Why Dijkstra's Algorithm Single Source Shortest Path in DSA Javascript?

The Big Idea

What if your phone could instantly find the fastest way home, no matter how complicated the roads are?

The Scenario

Imagine you are in a new city and want to find the shortest way to visit a friend's house from your hotel. You try to remember every street and calculate distances in your head.

This manual way is like trying to find the shortest path in a map without any help.

The Problem

Doing this by hand is slow and confusing. You might forget some streets or pick a longer route by mistake.

It's easy to get lost or waste time checking every possible path.

The Solution

Dijkstra's Algorithm helps by automatically finding the shortest path from one place to all others in a city map.

It checks paths step-by-step, always choosing the closest next stop, so you never waste time on longer routes.

Before vs After

✗ Before

function findShortestPath(graph, start) {
  // Try all paths manually
  // Keep track of distances by hand
  // Very slow and error-prone
}

✓ After

function dijkstra(graph, start) {
  const distances = {};
  const visited = new Set();
  // Initialize distances and find shortest paths efficiently
}

What It Enables

It makes finding the quickest route in complex maps easy and fast, even when there are many roads.

Real Life Example

GPS apps use Dijkstra's Algorithm to show you the fastest way home by calculating shortest paths on the road network.

Key Takeaways

Manual pathfinding is slow and error-prone.

Dijkstra's Algorithm finds shortest paths efficiently.

It is essential for navigation and routing applications.

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