {
    private array: (T | undefined)[];
    private front: number = 0;
    private rear: number = 0;
    private count: number = 0;
    private static readonly MIN_CAPACITY = 8;
 
    constructor() {
        this.array = new Array(AmortizedDeque.MIN_CAPACITY);
    }
 
    addRear(element: T): void {
        this.ensureCapacity();
        this.array[this.rear] = element;
        this.rear = (this.rear + 1) % this.array.length;
        this.count++;
    }
 
    removeFront(): T {
        if (this.isEmpty()) throw new Error("Empty");
        
        const element = this.array[this.front] as T;
        this.array[this.front] = undefined;
        this.front = (this.front + 1) % this.array.length;
        this.count--;
 
        this.tryShrink();  // Prevent memory waste
        return element;
    }
 
    private ensureCapacity(): void {
        if (this.count === this.array.length) {
            this.resize(this.array.length * 2);
        }
    }
 
    private tryShrink(): void {
        // Shrink if less than 25% utilized and above minimum capacity
        if (this.count < this.array.length / 4 && 
            this.array.length > AmortizedDeque.MIN_CAPACITY * 2) {
            this.resize(this.array.length / 2);
        }
    }
 
    private resize(newCapacity: number): void {
        const newArray = new Array(Math.max(newCapacity, AmortizedDeque.MIN_CAPACITY));
        
        for (let i = 0; i < this.count; i++) {
            newArray[i] = this.array[(this.front + i) % this.array.length];
        }
 
        this.array = newArray;
        this.front = 0;
        this.rear = this.count;
    }
 
    isEmpty(): boolean { return this.count === 0; }
    size(): number { return this.count; }
}

Shrinking Carefully

Standard Library Implementations

Let's examine how major languages implement deques in their standard libraries:

Python: collections.deque

Implementation: Block-based (linked list of 64-element blocks)
O(1) append/pop at both ends
O(n) random access (but still possible via indexing)
maxlen parameter for bounded deques with automatic overflow
Rotation support: rotate(n) in O(n)

Java: ArrayDeque

Implementation: Circular array
O(1) amortized for end operations
Does NOT support random access via index
Does NOT allow null elements
Generally faster than LinkedList for deque operations

C++: std::deque

Implementation: Block-based (array of arrays)
O(1) for end operations
O(1) random access!
Iterator stability: some insertions invalidate iterators
Memory: slightly more overhead than vector, less than list

Standard Library Deque Comparison
Feature	Python deque	Java ArrayDeque	C++ std::deque
Implementation	Block-based	Circular array	Block-based
Random access	O(1) but slow	Not available	O(1) fast
Insert middle	O(n)	Not available	O(n)
Memory efficiency	Good (blocks)	Excellent	Good (blocks)
Bounded option	Yes (maxlen)	No	No
Thread-safe	Atomic ops only	No	No
Null elements	Yes	No	Yes

stdlib_deque_usage
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from collections import deque
 
# Python deque is the recommended choice for queues and stacks
# per official Python documentation
 
# Basic usage
d = deque([1, 2, 3])
d.append(4)        # Right end
d.appendleft(0)    # Left end
d.pop()            # Right end: 4
d.popleft()        # Left end: 0
 
# Bounded deque (automatic overflow handling)
limited = deque(maxlen=3)
limited.append(1)  # [1]
limited.append(2)  # [1, 2]
limited.append(3)  # [1, 2, 3]
limited.append(4)  # [2, 3, 4]  -- 1 was dropped!
 
# Rotation (unique to Python deque)
d = deque([1, 2, 3, 4, 5])
d.rotate(2)        # [4, 5, 1, 2, 3]
d.rotate(-2)       # [1, 2, 3, 4, 5]
 
# Extend left (useful for prepending sequences)
d.extendleft([0, -1, -2])  # [-2, -1, 0, 1, 2, 3, 4, 5]
# Note: extendleft reverses the input order!
 
# Count occurrences
count = d.count(1)
 
# Reverse in-place
d.reverse()

Choose Your Language's Deque

Choosing the Right Implementation

Given the trade-offs we've analyzed, here's a decision framework for choosing a deque implementation:

When to Choose Circular Array

•Cache performance is critical — sequential access patterns, hot loops
•Memory is constrained — per-element overhead matters
•Elements are small — integers, short strings, small structs
•Random access is needed — indexed access beyond ends
•Approximate size is known — can pre-allocate to avoid resizing
•Latency variance is acceptable — occasional resize is OK

When to Choose Doubly-Linked List

•Worst-case latency matters — O(1) guaranteed, no resize spikes
•Elements are large — pointer overhead becomes negligible
•Middle insertions needed — can insert in O(1) with node reference
•Size is highly variable — grows and shrinks dramatically
•Real-time systems — predictable operation timing required
•Elements can't be moved — pointers to elements remain valid

Default Choice

Real-Time Exception

Thread-Safety Considerations

In concurrent applications, standard deque implementations are not thread-safe. Multiple threads accessing a deque simultaneously can cause data corruption, lost updates, or crashes.

Options for Concurrent Deques:

External Synchronization — Wrap operations in locks
Concurrent Deque Implementations — Use specialized thread-safe variants
Lock-Free Algorithms — Advanced techniques for contention-free access

Concurrent Deque Options by Language
Language	Class/Type	Characteristics
Java	ConcurrentLinkedDeque	Lock-free, unbounded, excellent for high contention
Java	LinkedBlockingDeque	Blocking, bounded, good for producer-consumer
Python	queue.deque + Lock	Manual locking (deque ops are atomic but not compound ops)
C++	tbb::concurrent_queue	Intel TBB library, high performance
Rust	crossbeam::deque	Work-stealing deque, excellent for task schedulers

concurrent_deque
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/**
 * Thread-safe deque wrapper (conceptual - TypeScript is single-threaded)
 * 
 * In multi-threaded environments (e.g., with Web Workers), 
 * you'd need synchronization primitives.
 */
class ThreadSafeDeque<T> {
    private deque = new Deque<T>();
    private lock = new Mutex();  // Conceptual
 
    async addFront(element: T): Promise<void> {
        await this.lock.acquire();
        try {
            this.deque.addFront(element);
        } finally {
            this.lock.release();
        }
    }
 
    async removeFront(): Promise<T> {
        await this.lock.acquire();
        try {
            return this.deque.removeFront();
        } finally {
            this.lock.release();
        }
    }
 
    // For high-contention scenarios, consider:
    // - Lock-free algorithms (compare-and-swap)
    // - Sharded deques (one per thread, steal from others)
    // - Read-write locks (multiple readers OR one writer)
}

Work-Stealing Deques

Summary: Implementation Considerations

Let's consolidate the key implementation insights:

Key Takeaways

•Two main implementations: Circular array (cache-friendly, random access) and doubly-linked list (O(1) worst-case, stable pointers)
•Standard libraries use block-based: Python and C++ use hybrid approaches combining benefits of both
•Cache locality matters: Array-based can be 5-10x faster for sequential access despite same Big-O
•Resizing strategy affects amortized cost: Doubling ensures O(1) amortized; fixed increments give O(n)
•Memory overhead varies: Linked lists use 2 pointers per element; arrays have no per-element overhead
•Thread-safety requires explicit handling: Standard deques are not thread-safe; use concurrent variants or locks
•Default to standard library: Unless you have specific requirements, use your language's deque implementation
•Profile before optimizing: Big-O hides constants; measure with realistic workloads

Module Complete:

You've now achieved a comprehensive understanding of the Double-Ended Queue (Deque):

What a deque is and its conceptual foundation
How to use all deque operations effectively
Why deques generalize stacks and queues
How to implement deques with different trade-offs

This knowledge prepares you for advanced algorithms like sliding window maximum, work-stealing schedulers, and 0-1 BFS that leverage deque capabilities.

Module Complete

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