Dijkstra's Algorithm Single Source Shortest Path in DSA Typescript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of Dijkstra's Algorithm?

To find the shortest path from a single starting point (source) to all other points (nodes) in a graph with non-negative edge weights.

intermediate

In Dijkstra's Algorithm, what data structure is commonly used to select the next node with the smallest distance?

A priority queue (often implemented as a min-heap) is used to efficiently pick the node with the smallest current distance.

intermediate

Why can't Dijkstra's Algorithm handle graphs with negative edge weights?

Because it assumes that once a node's shortest distance is found, it won't change. Negative edges can create shorter paths later, breaking this assumption.

advanced

What is the time complexity of Dijkstra's Algorithm using a min-heap and adjacency list?

O((V + E) log V), where V is the number of vertices and E is the number of edges.

intermediate

Explain the relaxation step in Dijkstra's Algorithm.

Relaxation means checking if the current known distance to a neighbor can be improved by going through the current node. If yes, update the neighbor's distance.

What type of graph edges does Dijkstra's Algorithm require?

A Only unweighted edges

B Non-negative edge weights

C Negative edge weights allowed

D Edges with zero weight only

Dijkstra's Algorithm works correctly only if all edge weights are zero or positive.

Which data structure helps efficiently pick the next closest node in Dijkstra's Algorithm?

A Stack

B Queue

C Priority Queue

D Linked List

A priority queue allows quick access to the node with the smallest tentative distance.

What happens during the relaxation step?

A We update the shortest distance to neighbors if a shorter path is found

B We remove nodes from the graph

C We add new edges to the graph

D We reset all distances to infinity

Relaxation updates the shortest known distances to neighbors if a better path is found.

What is the initial distance to the source node in Dijkstra's Algorithm?

A Negative one

B Infinity

C One

D Zero

The source node distance is zero because the distance from the source to itself is zero.

Which of these is NOT true about Dijkstra's Algorithm?

A It works with negative edge weights

B It updates distances through relaxation

C It uses a priority queue for efficiency

D It finds shortest paths from one source to all nodes

Dijkstra's Algorithm does not work correctly with negative edge weights.

Describe the main steps of Dijkstra's Algorithm for finding shortest paths.

Think about how you keep track of shortest distances and update them.

Explain why Dijkstra's Algorithm cannot handle negative edge weights and what could happen if it tries.

Consider what happens if a shorter path appears after a node is finalized.

Practice

1. What is the main purpose of Dijkstra's Algorithm in graph theory?

easy

A. To detect cycles in a graph

B. To find the minimum spanning tree of a graph

C. To find the shortest path from a single source to all other vertices

D. To sort the vertices in topological order

Solution

1. Step 1: Understand the algorithm's goal

   Dijkstra's Algorithm is designed to find the shortest distance from one starting point (source) to every other point in a graph.

2. Step 2: Compare with other graph algorithms

   Minimum spanning tree algorithms find a tree connecting all vertices with minimum total weight, cycle detection finds loops, and topological sort orders vertices. These are different tasks.

3. Final Answer:

   To find the shortest path from a single source to all other vertices -> Option C

4. Quick Check:

   Dijkstra's Algorithm = shortest paths from source [OK]

Hint: Dijkstra finds shortest paths from one start point [OK]

Common Mistakes:

• Confusing shortest path with minimum spanning tree
• Thinking it detects cycles
• Mixing it with topological sorting

2. Which of the following TypeScript code snippets correctly initializes the distance array for Dijkstra's Algorithm for a graph with 5 vertices?

easy

A. const dist: number[] = Array(5).fill(Infinity); dist[0] = 0;

B. const dist: number[] = new Array(5).map(() => 0);

C. const dist: number[] = Array(5).fill(0); dist[0] = Infinity;

D. const dist: number[] = Array(5).fill(null); dist[0] = 0;

Solution

1. Step 1: Understand initialization requirements

   Distances start as Infinity except the source vertex which is 0.

2. Step 2: Check each code snippet

   const dist: number[] = Array(5).fill(Infinity); dist[0] = 0; correctly fills with Infinity and sets source distance to 0. const dist: number[] = new Array(5).map(() => 0); fills with 0 incorrectly. const dist: number[] = Array(5).fill(0); dist[0] = Infinity; fills with 0 and sets source to Infinity, wrong. const dist: number[] = Array(5).fill(null); dist[0] = 0; fills with null, which is invalid for numeric distances.

3. Final Answer:

   const dist: number[] = Array(5).fill(Infinity); dist[0] = 0; -> Option A

4. Quick Check:

   Distance array init = Infinity except source 0 [OK]

Hint: Fill array with Infinity, set source distance to 0 [OK]

Common Mistakes:

• Using map on new Array without fill (returns undefined)
• Setting all distances to 0 initially
• Using null instead of numeric values

3. Given the following TypeScript code snippet implementing Dijkstra's Algorithm on a graph with edges: 0->1 (4), 0->2 (1), 2->1 (2), what is the final distance array after running from source vertex 0?

const graph = new Map>();
graph.set(0, [[1, 4], [2, 1]]);
graph.set(1, []);
graph.set(2, [[1, 2]]);

const dist = Array(3).fill(Infinity);
dist[0] = 0;
const visited = new Set<number>();

while (visited.size < 3) {
  let u = -1;
  let minDist = Infinity;
  for (let i = 0; i < 3; i++) {
    if (!visited.has(i) && dist[i] < minDist) {
      minDist = dist[i];
      u = i;
    }
  }
  if (u === -1) break;
  visited.add(u);
  for (const [v, w] of graph.get(u) || []) {
    if (dist[u] + w < dist[v]) {
      dist[v] = dist[u] + w;
    }
  }
}
console.log(dist);

medium

A. [0, 6, 1]

B. [0, 4, 1]

C. [Infinity, Infinity, Infinity]

D. [0, 3, 1]

Solution

1. Step 1: Initialize distances and start from source 0

   dist = [0, Infinity, Infinity], visited = {}.

2. Step 2: Pick unvisited vertex with smallest dist and update neighbors

   Pick 0: update dist[1] = 4, dist[2] = 1. visited = {0}. Pick 2 (dist=1): update dist[1] = min(4, 1+2=3) = 3. visited = {0,2}. Pick 1 (dist=3): no neighbors. visited = {0,1,2}.

3. Final Answer:

   [0, 3, 1] -> Option D

4. Quick Check:

   Distances after updates = [0, 3, 1] [OK]

Hint: Update neighbors only if new path is shorter [OK]

Common Mistakes:

• Not updating dist[1] from 4 to 3 via vertex 2
• Picking wrong vertex with smallest distance
• Ignoring visited set causing infinite loops

4. Identify the bug in the following TypeScript code snippet implementing Dijkstra's Algorithm:

const graph = new Map>();
graph.set(0, [[1, 7], [2, 9], [5, 14]]);
graph.set(1, [[2, 10], [3, 15]]);
graph.set(2, [[3, 11], [5, 2]]);
graph.set(3, [[4, 6]]);
graph.set(4, [[5, 9]]);
graph.set(5, []);
const dist: number[] = new Array(graph.size);
dist.fill(Infinity);
dist[0] = 0;
const visited = new Set<number>();

while (visited.size < graph.size) {
  let u = -1;
  let minDist = Infinity;
  for (let i = 0; i < graph.size; i++) {
    if (!visited.has(i) && dist[i] < minDist) {
      minDist = dist[i];
      u = i;
    }
  }
  visited.add(u);
  for (const [v, w] of graph.get(u) || []) {
    if (dist[u] + w < dist[v]) {
      dist[v] = dist[u] + w;
    }
  }
}

medium

A. dist array is not initialized before use

B. Adding vertex u to visited set before checking if u is -1

C. Using graph.size instead of graph.length in loop

D. Not updating dist[u] inside the loop

Solution

1. Step 1: Analyze the selection of vertex u

   If no unvisited vertex has finite dist, u remains -1.

2. Step 2: Check visited addition without validation

   Code adds u to visited without checking if u === -1, which can cause adding -1 and errors.

3. Final Answer:

   Adding vertex u to visited set before checking if u is -1 -> Option B

4. Quick Check:

   Check u !== -1 before visited.add(u) [OK]

Hint: Always verify selected vertex is valid before marking visited [OK]

Common Mistakes:

• Ignoring u === -1 case causing infinite loops
• Confusing graph.size with array length
• Not initializing dist array properly

5. You have a weighted graph with vertices 0 to 4 and edges: 0->1 (2), 0->2 (4), 1->2 (1), 1->3 (7), 2->4 (3), 3->4 (1). Using Dijkstra's Algorithm from source 0, what is the shortest path distance to vertex 4?

hard

A. 6

B. 7

C. 8

D. 9

Solution

1. Step 1: Calculate shortest distances step-by-step

   Start dist: [0,∞,∞,∞,∞] From 0: dist[1]=2, dist[2]=4 From 1: dist[2]=min(4,2+1=3)=3, dist[3]=2+7=9 From 2: dist[4]=3+3=6 From 3: dist[4]=min(6,9+1=10)=6

2. Step 2: Identify shortest distance to vertex 4

   Shortest distance to vertex 4 is 6 via path 0->1->2->4.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   Shortest path distance to 4 = 6 [OK]

Hint: Check all paths and pick minimum sum of edge weights [OK]

Common Mistakes:

• Ignoring shorter path via vertex 2
• Adding edges incorrectly
• Assuming direct edge is always shortest