Linked Lists, Stacks, and Queues

## Intuition

Building blocks of all data structures: Linked lists, stacks, and queues are like the atoms of data structures — almost every complex data structure is built from these primitives. Linked lists give dynamic sizing, stacks enforce LIFO order, and queues enforce FIFO order.

Why it matters: Understanding these fundamentals lets you implement more complex data structures from scratch and choose the right abstraction for the job. Stacks handle undo/redo, queues handle task scheduling, and linked lists handle dynamic collections.

The key insight: Linked lists trade space for flexibility — each node wastes pointer space but gains O(1) insertion/deletion. Arrays trade flexibility for cache efficiency — contiguous memory enables O(1) access but O(n) insertion.

Singly Linked Lists

A singly linked list is a sequence of nodes where each node contains a value and a reference to the Next node. The list is accessed through a head pointer; traversal requires following pointers from The head.

class ListNode:
    def __init__(self, val=0, next=None):
        self.val = val
        self.next = next

Insertion and Deletion

def insert_at_head(head, val):
    """Insert at the head of the list. O(1)."""
    return ListNode(val, head)

def insert_at_tail(head, val):
    """Insert at the tail. O(n) without tail pointer, O(1) with tail pointer."""
    new_node = ListNode(val)
    if head is None:
        return new_node
    current = head
    while current.next:
        current = current.next
    current.next = new_node
    return head

def delete_node(head, val):
    """Delete the first node with the given value. O(n)."""
    if head is None:
        return None
    if head.val == val:
        return head.next
    current = head
    while current.next:
        if current.next.val == val:
            current.next = current.next.next
            return head
        current = current.next
    return head

Position (given a pointer to the preceding node). The critical disadvantage is $O(n)$ random access And poor cache locality. In practice, arrays dominate because cache effects matter more than Theoretical complexity for typical data sizes.

Reversal

def reverse_linked_list(head):
    """
    Reverse a singly linked list iteratively.
    Time: O(n), Space: O(1)
    """
    prev = None
    current = head
    while current:
        next_node = current.next
        current.next = prev
        prev = current
        current = next_node
    return prev

def reverse_recursive(head):
    """
    Reverse a singly linked list recursively.
    Time: O(n), Space: O(n) — recursion stack
    """
    if head is None or head.next is None:
        return head
    new_head = reverse_recursive(head.next)
    head.next.next = head
    head.next = None
    return new_head

Cycle Detection (Floyd”s Tortoise and Hare)

Floyd’s algorithm uses two pointers moving at different speeds. If there is a cycle, the fast Pointer will eventually catch up to the slow pointer.

def has_cycle(head):
    """
    Detect if a linked list has a cycle.
    Time: O(n), Space: O(1)
    """
    slow = head
    fast = head
    while fast and fast.next:
        slow = slow.next
        fast = fast.next.next
        if slow == fast:
            return True
    return False

def find_cycle_start(head):
    """
    Find the node where the cycle begins.
    Time: O(n), Space: O(1)
    """
    slow = head
    fast = head
    while fast and fast.next:
        slow = slow.next
        fast = fast.next.next
        if slow == fast:
            # Reset slow to head, move both at speed 1
            slow = head
            while slow != fast:
                slow = slow.next
                fast = fast.next
            return slow
    return None

Why this works: Let $\mu$ be the distance from head to cycle start, and $\lambda$ be the cycle Length. When slow and fast meet, slow has travelled $\mu + a\lambda$ steps and fast has travelled $\mu + a\lambda + b\lambda$ steps for some non-negative integers $a, b$ . Since fast moves twice as Fast: $2(\mu + a\lambda) = \mu + a\lambda + b\lambda$ Which gives $\mu = (b - a)\lambda$ . So the Distance from the meeting point to the cycle start (going around the cycle) is exactly $\mu$ — the Same as the distance from head to cycle start.

Fast and Slow Pointer Technique

Beyond cycle detection, the fast/slow pointer pattern is useful for:

def find_middle(head):
    """
    Find the middle node of a linked list.
    If even length, returns the second middle node.
    Time: O(n), Space: O(1)
    """
    slow = head
    fast = head
    while fast and fast.next:
        slow = slow.next
        fast = fast.next.next
    return slow

def is_palindrome(head):
    """
    Check if a linked list is a palindrome.
    Time: O(n), Space: O(1)
    """
    if not head or not head.next:
        return True
    # Find middle
    slow = head
    fast = head
    while fast.next and fast.next.next:
        slow = slow.next
        fast = fast.next.next
    # Reverse second half
    second_half = reverse_linked_list(slow.next)
    # Compare
    first = head
    second = second_half
    result = True
    while second:
        if first.val != second.val:
            result = False
            break
        first = first.next
        second = second.next
    # Restore (optional, but good practice)
    reverse_linked_list(second_half)
    return result

Doubly Linked Lists

Each node has both a next and prev pointer.

class DoublyListNode:
    def __init__(self, val=0, prev=None, next=None):
        self.val = val
        self.prev = prev
        self.next = next

Doubly linked lists support $O(1)$ deletion given a pointer to the node (no need to find the Predecessor) and reverse traversal. The trade-off is extra memory per node (one pointer) and more Complex insertion/deletion logic.

def delete_doubly_node(node):
    """
    Delete a node from a doubly linked list given the node itself.
    Time: O(1)
    """
    if node.prev:
        node.prev.next = node.next
    if node.next:
        node.next.prev = node.prev
    node.prev = None
    node.next = None

Skip Lists

A skip list is a probabilistic data structure that provides $O(\log n)$ expected-time search, Insertion, and deletion — the same asymptotic complexity as a balanced BST, but with simpler Implementation.

The structure consists of multiple levels of linked lists. The bottom level contains all elements. Each higher level is a “fast lane” that contains a random subset of elements from the level below.

import random

class SkipListNode:
    def __init__(self, val, levels):
        self.val = val
        self.forward = [None] * levels

class SkipList:
    """
    Probabilistic skip list.
    Expected time: O(log n) for search, insert, delete
    Space: O(n) expected
    """
    MAX_LEVEL = 16
    P = 0.5  # probability of promoting to next level

    def __init__(self):
        self.head = SkipListNode(float('-inf'), self.MAX_LEVEL)
        self.level = 1

    def _random_level(self):
        level = 1
        while random.random() < self.P and level < self.MAX_LEVEL:
            level += 1
        return level

    def search(self, target):
        current = self.head
        for i in range(self.level - 1, -1, -1):
            while current.forward[i] and current.forward[i].val < target:
                current = current.forward[i]
        current = current.forward[0]
        return current is not None and current.val == target

    def insert(self, val):
        update = [None] * self.MAX_LEVEL
        current = self.head
        for i in range(self.level - 1, -1, -1):
            while current.forward[i] and current.forward[i].val < val:
                current = current.forward[i]
            update[i] = current

        new_level = self._random_level()
        if new_level > self.level:
            for i in range(self.level, new_level):
                update[i] = self.head
            self.level = new_level

        new_node = SkipListNode(val, new_level)
        for i in range(new_level):
            new_node.forward[i] = update[i].forward[i]
            update[i].forward[i] = new_node

Kernel (for process address space management). They are preferred over balanced BSTs in these Contexts because they are simpler to implement correctly in concurrent settings — insertion and Deletion only need to lock the nodes being modified, not the entire structure.

Stacks

A stack is a LIFO (Last In, First Out) data structure with two primary operations: push (add to Top) and pop (remove from top).

Implementation

class Stack:
    """
    Stack implemented with a dynamic array.
    All operations: O(1) amortised
    """
    def __init__(self):
        self.data = []

    def push(self, val):
        self.data.append(val)

    def pop(self):
        if self.is_empty():
            raise IndexError("pop from empty stack")
        return self.data.pop()

    def peek(self):
        if self.is_empty():
            raise IndexError("peek at empty stack")
        return self.data[-1]

    def is_empty(self):
        return len(self.data) == 0

    def size(self):
        return len(self.data)

Applications

Expression Evaluation (Shunting-Yard Algorithm):

Dijkstra’s shunting-yard algorithm converts infix notation to postfix (Reverse Polish Notation), Which can then be evaluated with a simple stack-based algorithm.

def evaluate_expression(expr):
    """
    Evaluate a simple arithmetic expression with +, -, *, /, (, ).
    Time: O(n), Space: O(n)
    """
    def precedence(op):
        if op in ('+', '-'):
            return 1
        if op in ('*', '/'):
            return 2
        return 0

    def apply_op(a, b, op):
        if op == '+': return a + b
        if op == '-': return a - b
        if op == '*': return a * b
        if op == '/': return a // b  # integer division

    values = []
    ops = []
    i = 0
    while i < len(expr):
        c = expr[i]
        if c == ' ':
            i += 1
            continue
        elif c.isdigit():
            val = 0
            while i < len(expr) and expr[i].isdigit():
                val = val * 10 + int(expr[i])
                i += 1
            values.append(val)
            continue
        elif c == '(':
            ops.append(c)
        elif c == ')':
            while ops and ops[-1] != '(':
                values.append(apply_op(values.pop(), values.pop(), ops.pop()))
            ops.pop()  # remove '('
        else:
            while ops and precedence(ops[-1]) >= precedence(c):
                values.append(apply_op(values.pop(), values.pop(), ops.pop()))
            ops.append(c)
        i += 1

    while ops:
        values.append(apply_op(values.pop(), values.pop(), ops.pop()))

    return values[0]

Bracket Matching:

def is_valid_parentheses(s):
    """
    Check if brackets are properly matched and nested.
    Time: O(n), Space: O(n)
    """
    stack = []
    matching = {')': "('', "]': "['', "}': "{''}
    for c in s:
        if c in "([{':
            stack.append(c)
        elif c in ')]}':
            if not stack or stack[-1] != matching[c]:
                return False
            stack.pop()
    return len(stack) == 0

Call Stack:

Every recursive function call is pushed onto the call stack. Deep recursion can cause stack overflow (the call stack has limited size, 1-8 MB). Iterative solutions using an explicit stack Avoid this limit.

Queues

A queue is a FIFO (First In, First Out) data structure with enqueue (add to back) and dequeue (remove from front) operations.

Circular Buffer Implementation

class CircularQueue:
    """
    Queue implemented as a circular buffer (ring buffer).
    All operations: O(1)
    Space: O(capacity)
    """
    def __init__(self, capacity):
        self.capacity = capacity
        self.data = [None] * capacity
        self.head = 0  # front of queue
        self.tail = 0  # one past the back of queue
        self.count = 0

    def enqueue(self, val):
        if self.count == self.capacity:
            raise IndexError("queue is full")
        self.data[self.tail] = val
        self.tail = (self.tail + 1) % self.capacity
        self.count += 1

    def dequeue(self):
        if self.is_empty():
            raise IndexError("dequeue from empty queue")
        val = self.data[self.head]
        self.head = (self.head + 1) % self.capacity
        self.count -= 1
        return val

    def peek(self):
        if self.is_empty():
            raise IndexError("peek at empty queue")
        return self.data[self.head]

    def is_empty(self):
        return self.count == 0

    def size(self):
        return self.count

Queues, audio playback buffers, log rotation, producer-consumer patterns, and pipe implementations. The key advantage is that enqueue and dequeue never require memory allocation or copying — they just Advance indices modulo the capacity.

Deques

A deque (double-ended queue) supports insertion and deletion at both ends in $O(1)$ .

from collections import deque

## Python's built-in deque — implemented as a doubly-linked list of fixed-size blocks
d = deque()
d.append(1)        # O(1) — enqueue back
d.appendleft(0)    # O(1) — enqueue front
d.pop()            # O(1) — dequeue back
d.popleft()        # O(1) — dequeue front
d[0]               # O(1) — access by index (slower than list for large deques)

Priority Queues

A priority queue supports insertion of elements with associated priorities and extraction of the Minimum (or maximum) priority element. The standard implementation uses a binary heap.

Binary Heap

A binary heap is a complete binary tree where every node is less than or equal to its children (min-heap) or greater than or equal to its children (max-heap). Stored as an array where for node at Index $i$ : parent is at $(i-1)/2$ Left child is at $2i+1$ Right child is at $2i+2$ .

class MinHeap:
    """
    Min-heap implementation.
    insert: O(log n), extract_min: O(log n), peek: O(1)
    Space: O(n)
    """
    def __init__(self):
        self.data = []

    def _parent(self, i):
        return (i - 1) // 2

    def _left(self, i):
        return 2 * i + 1

    def _right(self, i):
        return 2 * i + 2

    def _sift_up(self, i):
        while i > 0 and self.data[self._parent(i)] > self.data[i]:
            self.data[self._parent(i)], self.data[i] = self.data[i], self.data[self._parent(i)]
            i = self._parent(i)

    def _sift_down(self, i):
        n = len(self.data)
        while True:
            smallest = i
            left = self._left(i)
            right = self._right(i)
            if left < n and self.data[left] < self.data[smallest]:
                smallest = left
            if right < n and self.data[right] < self.data[smallest]:
                smallest = right
            if smallest == i:
                break
            self.data[i], self.data[smallest] = self.data[smallest], self.data[i]
            i = smallest

    def insert(self, val):
        self.data.append(val)
        self._sift_up(len(self.data) - 1)

    def extract_min(self):
        if not self.data:
            raise IndexError("extract from empty heap")
        min_val = self.data[0]
        self.data[0] = self.data[-1]
        self.data.pop()
        if self.data:
            self._sift_down(0)
        return min_val

    def peek(self):
        if not self.data:
            raise IndexError("peek at empty heap")
        return self.data[0]

    def heapify(self, arr):
        """Build a heap from an array. O(n)."""
        self.data = arr[:]
        for i in range(len(self.data) // 2 - 1, -1, -1):
            self._sift_down(i)

Operation	Binary Heap	Sorted Array	Unsorted Array
Insert	$O(\log n)$	$O(n)$	$O(1)$
Extract min	$O(\log n)$	$O(1)$	$O(n)$
Peek	$O(1)$	$O(1)$	$O(n)$
Build from array	$O(n)$	$O(n \log n)$	$O(1)$

Monotonic Stack

A monotonic stack maintains elements in either strictly increasing or strictly decreasing order. It Is used to find the next greater/lesser element, previous greater/lesser element, and similar Patterns.

Next Greater Element

def next_greater_element(arr):
    """
    Find the next greater element for each element in the array.
    If no greater element exists, result is -1.
    Time: O(n), Space: O(n)
    """
    n = len(arr)
    result = [-1] * n
    stack = []  # stores indices; values are monotonically decreasing

    for i in range(n):
        while stack and arr[stack[-1]] < arr[i]:
            idx = stack.pop()
            result[idx] = arr[i]
        stack.append(i)

    return result

Largest Rectangle in Histogram

def largest_rectangle_histogram(heights):
    """
    Find the area of the largest rectangle in a histogram.
    Time: O(n), Space: O(n)
    """
    stack = []  # indices of bars in increasing height order
    max_area = 0
    n = len(heights)

    for i in range(n + 1):
        # Use height 0 as sentinel for remaining bars
        current_height = heights[i] if i < n else 0
        while stack and heights[stack[-1]] > current_height:
            height = heights[stack.pop()]
            width = i if not stack else i - stack[-1] - 1
            max_area = max(max_area, height * width)
        stack.append(i)

    return max_area

Monotonic Queue

A monotonic queue (deque) maintains elements in monotonic order and is used for sliding window Maximum/minimum problems.

Sliding Window Maximum

from collections import deque

def sliding_window_maximum(arr, k):
    """
    Find the maximum in each sliding window of size k.
    Time: O(n), Space: O(k)
    """
    result = []
    dq = deque()  # stores indices; values are monotonically decreasing

    for i in range(len(arr)):
        # Remove elements outside the window
        while dq and dq[0] <= i - k:
            dq.popleft()

        # Remove elements smaller than current (they can never be the maximum)
        while dq and arr[dq[-1]] < arr[i]:
            dq.pop()

        dq.append(i)

        if i >= k - 1:
            result.append(arr[dq[0]])

    return result

Maximum of any future window that includes the current element. Removing them from the deque Maintains the invariant that the deque contains a decreasing sequence of values, and the maximum is Always at the front.

Union-Find (Disjoint Set Union)

Union-Find is a data structure that tracks a partition of elements into disjoint sets, supporting Two operations: find (which set does an element belong to?) and union (merge two sets).

class UnionFind:
    """
    Union-Find with path compression and union by rank.
    find: O(alpha(n)) amortised (inverse Ackermann, effectively O(1))
    union: O(alpha(n)) amortised
    Space: O(n)
    """
    def __init__(self, n):
        self.parent = list(range(n))
        self.rank = [0] * n
        self.count = n  # number of disjoint sets

    def find(self, x):
        if self.parent[x] != x:
            self.parent[x] = self.find(self.parent[x])  # path compression
        return self.parent[x]

    def union(self, x, y):
        root_x = self.find(x)
        root_y = self.find(y)
        if root_x == root_y:
            return False
        # Union by rank
        if self.rank[root_x] < self.rank[root_y]:
            root_x, root_y = root_y, root_x
        self.parent[root_y] = root_x
        if self.rank[root_x] == self.rank[root_y]:
            self.rank[root_x] += 1
        self.count -= 1
        return True

    def connected(self, x, y):
        return self.find(x) == self.find(y)

Time Complexity

The amortised time complexity of both find and union is $O(\alpha(n))$ Where $\alpha$ is the Inverse Ackermann function. For all practical values of $n$ (up to $2^{2^{2^{65536}}}$ ), $\alpha(n) \le 4$ . This is effectively constant time.

The two optimisations work together:

Path compression: During findMake every node on the path point directly to the root
Union by rank: During unionAttach the shorter tree under the root of the taller tree

Without either optimisation, find is $O(\log n)$ and union is $O(\log n)$ . Without both, worst Case is $O(n)$ .

Applications

def count_connected_components(n, edges):
    """
    Count connected components in an undirected graph.
    Time: O(n + m * alpha(n)), Space: O(n)
    where m = number of edges
    """
    uf = UnionFind(n)
    for u, v in edges:
        uf.union(u, v)
    return uf.count

def detect_cycle_undirected(n, edges):
    """
    Detect if an undirected graph has a cycle using Union-Find.
    Time: O(n + m * alpha(n)), Space: O(n)
    """
    uf = UnionFind(n)
    for u, v in edges:
        if uf.connected(u, v):
            return True
        uf.union(u, v)
    return False

Common Pitfalls

1. Losing the Head Pointer

The most common linked list bug is modifying the head pointer (e.g., during insertion at head or Reversal) and losing the reference to the entire list. Always use a dummy head node or return the New head explicitly. For operations that might modify the head, use:

dummy = ListNode(0, head)
## ... operate on dummy.next ...
return dummy.next

2. Null Pointer Dereference in Linked List Operations

Always check for None before accessing .next or .val. The pattern while current and current.next is safer than while current when you need to access current.next.val. Edge cases: empty list, single-node list, and the last node of the list.

3. Using a Stack When a Queue Is Needed (or Vice Versa)

Stacks are LIFO, queues are FIFO. Using the wrong one produces incorrect results for Ordering-sensitive problems. BFS requires a queue; DFS can use either a stack or recursion. Level-order traversal requires a queue. Expression evaluation uses a stack.

4. Forgetting to Handle the Sentinel in Monotonic Stack/Queue

Monotonic stack and queue algorithms often use a sentinel value (e.g., 0 for histogram heights, -infinity for next greater element) to flush remaining elements from the data structure. Forgetting the sentinel means elements at the end of the array are never processed.

5. Union-Find Without Path Compression

Without path compression, Union-Find degrades to $O(\log n)$ per operation. Without union by rank, It degrades to $O(n)$ worst case. Always use both optimisations. The code overhead is minimal (3 Extra lines) and the performance difference is enormous for large inputs.

6. Stack Overflow from Recursion Depth

Python’s default recursion limit is 1000. For linked lists with more than 1000 nodes, recursive Solutions (recursive reversal, recursive palindrome check) will crash with RecursionError. Use Iterative solutions for production code, or increase the limit with sys.setrecursionlimit() if you Are certain the input size is bounded.

7. Priority Queue Inefficiency for Update Operations

Standard binary heaps do not support efficient decrease-key operations (common in Dijkstra’s Algorithm). The workaround — insert a new entry and ignore stale entries — works but increases the Heap size. For algorithms that require frequent decrease-key, a Fibonacci heap provides $O(1)$ Amortised decrease-key, but has large constant factors and is rarely used in practice.

Summary

This topic covers the core concepts of linked lists, stacks, and queues, including underlying theory, practical implementation, and key applications.

Key concepts include:

Python data structures (lists, dicts, sets)
list comprehensions and generators
object-oriented Python
decorators and context managers
error handling with try/except

Understanding these concepts thoroughly is essential for both examinations and practical programming, and requires both theoretical knowledge and hands-on practice.

Worked Examples

Worked examples demonstrating the application of key concepts are covered in the detailed sub-pages linked above.

Cross-References

Hashing and Hash Tables: Covers hash-based data structures that provide O(1) average-case operations compared to linear structures.
Sorting Algorithms: Sorting techniques that can be applied to linked lists and arrays.
Dynamic Programming: Explores recursive problem-solving that builds on stack and queue concepts.

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