Dijkstra's Algorithm Single Source Shortest Path in DSA Javascript - 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 efficiently select the next node with the smallest tentative distance?

A priority queue (often implemented as a min-heap) is used to quickly 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 after visiting nodes, breaking this assumption.

advanced

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

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

beginner

Explain the role of the 'distance' array in Dijkstra's Algorithm.

It stores the shortest known distance from the source node to every other node. It updates as the algorithm finds shorter paths.

What type of graph edge weights can Dijkstra's Algorithm handle?

A Both negative and positive weights

B Only negative weights

C Only non-negative weights

D No weights (unweighted graphs only)

Dijkstra's Algorithm requires non-negative edge weights to guarantee correct shortest paths.

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

A Stack

B Priority Queue

C Linked List

D Queue

A priority queue (min-heap) allows quick access to the node with the smallest tentative distance.

What happens if you run Dijkstra's Algorithm on a graph with negative edge weights?

A It may give incorrect results

B It always finds the correct shortest paths

C It runs faster

D It ignores negative edges

Negative edges can cause the algorithm to miss shorter paths found later, leading to wrong answers.

What does the 'distance' array represent in Dijkstra's Algorithm?

A Shortest known distance from source to each node

B Total weight of all edges

C Number of edges from source

D Visited nodes count

The distance array keeps track of the shortest distance found so far from the source to each node.

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

A One

B Infinity

C Negative infinity

D Zero

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

Describe step-by-step how Dijkstra's Algorithm finds the shortest paths from a source node.

Think about how the algorithm updates distances and picks nodes.

Explain why Dijkstra's Algorithm cannot be used with graphs containing negative edge weights and what alternative algorithm can be used instead.

Focus on the assumption about shortest paths and mention a known alternative.

Practice

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

easy

A. To sort vertices in topological order

B. To find the maximum flow in a network

C. To detect cycles in a graph

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

Solution

1. Step 1: Understand the algorithm's goal

   Dijkstra's Algorithm is designed to find the shortest paths from one starting point (source) to all other points in a graph with non-negative edge weights.

2. Step 2: Compare with other graph algorithms

   Other algorithms like max flow or cycle detection serve different purposes, so they do not match Dijkstra's goal.

3. Final Answer:

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

4. Quick Check:

   Shortest path from one source = To find the shortest path from a single source to all other vertices [OK]

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

Common Mistakes:

• Confusing Dijkstra with max flow algorithms
• Thinking it detects cycles
• Assuming it sorts vertices

2. Which of the following is the correct way to initialize the distance array in Dijkstra's Algorithm in JavaScript?

easy

A. const dist = new Array(n).fill(Infinity); dist[source] = 0;

B. const dist = Array(n).fill(0); dist[source] = Infinity;

C. const dist = new Array(n).fill(null); dist[source] = 0;

D. const dist = new Array(n).fill(-1); dist[source] = 0;

Solution

1. Step 1: Understand distance initialization

   Distances to all nodes except the source should start as Infinity to represent unknown paths, and the source distance is 0.

2. Step 2: Check each option

   const dist = new Array(n).fill(Infinity); dist[source] = 0; correctly sets all distances to Infinity and source to 0. Others either set wrong initial values or invert source distance.

3. Final Answer:

   const dist = new Array(n).fill(Infinity); dist[source] = 0; -> Option A

4. Quick Check:

   Distance array starts with Infinity except source = const dist = new Array(n).fill(Infinity); dist[source] = 0; [OK]

Hint: Start distances as Infinity, source as 0 [OK]

Common Mistakes:

• Setting source distance to Infinity
• Using 0 for all distances initially
• Using null or -1 incorrectly

3. Consider the following JavaScript code snippet implementing part of Dijkstra's Algorithm. What will be the output of the distance array after running this code?

const graph = {
  0: [{node:1, weight:4}, {node:2, weight:1}],
  1: [{node:3, weight:1}],
  2: [{node:1, weight:2}, {node:3, weight:5}],
  3: []
};
const dist = [0, Infinity, Infinity, Infinity];
const visited = new Set();

function dijkstra() {
  while (visited.size < 4) {
    let u = -1;
    let minDist = Infinity;
    for (let i = 0; i < 4; i++) {
      if (!visited.has(i) && dist[i] < minDist) {
        minDist = dist[i];
        u = i;
      }
    }
    visited.add(u);
    for (const neighbor of graph[u]) {
      if (dist[u] + neighbor.weight < dist[neighbor.node]) {
        dist[neighbor.node] = dist[u] + neighbor.weight;
      }
    }
  }
}
dijkstra();
console.log(dist);

medium

A. [0, 4, 1, 6]

B. [0, 3, 1, 4]

C. [0, 2, 1, 3]

D. [0, 5, 1, 7]

Solution

1. Step 1: Trace initial distances and first iteration

   Start dist = [0, Infinity, Infinity, Infinity]. Pick node 0 (dist=0). Update neighbors: node 1 = 4, node 2 = 1.

2. Step 2: Next iterations and updates

   Pick node 2 (dist=1). Update node 1: min(4, 1+2=3) = 3; node 3: min(Infinity, 1+5=6) = 6. Pick node 1 (dist=3). Update node 3: min(6, 3+1=4) = 4. Pick node 3 (dist=4). No neighbors.

3. Final Answer:

   [0, 3, 1, 4] -> Option B

4. Quick Check:

   Distances after updates = [0, 3, 1, 4] [OK]

Hint: Pick unvisited node with smallest dist, update neighbors [OK]

Common Mistakes:

• Not updating distances correctly
• Choosing wrong next node
• Ignoring neighbor updates

4. The following code snippet is intended to implement Dijkstra's Algorithm but contains a bug. What is the bug?

const dist = new Array(n).fill(Infinity);
dist[source] = 0;
const visited = new Set();

while (visited.size < n) {
  let u = -1;
  let minDist = Infinity;
  for (let i = 0; i < n; i++) {
    if (dist[i] < minDist) {
      minDist = dist[i];
      u = i;
    }
  }
  visited.add(u);
  for (const neighbor of graph[u]) {
    if (dist[u] + neighbor.weight < dist[neighbor.node]) {
      dist[neighbor.node] = dist[u] + neighbor.weight;
    }
  }
}

medium

A. The visited set is not updated inside the loop

B. The distance array is not initialized correctly

C. The code does not check if node i is already visited before selecting u

D. The neighbor weights are not added to dist[u]

Solution

1. Step 1: Analyze node selection loop

   The loop selects node u with smallest dist[i] but does not check if i is already visited, so it may pick visited nodes again.

2. Step 2: Understand impact of missing visited check

   Without skipping visited nodes, the algorithm can loop incorrectly or update distances wrongly, breaking correctness.

3. Final Answer:

   The code does not check if node i is already visited before selecting u -> Option C

4. Quick Check:

   Must skip visited nodes when picking u = The code does not check if node i is already visited before selecting u [OK]

Hint: Always skip visited nodes when choosing next node [OK]

Common Mistakes:

• Forgetting to check visited nodes in selection
• Incorrect distance initialization
• Not updating visited set

5. You have a weighted graph with 5 nodes (0 to 4). The edges are:

• 0 -> 1 (2)
• 0 -> 2 (4)
• 1 -> 2 (1)
• 1 -> 3 (7)
• 2 -> 4 (3)
• 3 -> 4 (1)

Using Dijkstra's Algorithm from source node 0, what is the shortest distance to node 4?

hard

A. 6

B. 7

C. 8

D. 5

Solution

1. Step 1: Trace shortest paths from node 0

   Start dist: 0=0, others=Infinity. From 0: dist[1]=2, dist[2]=4.

2. Step 2: Update distances via neighbors

   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.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   Shortest distance to node 4 = 6 [OK]

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

Common Mistakes:

• Ignoring shorter path through node 2
• Adding wrong edge weights
• Not updating distances properly