Divide and Conquer Algorithm: Concept and Examples

Introduction to Arrays

Arrays are one of the most fundamental data structures in programming. Think of them like a shelf with multiple boxes, where each box holds an item. Arrays store elements of the same type in contiguous memory locations, making them efficient for accessing and manipulating data. Whether you’re building a simple to-do list app or a complex machine learning model, arrays are indispensable tools.

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What Are Arrays?

Definition and Basic Structure

An array is a collection of items stored at contiguous memory locations. Each item is called an element, and its position is defined by an index (usually starting at 0). For example, an array of integers [5, 3, 9] has three elements, accessible via indexes 0, 1, and 2.

Arrays are static in size in many languages (like Java), meaning their length is fixed upon creation. However, dynamic arrays (like Python lists) can resize automatically. Understanding these distinctions is vital for tackling top DSA interview questions.

Memory Representation

Under the hood, an array of type T and length N occupies N × sizeof(T) bytes in memory. The address of the element at index i can be computed as:

address_of(A[i]) = base_address + i * sizeof(T)

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Array Indexing and Bounds

Most languages enforce bounds checking in some form:

• Unchecked (C, C++): Accessing out-of-bounds may lead to undefined behavior.
  
  
• Checked (Java, Python): Raises runtime exceptions (e.g., ArrayIndexOutOfBoundsException, IndexError).
  
  

Always validate indices to prevent segmentation faults or runtime errors. In performance-critical code, bounds checks may be elided by the compiler under certain optimizations.

Initialization and Default Values

When an array is created, each element must hold a value:

• Zero-initialized (C statically allocated arrays).
  
  
• Default-initialized (Java: 0 for numbers, null for objects).
  
  
• Uninitialized (C automatic arrays may contain garbage values if not explicitly set).
  
  

Language-Specific Implementations

• C: int arr[10];—static allocation on stack (or global); manual memory management with malloc for dynamic arrays.
  
  
• C++: std::vector<int> v;—dynamic, resizable, encapsulates memory and bounds checks (at()).
  
  
• Java: int[] arr = new int[10];—heap allocation, fixed size.
  
  
• Python: list—dynamic, heterogeneous, high-level operations (append, slicing).
  
  

Types of Arrays

One-Dimensional Arrays

A 1D array is the simplest form, storing elements in a linear sequence. It’s ideal for lists like student grades or temperature readings.

Example:

temperatures = [72, 68, 75, 80]

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Multi-Dimensional Arrays

These are arrays of arrays. A 2D array (matrix) is common in image processing or game boards.

Two-Dimensional Arrays

Used to represent grids, spreadsheets, images, and more. Each element is accessed via two indices:

int[][] chessboard = new int[8][8];

Row-major vs. column-major storage affects cache performance and iteration order. Learn grid handling in our web development path.

Higher-Dimensional Arrays

3D arrays model volumetric data (e.g., CT scans), time-series of matrices, or tensors in deep learning frameworks. Operations on these structures often leverage specialized libraries (e.g., TensorFlow, PyTorch).

Jagged (Ragged) Arrays

Arrays whose rows can have different lengths, common in languages like C# and Java:

int[][] jagged = new int[3][];

jagged[0] = new int[2];

jagged[1] = new int[5];

Jagged arrays save memory when rows have varying sizes but incur additional indexing overhead.

Dynamic vs. Static Arrays

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Resizing Strategies

• Doubling: Allocate new capacity = 2 × old capacity (common in C++ std::vector).
  
  
• Incremental: Allocate small fixed increments (inefficient for large data growth).
  
  

Shrink-to-fit: Release unused capacity after removals to free memory.