Adjacency List Representation in DSA Java - Time & Space Complexity

Time Complexity: Adjacency List Representation

O(n)

Understanding Time Complexity

We want to understand how fast operations run when using adjacency lists to represent graphs.

Specifically, how the time to access or traverse connections changes as the graph grows.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

// Adding edges to an adjacency list for a graph
List<List<Integer>> adjList = new ArrayList<>();
for (int i = 0; i < n; i++) {
    adjList.add(new ArrayList<>());
}

// Add edge from u to v
adjList.get(u).add(v);
    

This code creates an adjacency list for a graph with n vertices and adds an edge from vertex u to vertex v.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Loop runs n times to create empty lists for each vertex.
• How many times: Exactly n times for initialization; adding an edge is a single operation.

How Execution Grows With Input

As the number of vertices n grows, creating the adjacency list takes time proportional to n.

Input Size (n)	Approx. Operations
10	10 (creating 10 lists)
100	100 (creating 100 lists)
1000	1000 (creating 1000 lists)

Pattern observation: The time grows linearly with the number of vertices.

Final Time Complexity

Time Complexity: O(n)

This means the time to set up the adjacency list grows directly with the number of vertices.

Common Mistake

[X] Wrong: "Adding an edge takes O(n) time because we have to search the list."

[OK] Correct: Adding an edge is just appending to a list, which is O(1) on average, not searching.

Interview Connect

Understanding adjacency lists helps you explain graph storage clearly and analyze graph algorithms efficiently.

Self-Check

"What if we used an adjacency matrix instead? How would the time complexity for adding edges change?"

Practice

1. What does an adjacency list represent in graph data structures?

easy

A. A list where each vertex stores a list of its adjacent vertices

B. A matrix showing connections between all pairs of vertices

C. A list of all edges without vertex connections

D. A list storing only isolated vertices

Solution

1. Step 1: Understand adjacency list concept

   An adjacency list stores each vertex with a list of vertices it connects to directly.

2. Step 2: Compare with other graph representations

   Unlike adjacency matrix, adjacency list focuses on neighbors per vertex, not all pairs.

3. Final Answer:

   A list where each vertex stores a list of its adjacent vertices -> Option A

4. Quick Check:

   Adjacency list = vertex with neighbors [OK]

Hint: Adjacency list = vertex + its neighbors list [OK]

Common Mistakes:

• Confusing adjacency list with adjacency matrix
• Thinking it stores edges only without vertices
• Assuming it stores isolated vertices only

2. Which of the following is the correct Java syntax to declare an adjacency list for a graph with 5 vertices?

easy

A. ArrayList<Integer>[] graph = new ArrayList[5];

B. ArrayList<Integer> graph = new ArrayList(5);

C. List<Integer> graph = new List[5];

D. ArrayList[] graph = new ArrayList<Integer>(5);

Solution

1. Step 1: Understand adjacency list declaration

   We need an array of ArrayLists, one for each vertex, so ArrayList<Integer>[] is correct.

2. Step 2: Check syntax correctness

   ArrayList<Integer>[] graph = new ArrayList[5]; correctly declares an array of ArrayLists with size 5. Others misuse generics or types.

3. Final Answer:

   ArrayList<Integer>[] graph = new ArrayList[5]; -> Option A

4. Quick Check:

   Array of ArrayLists = ArrayList<Integer>[] graph = new ArrayList[5]; [OK]

Hint: Adjacency list = array of ArrayLists in Java [OK]

Common Mistakes:

• Using List instead of ArrayList for array declaration
• Trying to instantiate generic arrays directly
• Missing generic type in ArrayList declaration

3. Given the following Java code snippet for an undirected graph adjacency list, what will be the output after adding edges?

ArrayList<Integer>[] graph = new ArrayList[3];
for (int i = 0; i < 3; i++) graph[i] = new ArrayList<>();
graph[0].add(1);
graph[1].add(0);
graph[1].add(2);
graph[2].add(1);
System.out.println(graph[1]);

medium

A. [0, 1]

B. [1, 2]

C. [2]

D. [0, 2]

Solution

1. Step 1: Trace additions to graph[1]

   graph[1] adds 0 and 2, so its list contains [0, 2].

2. Step 2: Confirm output of System.out.println(graph[1])

   Printing graph[1] shows the list [0, 2].

3. Final Answer:

   [0, 2] -> Option D

4. Quick Check:

   graph[1] = [0, 2] [OK]

Hint: Check which vertices are added to graph[1] list [OK]

Common Mistakes:

• Confusing indices and values in adjacency list
• Forgetting undirected edges require adding both ways
• Misreading the printed list order

4. Identify the error in this Java code snippet that tries to create an adjacency list for a graph with 4 vertices:

ArrayList<Integer>[] graph = new ArrayList[4];
for (int i = 0; i <= 4; i++) {
    graph[i] = new ArrayList<>();
}

medium

A. Missing generic type in ArrayList declaration

B. ArrayList cannot be used for adjacency list

C. Loop condition should be i < 4, not i <= 4

D. Array size should be 5 instead of 4

Solution

1. Step 1: Analyze loop range and array size

   Array size is 4, valid indices are 0 to 3. Loop uses i <= 4, which accesses index 4 (out of bounds).

2. Step 2: Identify correct loop condition

   Loop should run while i < 4 to avoid ArrayIndexOutOfBoundsException.

3. Final Answer:

   Loop condition should be i < 4, not i <= 4 -> Option C

4. Quick Check:

   Array indices 0 to 3, loop must use i < 4 [OK]

Hint: Array indices go from 0 to size-1 [OK]

Common Mistakes:

• Using <= instead of < in loop condition
• Assuming array size 4 means indices 0 to 4
• Ignoring generic type warnings

5. You want to represent a weighted directed graph using adjacency lists in Java. Which data structure is best to store the edges for each vertex?

hard

A. ArrayList<Integer> storing only destination vertices

B. ArrayList> storing destination and weight

C. ArrayList storing all edges as lists

D. ArrayList<Boolean> storing if edge exists or not

Solution

1. Step 1: Understand weighted directed graph needs

   Each edge has a destination vertex and a weight value, so both must be stored.

2. Step 2: Choose data structure to store pairs

   Using ArrayList of Pair allows storing destination and weight together.

3. Step 3: Evaluate other options

   ArrayList<Integer> storing only destination vertices stores only destination, missing weight; ArrayList storing all edges as lists is nested lists without clear pair; ArrayList<Boolean> storing if edge exists or not stores booleans, not weights.

4. Final Answer:

   ArrayList> storing destination and weight -> Option B

5. Quick Check:

   Weighted edges need destination + weight pairs [OK]

Hint: Store edges as pairs (destination, weight) in adjacency list [OK]

Common Mistakes:

• Storing only destination without weight
• Using nested lists without clear structure
• Using booleans which can't hold weights