Data Structures & AlgorithmsGraph Representation — Adjacency Matrix

Graph Representation — Adjacency Matrix

LevelIntermediate

Duration60 mins

TopicGraph Representation — Adjacency Matrix

2 / 4

Space Complexity — O(V²)

The Price of Constant-Time Access

The adjacency matrix representation offers something remarkable: O(1) edge existence checking. Want to know if vertices 7 and 42 are connected? One array lookup. Done.

But this power comes at a cost. The adjacency matrix unconditionally allocates space for every possible edge in the graph—whether those edges exist or not. For a graph with V vertices, that means V × V = V² cells, always.

This page examines the space implications in depth. We'll analyze why this quadratic space requirement exists, quantify its impact on real systems, and develop intuition for when O(V²) space is acceptable versus prohibitive.

What You Will Learn

By the end of this page, you will understand exactly why adjacency matrices require O(V²) space, how to calculate memory usage for specific graphs, the difference between sparse and dense graphs from a space perspective, and when the quadratic space cost becomes a fundamental barrier.

Why O(V²) Space Is Inevitable

The O(V²) space requirement isn't an implementation choice—it's a mathematical consequence of the representation itself.

The Fundamental Guarantee:

An adjacency matrix provides O(1) edge lookup by using direct indexing. To check if edge (i, j) exists, we access matrix[i][j]. This works because:

Every vertex pair (i, j) has a dedicated memory location
The location is computable in constant time: base_address + i × V + j
Reading that location takes constant time

But to have a dedicated slot for every pair, we need exactly V × V = V² slots. There's no avoiding this—it's the very structure that enables O(1) lookup.

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Memory Layout of a V×V Adjacency Matrix:
 
For V = 4 vertices (0, 1, 2, 3):
 
Memory addresses (assuming contiguous allocation):
┌────────┬────────┬────────┬────────┐
│ [0][0] │ [0][1] │ [0][2] │ [0][3] │  Row 0: addresses 0-3
├────────┼────────┼────────┼────────┤
│ [1][0] │ [1][1] │ [1][2] │ [1][3] │  Row 1: addresses 4-7
├────────┼────────┼────────┼────────┤
│ [2][0] │ [2][1] │ [2][2] │ [2][3] │  Row 2: addresses 8-11
├────────┼────────┼────────┼────────┤
│ [3][0] │ [3][1] │ [3][2] │ [3][3] │  Row 3: addresses 12-15
└────────┴────────┴────────┴────────┘
 
Total cells: 4 × 4 = 16 = V²
 
To find matrix[i][j]: 
  address = base + (i × V) + j
  
Example: matrix[2][3] = base + (2 × 4) + 3 = base + 11
  → O(1) calculation, O(1) access

The Space-Time Tradeoff

This is a classic space-time tradeoff. We 'pay' with O(V²) space to 'buy' O(1) edge lookup. Alternative representations (like adjacency lists) use less space but sacrifice constant-time edge checking. Neither is inherently better—the right choice depends on your specific graph and operations.

Calculating Actual Memory Usage

O(V²) is asymptotic notation—it tells us how space grows, not the exact bytes consumed. Let's calculate precise memory requirements.

Formula for Memory:

Memory = V² × (bytes per cell) + (overhead)

The 'bytes per cell' depends on what we store:

Unweighted graph (boolean/bit): 1 bit to 1 byte per cell
Integer weights: 4 bytes (32-bit) or 8 bytes (64-bit) per cell
Floating-point weights: 4 bytes (float) or 8 bytes (double) per cell

Memory Requirements for Different Graph Sizes
Vertices (V)	V²	1 byte/cell	4 bytes/cell	8 bytes/cell
100	10,000	10 KB	40 KB	80 KB
1,000	1,000,000	1 MB	4 MB	8 MB
10,000	100,000,000	100 MB	400 MB	800 MB
50,000	2,500,000,000	2.5 GB	10 GB	20 GB
100,000	10,000,000,000	10 GB	40 GB	80 GB
1,000,000	10¹²	1 TB	4 TB	8 TB

The V² Growth Rate Is Brutal

Notice how quickly memory requirements explode. A 10x increase in vertices means 100x more memory. A million-vertex graph—common in social networks—would require terabytes of RAM for an adjacency matrix. This is why adjacency matrices are impractical for large graphs.

Practical Example: Social Network

Consider Facebook with ~3 billion users. An adjacency matrix representation would require:

3,000,000,000² × 1 byte = 9 × 10¹⁸ bytes = 9 exabytes

That's approximately 9 million terabytes—more storage than exists in most data centers combined. Obviously, Facebook doesn't use adjacency matrices.

Practical Example: Local Road Network

Consider a city's road network with 10,000 intersections:

10,000² × 4 bytes = 400,000,000 bytes = 400 MB

This is manageable for a modern computer. An adjacency matrix might be acceptable here, especially if we need frequent O(1) edge checks.

Sparse vs Dense Graphs: The Critical Distinction

The efficiency of an adjacency matrix depends fundamentally on how many edges the graph actually has. This brings us to a crucial classification:

Dense Graph: A graph where E (number of edges) is close to the maximum possible V². In the extreme, a complete graph has exactly V(V-1)/2 edges (undirected) or V(V-1) edges (directed).

Sparse Graph: A graph where E is much smaller than V²—often E = O(V) or E = O(V log V). Most real-world graphs are sparse.

Why This Matters:

An adjacency matrix always uses V² space. For a dense graph, most of those cells contain meaningful data (edges). For a sparse graph, most cells are zeros—wasted space.

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Comparison: Dense vs Sparse Graph Space Efficiency
 
Dense Graph (V = 100, E ≈ 4,500 edges — nearly complete):
┌────────────────────────────────────────┐
│ ██████████████████████████████████████ │
│ ██████████  Matrix cells  ███████████ │
│ ██████████  (V² = 10,000) ███████████ │
│ ██████████████████████████████████████ │
└────────────────────────────────────────┘
Useful data: 4,500/10,000 = 45% → GOOD use of space
 
Sparse Graph (V = 100, E = 200 edges — typical real-world):
┌────────────────────────────────────────┐
│ ░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░ │
│ ░░░░░░░░░░  Matrix cells  ░░░░░░░░░░░ │
│ ░░░░░░░░░░  (V² = 10,000) ░░░░░░░░░░░ │
│ ░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░ │
└────────────────────────────────────────┘
Useful data: 200/10,000 = 2% → TERRIBLE use of space
 
█ = edge exists (useful data)
░ = no edge (wasted space)

Dense Graph Examples

•Complete graphs — Every vertex connects to every other
•Small, highly connected networks — Dense communication networks
•Correlation matrices — Every variable potentially correlates with every other
•Game theory payoff matrices — Every player pair has interactions
•Distance matrices — Distances between all pairs of locations

Sparse Graph Examples

•Social networks — Average person has ~hundreds of friends, not millions
•Road networks — Each intersection connects to ~4 roads
•The internet — Each page links to ~tens of pages, not billions
•Dependency graphs — Each package depends on ~10-100 others
•Biological networks — Proteins interact with a small subset

The Sparsity Threshold

A common heuristic: if E < V²/2 (less than half of possible edges exist), the graph is 'sparse enough' that adjacency lists often outperform matrices in space efficiency. Most practical graphs are far sparser than this threshold.

The Wasted Space Problem

For sparse graphs, the adjacency matrix is monumentally wasteful. Let's quantify this waste.

Space Efficiency Ratio:

Efficiency = (actual edges) / (matrix cells) = E / V²

For typical sparse graphs where E = O(V):

Efficiency = O(V) / V² = O(1/V)

As the graph grows larger, efficiency approaches zero. You're storing increasingly more zeros for each actual edge.

Space Efficiency: Adjacency Matrix for Sparse Graphs (E = 3V)
Vertices (V)	Edges (E = 3V)	Matrix Cells (V²)	Efficiency	Wasted %
100	300	10,000	3%	97%
1,000	3,000	1,000,000	0.3%	99.7%
10,000	30,000	100,000,000	0.03%	99.97%
100,000	300,000	10,000,000,000	0.003%	99.997%

The Mathematics of Waste:

For a graph with V = 10,000 vertices and E = 30,000 edges (a typical sparse graph with average degree 6):

Matrix stores: 10,000 × 10,000 = 100,000,000 cells
Actual data: 30,000 edges
Zeros stored: 99,970,000 cells
Using 4 bytes per cell: 400 MB total, only 120 KB useful

We're using 3,333 times more memory than necessary to store the actual edge information. This is the core inefficiency of adjacency matrices for sparse graphs.

Memory Isn't Just About Capacity

Even if you have enough RAM, wasted memory has real costs: increased cache misses (the matrix doesn't fit in CPU cache), longer initialization time (O(V²) to zero-fill), and wasted memory bandwidth. These factors can significantly slow down even cache-friendly operations.

Space Optimization for Undirected Graphs

For undirected graphs, the adjacency matrix is symmetric: A[i][j] = A[j][i]. This means half the matrix is redundant. Can we save space by storing only half?

Yes, but with caveats.

Option 1: Store only the upper (or lower) triangular matrix

Instead of V² cells, store only V(V+1)/2 cells—roughly half. This saves ~50% space but complicates indexing:

// To access element (i, j) where i ≤ j:
index = i * V - i * (i - 1) / 2 + (j - i)

This index calculation is no longer O(1) in a trivial sense—it requires arithmetic that may impact performance.

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Full Matrix vs Triangular Storage:
 
Full V×V Matrix (V=4):          Upper Triangular (10 cells):
    0  1  2  3                  
  ┌───┬───┬───┬───┐             Stored as 1D array:
0 │ 0 │ 1 │ 0 │ 1 │             [A00, A01, A02, A03, A11, A12, A13, A22, A23, A33]
  ├───┼───┼───┼───┤               ↓
1 │ 1 │ 0 │ 1 │ 0 │             [  0,   1,   0,   1,   0,   1,   0,   0,   1,   0]
  ├───┼───┼───┼───┤
2 │ 0 │ 1 │ 0 │ 1 │             Space: 4×5/2 = 10 cells (vs 16 for full matrix)
  ├───┼───┼───┼───┤             Savings: 37.5% for V=4, approaches 50% as V→∞
3 │ 1 │ 0 │ 1 │ 0 │
  └───┴───┴───┴───┘

Option 2: Bit-packing for unweighted graphs

For unweighted graphs (binary adjacency), we can pack 8 edges into a single byte or 64 edges into a 64-bit integer. This reduces memory by a factor of 8 or 64:

Space = V² / 8 bytes (bit-packed)

For a 10,000-vertex graph: 100 million bits = 12.5 MB instead of 100 MB.

Tradeoff: Accessing individual bits requires bitwise operations (shifting, masking), adding constant overhead to each access. Still O(1), but with higher constants.

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/**
 * Bit-packed adjacency matrix for unweighted graphs.
 * Stores 64 edge flags per 64-bit integer.
 */
class BitPackedAdjacencyMatrix {
    private bits: BigUint64Array;
    private readonly vertices: number;
    private readonly wordsPerRow: number;
 
    constructor(vertices: number) {
        this.vertices = vertices;
        // Each row needs V bits, stored in groups of 64
        this.wordsPerRow = Math.ceil(vertices / 64);
        // Total 64-bit words needed
        const totalWords = vertices * this.wordsPerRow;
        this.bits = new BigUint64Array(totalWords);
    }
 
    /**
     * Set edge between i and j.
     * For undirected graphs, call for both (i,j) and (j,i).
     */
    setEdge(i: number, j: number): void {
        const wordIndex = i * this.wordsPerRow + Math.floor(j / 64);
        const bitIndex = j % 64;
        this.bits[wordIndex] |= (1n << BigInt(bitIndex));
    }
 
    /**
     * Check if edge exists between i and j.
     * Time: O(1) with bit operations overhead
     */
    hasEdge(i: number, j: number): boolean {
        const wordIndex = i * this.wordsPerRow + Math.floor(j / 64);
        const bitIndex = j % 64;
        return (this.bits[wordIndex] & (1n << BigInt(bitIndex))) !== 0n;
    }
 
    /**
     * Memory usage in bytes.
     */
    memoryUsage(): number {
        return this.bits.length * 8; // 8 bytes per BigUint64
    }
}
 
// Example: 10,000 vertices
const graph = new BitPackedAdjacencyMatrix(10000);
console.log(`Memory: ${graph.memoryUsage() / 1024 / 1024} MB`);
// Output: ~12.5 MB (vs ~100 MB for byte-per-cell)

Optimizations Don't Change the Complexity

These optimizations reduce memory by constant factors (2x for triangular, 8x for bit-packing, 16x combined). The space is still O(V²). For truly large sparse graphs, you need a fundamentally different representation—adjacency lists—which we'll cover in the next module.

Memory Allocation and Initialization

Creating an adjacency matrix isn't just about the space it occupies—it's also about the time to allocate and initialize that space.

Initialization Time: O(V²)

When you create a new adjacency matrix, you must initialize V² cells to their default value (0, ∞, or false). Even if your graph has only V edges, you spend O(V²) time on initialization.

This can dominate runtime for algorithms on sparse graphs:

Graph copy: O(V²) even if only O(V) edges exist
Repeated graph construction: O(V²) per construction

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// Creating an adjacency matrix - O(V²) time
function createAdjacencyMatrix(vertices: number): number[][] {
    // O(V) to create V rows
    const matrix: number[][] = new Array(vertices);
    
    for (let i = 0; i < vertices; i++) {
        // O(V) to create and fill each row with zeros
        matrix[i] = new Array(vertices).fill(0);
    }
    
    // Total: O(V × V) = O(V²)
    return matrix;
}
 
// Time comparison for initialization:
console.time("V=1,000");
createAdjacencyMatrix(1000);  // 1 million cells
console.timeEnd("V=1,000");   // ~1-2ms
 
console.time("V=10,000");
createAdjacencyMatrix(10000); // 100 million cells
console.timeEnd("V=10,000");  // ~100-200ms
 
console.time("V=50,000");
createAdjacencyMatrix(50000); // 2.5 billion cells
console.timeEnd("V=50,000");  // ~5-10 seconds (or crash!)

Memory Allocation Considerations:

Contiguous vs Non-Contiguous: A 2D array in many languages is an array of arrays—not contiguous memory. This can hurt cache performance. For critical applications, consider using a single 1D array of size V².
Virtual Memory: The OS may allocate virtual addresses without backing RAM until accessed. Accessing memory sequentially during initialization causes page faults.
Language-Specific Behavior: Some languages (Java, Python) have significant per-object overhead. A 'compact' numpy array is far more efficient than a Python list-of-lists.

Amortization Strategy

If you need to construct many small matrices, consider maintaining a pool of pre-allocated matrices. Reset them (O(V²) time) rather than allocating new ones. For very large matrices accessed sparsely, lazy initialization (allocate rows on first access) can help—but this adds complexity.

Space Comparison: Matrix vs List

To truly appreciate the O(V²) cost, let's compare with the alternative representation: adjacency lists.

Adjacency List Space Complexity: O(V + E)

An adjacency list stores:

One list per vertex: O(V) lists
Each edge appears once (directed) or twice (undirected): O(E) edge entries

For sparse graphs where E = O(V), adjacency lists use O(V) space. For dense graphs where E = O(V²), they use O(V²) space—but not more.

Space Comparison: Matrix vs List for Various Graphs
Graph Type	V	E	Matrix (V²)	List (V + E)	Matrix/List Ratio
Path (V nodes)	1,000	999	1,000,000	1,999	500x
Tree (V nodes)	1,000	999	1,000,000	1,999	500x
Sparse (E = 3V)	1,000	3,000	1,000,000	4,000	250x
Medium (E = V log V)	1,000	10,000	1,000,000	11,000	91x
Dense (E = V²/2)	1,000	500,000	1,000,000	501,000	2x
Complete	1,000	999,000	1,000,000	1,000,000	1x

Key Insight:

For typical sparse graphs (E = O(V)), adjacency matrices waste hundreds to thousands of times more memory than adjacency lists. The waste grows linearly with V.

For dense graphs (E approaching V²), both representations use roughly the same space. The matrix's overhead becomes negligible.

The Crossover Point:

Matrices become space-competitive when E > V²/c for some small constant c. In practice, if more than ~10-20% of possible edges exist, the matrix overhead is less significant.

Don't Forget Hidden Constants

Adjacency lists have their own overhead: pointer storage, dynamic array growth, object headers. Each edge in a list might cost 8-24 bytes of metadata beyond the vertex ID. But even with 10x overhead per edge, lists dramatically outperform matrices for sparse graphs.

Summary: Understanding O(V²) Space

We've thoroughly analyzed the space characteristics of adjacency matrices. Let's consolidate the key insights:

Key Takeaways

•O(V²) space is inherent — The structure that enables O(1) lookup requires a dedicated cell for every possible edge.
•Memory grows quadratically — 10× more vertices means 100× more memory. This becomes prohibitive quickly.
•Sparse graphs waste space — Most real-world graphs are sparse, making matrices store 97-99%+ zeros.
•Dense graphs are efficient — When E approaches V², the matrix uses its space well.
•Optimizations help but don't change complexity — Triangular storage saves 50%, bit-packing saves 8×, but it's still O(V²).
•Initialization is also O(V²) — Creating a matrix takes quadratic time even for sparse graphs.
•Adjacency lists are space-superior for sparse graphs — They use O(V + E) space, which can be orders of magnitude less.

What's Next:

We've established the space cost. But what do we get in return? The next page explores the key benefit of adjacency matrices: O(1) edge lookup—and analyzes when this constant-time access justifies the quadratic space investment.

Space Analysis Complete

You now understand exactly why adjacency matrices use O(V²) space, how to calculate concrete memory requirements, and why this is problematic for sparse graphs. This foundation is essential for making informed representation choices. Next: the O(1) edge lookup that justifies this cost.

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Data Structures & AlgorithmsGraph Representation — Adjacency Matrix

Graph Representation — Adjacency Matrix

LevelIntermediate

Duration60 mins

TopicGraph Representation — Adjacency Matrix

2 / 4

Space Complexity — O(V²)

The Price of Constant-Time Access

The adjacency matrix representation offers something remarkable: O(1) edge existence checking. Want to know if vertices 7 and 42 are connected? One array lookup. Done.

What You Will Learn

Why O(V²) Space Is Inevitable

The O(V²) space requirement isn't an implementation choice—it's a mathematical consequence of the representation itself.

The Fundamental Guarantee:

An adjacency matrix provides O(1) edge lookup by using direct indexing. To check if edge (i, j) exists, we access matrix[i][j]. This works because:

Every vertex pair (i, j) has a dedicated memory location
The location is computable in constant time: base_address + i × V + j
Reading that location takes constant time

But to have a dedicated slot for every pair, we need exactly V × V = V² slots. There's no avoiding this—it's the very structure that enables O(1) lookup.

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Memory Layout of a V×V Adjacency Matrix:
 
For V = 4 vertices (0, 1, 2, 3):
 
Memory addresses (assuming contiguous allocation):
┌────────┬────────┬────────┬────────┐
│ [0][0] │ [0][1] │ [0][2] │ [0][3] │  Row 0: addresses 0-3
├────────┼────────┼────────┼────────┤
│ [1][0] │ [1][1] │ [1][2] │ [1][3] │  Row 1: addresses 4-7
├────────┼────────┼────────┼────────┤
│ [2][0] │ [2][1] │ [2][2] │ [2][3] │  Row 2: addresses 8-11
├────────┼────────┼────────┼────────┤
│ [3][0] │ [3][1] │ [3][2] │ [3][3] │  Row 3: addresses 12-15
└────────┴────────┴────────┴────────┘
 
Total cells: 4 × 4 = 16 = V²
 
To find matrix[i][j]: 
  address = base + (i × V) + j
  
Example: matrix[2][3] = base + (2 × 4) + 3 = base + 11
  → O(1) calculation, O(1) access

The Space-Time Tradeoff

Calculating Actual Memory Usage

O(V²) is asymptotic notation—it tells us how space grows, not the exact bytes consumed. Let's calculate precise memory requirements.

Formula for Memory:

Memory = V² × (bytes per cell) + (overhead)

The 'bytes per cell' depends on what we store:

Unweighted graph (boolean/bit): 1 bit to 1 byte per cell
Integer weights: 4 bytes (32-bit) or 8 bytes (64-bit) per cell
Floating-point weights: 4 bytes (float) or 8 bytes (double) per cell

Memory Requirements for Different Graph Sizes
Vertices (V)	V²	1 byte/cell	4 bytes/cell	8 bytes/cell
100	10,000	10 KB	40 KB	80 KB
1,000	1,000,000	1 MB	4 MB	8 MB
10,000	100,000,000	100 MB	400 MB	800 MB
50,000	2,500,000,000	2.5 GB	10 GB	20 GB
100,000	10,000,000,000	10 GB	40 GB	80 GB
1,000,000	10¹²	1 TB	4 TB	8 TB

The V² Growth Rate Is Brutal

Practical Example: Social Network

Consider Facebook with ~3 billion users. An adjacency matrix representation would require:

3,000,000,000² × 1 byte = 9 × 10¹⁸ bytes = 9 exabytes

That's approximately 9 million terabytes—more storage than exists in most data centers combined. Obviously, Facebook doesn't use adjacency matrices.

Practical Example: Local Road Network

Consider a city's road network with 10,000 intersections:

10,000² × 4 bytes = 400,000,000 bytes = 400 MB

This is manageable for a modern computer. An adjacency matrix might be acceptable here, especially if we need frequent O(1) edge checks.

Sparse vs Dense Graphs: The Critical Distinction

The efficiency of an adjacency matrix depends fundamentally on how many edges the graph actually has. This brings us to a crucial classification:

Dense Graph: A graph where E (number of edges) is close to the maximum possible V². In the extreme, a complete graph has exactly V(V-1)/2 edges (undirected) or V(V-1) edges (directed).

Sparse Graph: A graph where E is much smaller than V²—often E = O(V) or E = O(V log V). Most real-world graphs are sparse.

Why This Matters:

An adjacency matrix always uses V² space. For a dense graph, most of those cells contain meaningful data (edges). For a sparse graph, most cells are zeros—wasted space.

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Comparison: Dense vs Sparse Graph Space Efficiency
 
Dense Graph (V = 100, E ≈ 4,500 edges — nearly complete):
┌────────────────────────────────────────┐
│ ██████████████████████████████████████ │
│ ██████████  Matrix cells  ███████████ │
│ ██████████  (V² = 10,000) ███████████ │
│ ██████████████████████████████████████ │
└────────────────────────────────────────┘
Useful data: 4,500/10,000 = 45% → GOOD use of space
 
Sparse Graph (V = 100, E = 200 edges — typical real-world):
┌────────────────────────────────────────┐
│ ░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░ │
│ ░░░░░░░░░░  Matrix cells  ░░░░░░░░░░░ │
│ ░░░░░░░░░░  (V² = 10,000) ░░░░░░░░░░░ │
│ ░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░ │
└────────────────────────────────────────┘
Useful data: 200/10,000 = 2% → TERRIBLE use of space
 
█ = edge exists (useful data)
░ = no edge (wasted space)

Dense Graph Examples

•Complete graphs — Every vertex connects to every other
•Small, highly connected networks — Dense communication networks
•Correlation matrices — Every variable potentially correlates with every other
•Game theory payoff matrices — Every player pair has interactions
•Distance matrices — Distances between all pairs of locations

Sparse Graph Examples

•Social networks — Average person has ~hundreds of friends, not millions
•Road networks — Each intersection connects to ~4 roads
•The internet — Each page links to ~tens of pages, not billions
•Dependency graphs — Each package depends on ~10-100 others
•Biological networks — Proteins interact with a small subset

The Sparsity Threshold

The Wasted Space Problem

For sparse graphs, the adjacency matrix is monumentally wasteful. Let's quantify this waste.

Space Efficiency Ratio:

Efficiency = (actual edges) / (matrix cells) = E / V²

For typical sparse graphs where E = O(V):

Efficiency = O(V) / V² = O(1/V)

As the graph grows larger, efficiency approaches zero. You're storing increasingly more zeros for each actual edge.

Space Efficiency: Adjacency Matrix for Sparse Graphs (E = 3V)
Vertices (V)	Edges (E = 3V)	Matrix Cells (V²)	Efficiency	Wasted %
100	300	10,000	3%	97%
1,000	3,000	1,000,000	0.3%	99.7%
10,000	30,000	100,000,000	0.03%	99.97%
100,000	300,000	10,000,000,000	0.003%	99.997%

The Mathematics of Waste:

For a graph with V = 10,000 vertices and E = 30,000 edges (a typical sparse graph with average degree 6):

Matrix stores: 10,000 × 10,000 = 100,000,000 cells
Actual data: 30,000 edges
Zeros stored: 99,970,000 cells
Using 4 bytes per cell: 400 MB total, only 120 KB useful

We're using 3,333 times more memory than necessary to store the actual edge information. This is the core inefficiency of adjacency matrices for sparse graphs.

Memory Isn't Just About Capacity

Space Optimization for Undirected Graphs

For undirected graphs, the adjacency matrix is symmetric: A[i][j] = A[j][i]. This means half the matrix is redundant. Can we save space by storing only half?

Yes, but with caveats.

Option 1: Store only the upper (or lower) triangular matrix

Instead of V² cells, store only V(V+1)/2 cells—roughly half. This saves ~50% space but complicates indexing:

// To access element (i, j) where i ≤ j:
index = i * V - i * (i - 1) / 2 + (j - i)

This index calculation is no longer O(1) in a trivial sense—it requires arithmetic that may impact performance.

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Full Matrix vs Triangular Storage:
 
Full V×V Matrix (V=4):          Upper Triangular (10 cells):
    0  1  2  3                  
  ┌───┬───┬───┬───┐             Stored as 1D array:
0 │ 0 │ 1 │ 0 │ 1 │             [A00, A01, A02, A03, A11, A12, A13, A22, A23, A33]
  ├───┼───┼───┼───┤               ↓
1 │ 1 │ 0 │ 1 │ 0 │             [  0,   1,   0,   1,   0,   1,   0,   0,   1,   0]
  ├───┼───┼───┼───┤
2 │ 0 │ 1 │ 0 │ 1 │             Space: 4×5/2 = 10 cells (vs 16 for full matrix)
  ├───┼───┼───┼───┤             Savings: 37.5% for V=4, approaches 50% as V→∞
3 │ 1 │ 0 │ 1 │ 0 │
  └───┴───┴───┴───┘

Option 2: Bit-packing for unweighted graphs

For unweighted graphs (binary adjacency), we can pack 8 edges into a single byte or 64 edges into a 64-bit integer. This reduces memory by a factor of 8 or 64:

Space = V² / 8 bytes (bit-packed)

For a 10,000-vertex graph: 100 million bits = 12.5 MB instead of 100 MB.

Tradeoff: Accessing individual bits requires bitwise operations (shifting, masking), adding constant overhead to each access. Still O(1), but with higher constants.

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/**
 * Bit-packed adjacency matrix for unweighted graphs.
 * Stores 64 edge flags per 64-bit integer.
 */
class BitPackedAdjacencyMatrix {
    private bits: BigUint64Array;
    private readonly vertices: number;
    private readonly wordsPerRow: number;
 
    constructor(vertices: number) {
        this.vertices = vertices;
        // Each row needs V bits, stored in groups of 64
        this.wordsPerRow = Math.ceil(vertices / 64);
        // Total 64-bit words needed
        const totalWords = vertices * this.wordsPerRow;
        this.bits = new BigUint64Array(totalWords);
    }
 
    /**
     * Set edge between i and j.
     * For undirected graphs, call for both (i,j) and (j,i).
     */
    setEdge(i: number, j: number): void {
        const wordIndex = i * this.wordsPerRow + Math.floor(j / 64);
        const bitIndex = j % 64;
        this.bits[wordIndex] |= (1n << BigInt(bitIndex));
    }
 
    /**
     * Check if edge exists between i and j.
     * Time: O(1) with bit operations overhead
     */
    hasEdge(i: number, j: number): boolean {
        const wordIndex = i * this.wordsPerRow + Math.floor(j / 64);
        const bitIndex = j % 64;
        return (this.bits[wordIndex] & (1n << BigInt(bitIndex))) !== 0n;
    }
 
    /**
     * Memory usage in bytes.
     */
    memoryUsage(): number {
        return this.bits.length * 8; // 8 bytes per BigUint64
    }
}
 
// Example: 10,000 vertices
const graph = new BitPackedAdjacencyMatrix(10000);
console.log(`Memory: ${graph.memoryUsage() / 1024 / 1024} MB`);
// Output: ~12.5 MB (vs ~100 MB for byte-per-cell)

Optimizations Don't Change the Complexity

Memory Allocation and Initialization

Creating an adjacency matrix isn't just about the space it occupies—it's also about the time to allocate and initialize that space.

Initialization Time: O(V²)

When you create a new adjacency matrix, you must initialize V² cells to their default value (0, ∞, or false). Even if your graph has only V edges, you spend O(V²) time on initialization.

This can dominate runtime for algorithms on sparse graphs:

Graph copy: O(V²) even if only O(V) edges exist
Repeated graph construction: O(V²) per construction

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// Creating an adjacency matrix - O(V²) time
function createAdjacencyMatrix(vertices: number): number[][] {
    // O(V) to create V rows
    const matrix: number[][] = new Array(vertices);
    
    for (let i = 0; i < vertices; i++) {
        // O(V) to create and fill each row with zeros
        matrix[i] = new Array(vertices).fill(0);
    }
    
    // Total: O(V × V) = O(V²)
    return matrix;
}
 
// Time comparison for initialization:
console.time("V=1,000");
createAdjacencyMatrix(1000);  // 1 million cells
console.timeEnd("V=1,000");   // ~1-2ms
 
console.time("V=10,000");
createAdjacencyMatrix(10000); // 100 million cells
console.timeEnd("V=10,000");  // ~100-200ms
 
console.time("V=50,000");
createAdjacencyMatrix(50000); // 2.5 billion cells
console.timeEnd("V=50,000");  // ~5-10 seconds (or crash!)

Memory Allocation Considerations:

Contiguous vs Non-Contiguous: A 2D array in many languages is an array of arrays—not contiguous memory. This can hurt cache performance. For critical applications, consider using a single 1D array of size V².
Virtual Memory: The OS may allocate virtual addresses without backing RAM until accessed. Accessing memory sequentially during initialization causes page faults.
Language-Specific Behavior: Some languages (Java, Python) have significant per-object overhead. A 'compact' numpy array is far more efficient than a Python list-of-lists.

Amortization Strategy

Space Comparison: Matrix vs List

To truly appreciate the O(V²) cost, let's compare with the alternative representation: adjacency lists.

Adjacency List Space Complexity: O(V + E)

An adjacency list stores:

One list per vertex: O(V) lists
Each edge appears once (directed) or twice (undirected): O(E) edge entries

For sparse graphs where E = O(V), adjacency lists use O(V) space. For dense graphs where E = O(V²), they use O(V²) space—but not more.

Space Comparison: Matrix vs List for Various Graphs
Graph Type	V	E	Matrix (V²)	List (V + E)	Matrix/List Ratio
Path (V nodes)	1,000	999	1,000,000	1,999	500x
Tree (V nodes)	1,000	999	1,000,000	1,999	500x
Sparse (E = 3V)	1,000	3,000	1,000,000	4,000	250x
Medium (E = V log V)	1,000	10,000	1,000,000	11,000	91x
Dense (E = V²/2)	1,000	500,000	1,000,000	501,000	2x
Complete	1,000	999,000	1,000,000	1,000,000	1x

Key Insight:

For typical sparse graphs (E = O(V)), adjacency matrices waste hundreds to thousands of times more memory than adjacency lists. The waste grows linearly with V.

For dense graphs (E approaching V²), both representations use roughly the same space. The matrix's overhead becomes negligible.

The Crossover Point:

Matrices become space-competitive when E > V²/c for some small constant c. In practice, if more than ~10-20% of possible edges exist, the matrix overhead is less significant.

Don't Forget Hidden Constants

Summary: Understanding O(V²) Space

We've thoroughly analyzed the space characteristics of adjacency matrices. Let's consolidate the key insights:

Key Takeaways

•O(V²) space is inherent — The structure that enables O(1) lookup requires a dedicated cell for every possible edge.
•Memory grows quadratically — 10× more vertices means 100× more memory. This becomes prohibitive quickly.
•Sparse graphs waste space — Most real-world graphs are sparse, making matrices store 97-99%+ zeros.
•Dense graphs are efficient — When E approaches V², the matrix uses its space well.
•Optimizations help but don't change complexity — Triangular storage saves 50%, bit-packing saves 8×, but it's still O(V²).
•Initialization is also O(V²) — Creating a matrix takes quadratic time even for sparse graphs.
•Adjacency lists are space-superior for sparse graphs — They use O(V + E) space, which can be orders of magnitude less.

What's Next:

Space Analysis Complete

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