Data Structures: Arrays and Linked Lists

Arrays and linked lists are foundational linear data structures. They solve the same broad problem, storing an ordered collection of elements, but they do so with different memory layouts and different performance trade-offs. Understanding how they store data and how common operations behave is essential for writing efficient programs and for choosing the right structure under real constraints such as cache behavior, dynamic growth, and predictable performance.

What “linear” means in practice

A linear data structure represents a sequence: first element, second element, and so on. You can traverse from one element to the next in order, and many operations are defined in terms of positions (index 0, index 1, or “the node after this one”).

The core difference between arrays and linked lists is how they map that sequence into memory:

• Arrays use sequential storage (contiguous memory).
• Linked lists use dynamic allocation (elements can be scattered in memory and connected by pointers or references).

That single distinction drives most time and space complexity differences.

Arrays: sequential storage and fast indexing

An array stores elements in a contiguous block of memory. If the first element begins at address $A$ and each element has size $s$, the element at index $i$ is located at:

$$\text{address}(i)=A+i\cdot s$$

This direct addressing is why arrays provide constant-time random access.

Core operations and complexity

Arrays are typically evaluated by the time complexity of common operations:

• Access by index: $O(1)$

Jump straight to position $i$.

• Search (unsorted): $O(n)$

Scan until you find the target.

• Insert or delete at end: often $O(1)$, but can be $O(n)$ when resizing is required

This depends on whether the array has spare capacity.

• Insert or delete at arbitrary position: $O(n)$

Elements after the position must shift left or right to keep the array contiguous.

A practical example: inserting an item at the beginning of a list of 1,000,000 integers in an array requires shifting 1,000,000 elements. That is expensive even though the elements themselves are small.

Dynamic arrays and resizing behavior

Many languages provide dynamic arrays (for example, Python lists, Java ArrayList, C++ std::vector). Internally, they still rely on a contiguous array, but they manage capacity to support growth.

When capacity is exhausted, the structure allocates a larger block and copies elements. A common strategy is to grow by a constant factor (often around 2). This yields:

• Append: amortized $O(1)$

Most appends are constant-time; occasional resizes cost $O(n)$.

• Memory overhead: unused capacity

Extra reserved space improves performance but increases memory consumption.

This is an explicit time/space trade-off: keep slack space to avoid frequent reallocations.

Real-world performance: cache locality

Arrays often outperform linked lists in practice for traversal and bulk operations because contiguous memory benefits CPU caching. Iterating through an array typically produces fewer cache misses than chasing pointers across unrelated memory locations.

If you do a lot of sequential reads, arrays are hard to beat.

Linked lists: dynamic allocation and flexible updates

A linked list stores elements in nodes. Each node contains the data and a pointer (or reference) to the next node. In a doubly linked list, nodes also include a pointer to the previous node.

Because nodes are allocated individually, they do not need contiguous memory. This makes growing and shrinking straightforward without global reallocations.

Types of linked lists

• Singly linked list: each node points to the next

Simple, lower overhead than doubly linked, but cannot traverse backward efficiently.

• Doubly linked list: each node points to next and previous

Easier deletions when you already have a node reference, supports backward traversal, costs more memory per node.

Some implementations also use a tail pointer for efficient append operations.

Core operations and complexity

• Access by position (i-th element): $O(n)$

No direct addressing; you must walk from the head node.

• Search: $O(n)$

Same as arrays for unsorted data, but with higher constant factors due to pointer chasing.

• Insert or delete at head: $O(1)$

Adjust one or two pointers.

• Insert or delete after a known node: $O(1)$

This is a key advantage: once you have a node reference, updates are cheap.

• Insert or delete by position: typically $O(n)$

You still must traverse to reach the position.

A subtle but important point: linked lists shine when your algorithm already traverses and maintains references to nodes as it goes. If you repeatedly insert and delete in the middle while iterating, a linked list can avoid the shifting cost arrays incur.

Space overhead and fragmentation

Each node stores not only data but also at least one pointer. That overhead can be significant, especially for small element types. Additionally, separate allocations can lead to heap fragmentation and reduced cache performance.

So while linked lists avoid unused capacity, they often pay in per-element overhead and real-world speed.

Comparing arrays and linked lists: practical trade-offs

When arrays are typically the better choice

Arrays (including dynamic arrays) are usually preferred when:

• You need fast random access by index.
• Your workload is heavy on iteration and reading.
• You care about cache-friendly performance.
• Inserts and deletes are mostly at the end, or infrequent.

Example: storing pixels in an image row, keeping a list of records you frequently index into, or buffering samples for numeric computation.

When linked lists make sense

Linked lists are a reasonable choice when:

• You perform many insertions and deletions near the head or around the current position during traversal.
• You frequently splice sequences (in some environments and implementations).
• You want to avoid resizing and copying large contiguous blocks.

Example: maintaining a sequence where you repeatedly remove and insert items while scanning through it, and you keep direct node references as part of the algorithm.

In many application-level scenarios, though, developers reach for linked lists out of habit when a dynamic array would be simpler and faster. The decision should be driven by operation patterns, not by the idea that linked lists are “more dynamic.”

Operations complexity at a glance

The headline complexity differences stem from layout:

• Arrays: direct indexing, expensive mid-sequence updates due to shifting.
• Linked lists: cheap local pointer updates, expensive positional access due to traversal.

A useful mental model is:

• If you often ask “give me element i,” arrays match that question.
• If you often ask “insert after this element I am holding,” linked lists match that question.

Implementation considerations that affect design

Error-prone edges in linked lists

Linked list implementations demand careful handling of:

• Empty list cases (head is null)
• Single element lists
• Deleting the head or tail
• Keeping head and tail pointers consistent
• In doubly linked lists, maintaining both next and previous links

Mistakes often show up as lost nodes (memory leaks in manual-memory languages) or cycles and broken chains.

Arrays and capacity management

For arrays, the tricky parts are:

• Choosing a growth strategy (capacity doubling is common)
• Avoiding excessive copying in performance-critical paths
• Understanding that “size” and “capacity” are different concepts

Choosing based on data and workload

The best choice depends on:

• Update frequency and where updates occur
• Need for indexing
• Memory limits and overhead tolerance
• Typical traversal patterns

If performance matters, measure with realistic inputs. On modern hardware, cache effects and allocation costs can dominate theoretical $O(1)$ versus $O(n)$ arguments.

Summary: a clear rule of thumb

Arrays provide compact sequential storage with excellent indexing and traversal performance, at the cost of expensive insertions and deletions away from the end and occasional resizing copies. Linked lists provide flexible dynamic allocation with fast local insertions and deletions once you have a node reference, at the cost of slow indexing, higher memory overhead, and weaker cache locality.

Choosing between arrays and linked lists is less about which is “better” and more about aligning the structure with your operations and constraints. When you understand how memory layout drives complexity, the right choice becomes straightforward.