Graph Terminology Vertices Edges Directed Undirected Weighted in DSA Typescript - Time & Space Complexity

Time Complexity: Graph Terminology Vertices Edges Directed Undirected Weighted

O(V + E)

Understanding Time Complexity

Graphs are made of points and connections. We want to understand how the work grows when we explore these connections.

How does the number of points and connections affect the time to visit or check them?

Scenario Under Consideration

Analyze the time complexity of this simple graph traversal code.

function traverseGraph(graph: Map): void {
  for (const vertex of graph.keys()) {
    const edges = graph.get(vertex) || [];
    for (const neighbor of edges) {
      // process edge from vertex to neighbor
    }
  }
}
    

This code visits every vertex and its connected edges once.

Identify Repeating Operations

Look at the loops that repeat work:

• Primary operation: Visiting each vertex and then each edge connected to it.
• How many times: Outer loop runs once per vertex (V times), inner loop runs once per edge connected to that vertex.

How Execution Grows With Input

As the graph grows, the work grows with the number of vertices and edges.

Input Size (V vertices, E edges)	Approx. Operations
10 vertices, 15 edges	~25 operations (10 + 15)
100 vertices, 200 edges	~300 operations (100 + 200)
1000 vertices, 5000 edges	~6000 operations (1000 + 5000)

Pattern observation: The total work grows roughly by adding vertices and edges together.

Final Time Complexity

Time Complexity: O(V + E)

This means the time to explore the graph grows with the total number of points plus connections.

Common Mistake

[X] Wrong: "The time depends only on the number of vertices."

[OK] Correct: Edges also matter because each connection must be checked, so ignoring edges misses part of the work.

Interview Connect

Understanding how vertices and edges affect time helps you explain graph algorithms clearly and confidently in interviews.

Self-Check

"What if the graph was directed instead of undirected? How would the time complexity change?"

Practice

1. In a graph, what do we call the points or nodes where edges meet?

easy

A. Edges

B. Weights

C. Vertices

D. Paths

Solution

1. Step 1: Understand graph components

   A graph consists of points called nodes and connections called edges.

2. Step 2: Identify the term for nodes

   The points or nodes where edges meet are called vertices.

3. Final Answer:

   Vertices -> Option C

4. Quick Check:

   Nodes = Vertices [OK]

Hint: Nodes in graphs are called vertices [OK]

Common Mistakes:

• Confusing edges with vertices
• Thinking weights are nodes
• Mixing paths with nodes

2. Which of the following is the correct way to represent a weighted edge from vertex 1 to vertex 2 with weight 5 in TypeScript?

easy

A. const edge = { 1: 2, 5 };

B. const edge = [1, 2, 5];

C. const edge = (1, 2, 5);

D. const edge = { from: 1, to: 2, weight: 5 };

Solution

1. Step 1: Understand object syntax in TypeScript

   Objects use key-value pairs with curly braces and colons.

2. Step 2: Check each option's syntax

   const edge = { from: 1, to: 2, weight: 5 }; correctly uses keys 'from', 'to', and 'weight' with values. const edge = [1, 2, 5]; is an array, valid but less descriptive. const edge = (1, 2, 5); uses parentheses, invalid for object. const edge = { 1: 2, 5 }; has invalid syntax.

3. Final Answer:

   const edge = { from: 1, to: 2, weight: 5 }; -> Option D

4. Quick Check:

   Object with keys and values = const edge = { from: 1, to: 2, weight: 5 }; [OK]

Hint: Use curly braces and key:value pairs for objects [OK]

Common Mistakes:

• Using parentheses instead of braces
• Incorrect object key syntax
• Confusing arrays with objects

3. What will be the output of the following TypeScript code that creates a directed weighted graph and prints its edges?

type Edge = { from: number; to: number; weight: number };
const graph: Edge[] = [
  { from: 1, to: 2, weight: 3 },
  { from: 2, to: 3, weight: 4 },
  { from: 3, to: 1, weight: 5 }
];
for (const edge of graph) {
  console.log(`${edge.from} -> ${edge.to} (weight: ${edge.weight})`);
}

medium

A. 1 -> 2 (weight: 3)
2 -> 3 (weight: 4)
3 -> 1 (weight: 5)

B. 1 -> 2 (weight: 5)
2 -> 3 (weight: 3)
3 -> 1 (weight: 4)

C. 1 -> 3 (weight: 5)
2 -> 1 (weight: 4)
3 -> 2 (weight: 3)

D. Error: Invalid graph definition

Solution

1. Step 1: Read the graph array

   The graph array has edges: 1->2 with weight 3, 2->3 with weight 4, 3->1 with weight 5.

2. Step 2: Trace the for loop output

   The loop prints each edge in the format 'from -> to (weight: weight)'. So output matches 1 -> 2 (weight: 3) 2 -> 3 (weight: 4) 3 -> 1 (weight: 5) exactly.

3. Final Answer:

   1 -> 2 (weight: 3) 2 -> 3 (weight: 4) 3 -> 1 (weight: 5) -> Option A

4. Quick Check:

   Print edges as defined = 1 -> 2 (weight: 3) 2 -> 3 (weight: 4) 3 -> 1 (weight: 5) [OK]

Hint: Match printed edges exactly as in array order [OK]

Common Mistakes:

• Mixing up weights and vertices
• Reversing edge directions
• Assuming syntax error

4. Identify the error in the following TypeScript code that tries to create an undirected weighted graph:

type Edge = { from: number; to: number; weight: number };
const graph: Edge[] = [
  { from: 1, to: 2, weight: 7 },
  { from: 2, to: 1, weight: 7 },
  { from: 2, to: 3, weight: 5 }
];
function addEdge(from: number, to: number, weight: number) {
  graph.push({ from, to, weight });
  graph.push({ from: to, to: from, weight });
}
addEdge(3, 4, 6);
console.log(graph.length);

medium

A. No error; graph length will be 5

B. Error: Duplicate edges cause conflict

C. Error: addEdge function pushes edges incorrectly

D. Error: graph array is immutable

Solution

1. Step 1: Count initial edges

   Initially, graph has 3 edges: 1->2, 2->1, 2->3.

2. Step 2: Analyze addEdge function call

   addEdge(3, 4, 6) pushes { from: 3, to: 4, weight: 6 } and { from: 4, to: 3, weight: 6 }, adding 2 more edges for undirected graph.

3. Step 3: Calculate final length

   3 initial + 2 added = 5. No errors occur; code runs successfully.

4. Final Answer:

   No error; graph length will be 5 -> Option A

5. Quick Check:

   3 + 2 = 5 [OK]

Hint: Count edges before and after addEdge call [OK]

Common Mistakes:

• Assuming duplicate edges cause error
• Miscounting edges after push
• Thinking graph is immutable

5. You have a weighted undirected graph represented as an adjacency list in TypeScript:

const graph: Record = {
  1: [{ node: 2, weight: 4 }, { node: 3, weight: 1 }],
  2: [{ node: 1, weight: 4 }, { node: 3, weight: 2 }, { node: 4, weight: 5 }],
  3: [{ node: 1, weight: 1 }, { node: 2, weight: 2 }, { node: 4, weight: 8 }],
  4: [{ node: 2, weight: 5 }, { node: 3, weight: 8 }]
};

// How to find the total weight of edges connected to vertex 2?

Which code snippet correctly calculates the total weight of edges connected to vertex 2?

hard

A. const total = graph[2].map(edge => edge.weight).length;

B. const total = graph[2].reduce((sum, edge) => sum + edge.weight, 0);

C. const total = graph[2].filter(edge => edge.weight > 0);

D. const total = graph.reduce((sum, edges) => sum + edges.weight, 0);

Solution

1. Step 1: Understand adjacency list structure

   graph[2] is an array of edges connected to vertex 2, each with a weight property.

2. Step 2: Calculate total weight

   Use reduce to sum all edge weights in graph[2]. const total = graph[2].reduce((sum, edge) => sum + edge.weight, 0); correctly sums weights. const total = graph[2].map(edge => edge.weight).length; counts edges, not sum. const total = graph[2].filter(edge => edge.weight > 0); filters edges, no sum. const total = graph.reduce((sum, edges) => sum + edges.weight, 0); tries to reduce graph object, invalid.

3. Final Answer:

   const total = graph[2].reduce((sum, edge) => sum + edge.weight, 0); -> Option B

4. Quick Check:

   Sum weights with reduce = const total = graph[2].reduce((sum, edge) => sum + edge.weight, 0); [OK]

Hint: Use reduce to sum weights in adjacency list [OK]

Common Mistakes:

• Using map length instead of sum
• Filtering without summing
• Reducing wrong data structure