Navigating the Maze: A Step-by-Step Walkthrough of Dijkstra's Algorithm | Finding the Shortest Route: Dijkstra's Algorithm and Beyond | Advanced Computer Algorithms: Data Structures, Graph Theory, and Complexity Analysis for University-Level Computer Science

Now that we've grappled with the fundamental idea of finding the shortest path, let's roll up our sleeves and dive into Dijkstra's algorithm. Imagine you're navigating a city with varying road lengths. Dijkstra's algorithm is like having a super-efficient GPS that always finds the quickest way from your current location to any other point, by systematically exploring the city.

At its core, Dijkstra's algorithm works by maintaining a set of visited nodes and a set of unvisited nodes. It iteratively selects the unvisited node with the smallest known distance from the starting node and marks it as visited. This process continues until all reachable nodes have been visited, or until the destination node has been reached.

To keep track of distances, we'll use two primary data structures: a distance array (or map) to store the shortest distance found so far from the start node to every other node, and a way to keep track of which nodes are unvisited. We'll initialize all distances to infinity, except for the starting node, which has a distance of 0.

const distances = {};
const unvisited = new Set();

// Initialize distances and unvisited set (assuming 'graph' is your adjacency list)
for (const node in graph) {
  distances[node] = Infinity;
  unvisited.add(node);
}
distances[startNode] = 0;

The heart of the algorithm is a loop that continues as long as there are unvisited nodes. In each iteration, we find the unvisited node with the minimum distance. This node becomes our current node.

while (unvisited.size > 0) {
  let currentNode = null;
  let minDistance = Infinity;

  for (const node of unvisited) {
    if (distances[node] < minDistance) {
      minDistance = distances[node];
      currentNode = node;
    }
  }

  // If no reachable unvisited node is found, we can break.
  if (currentNode === null) break;

  unvisited.delete(currentNode);

  // ... process neighbors ...
}

Once we've selected our currentNode, we examine its neighbors. For each neighbor, we calculate a potential new shortest distance from the start node through the currentNode. If this new distance is shorter than the current recorded distance to the neighbor, we update the neighbor's distance.

for (const neighbor in graph[currentNode]) {
  const weight = graph[currentNode][neighbor];
  const distanceThroughCurrent = distances[currentNode] + weight;

  if (distanceThroughCurrent < distances[neighbor]) {
    distances[neighbor] = distanceThroughCurrent;
    // Optionally, you might also store the 'previous' node to reconstruct the path
    // previous[neighbor] = currentNode;
  }
}

This process repeats. With each step, we 'commit' to the shortest path to the selected node, effectively expanding our known shortest paths outwards from the start. This is why Dijkstra's algorithm is often visualized as a wave expanding from the starting point.

graph TD
    A[Start Node] --> B{Select unvisited node with min distance};
    B --> C{Process Neighbors};
    C --> D{Update distances if shorter path found};
    D --> B;
    B -- No unvisited nodes or destination reached --> E[Algorithm Ends];

Let's walk through a small example to solidify this. Imagine a graph with nodes A, B, C, and D, and we want to find the shortest path from A.

Initial state: Distances: {A: 0, B: Infinity, C: Infinity, D: Infinity} Unvisited: {A, B, C, D}

Step 1:

Select A (distance 0).
Neighbors of A: B (weight 2), C (weight 4).
Update distances: {A: 0, B: 2, C: 4, D: Infinity}.
Unvisited: {B, C, D}

Step 2:

Select B (distance 2).
Neighbors of B: C (weight 1).
Update distances: {A: 0, B: 2, C: 4 (original) vs. 2+1=3 (new) -> C: 3, D: Infinity}.
Unvisited: {C, D}

Step 3:

Select C (distance 3).
Neighbors of C: D (weight 5).
Update distances: {A: 0, B: 2, C: 3, D: 3+5=8}.
Unvisited: {D}

Step 4:

Select D (distance 8).
No unvisited neighbors.
Unvisited: {}

Algorithm ends. Shortest distances from A are: A: 0, B: 2, C: 3, D: 8.

This step-by-step process, with its focus on greedily choosing the closest unvisited node, is the essence of Dijkstra's algorithm. While this basic implementation is conceptually clear, more advanced data structures like priority queues can significantly optimize its performance for larger graphs.

const distances = {}; const unvisited = new Set(); // Initialize distances and unvisited set (assuming 'graph' is your adjacency list) for (const node in graph) { distances[node] = Infinity; unvisited.add(node); } distances[startNode] = 0;

while (unvisited.size > 0) { let currentNode = null; let minDistance = Infinity; for (const node of unvisited) { if (distances[node] < minDistance) { minDistance = distances[node]; currentNode = node; } } // If no reachable unvisited node is found, we can break. if (currentNode === null) break; unvisited.delete(currentNode); // ... process neighbors ... }

for (const neighbor in graph[currentNode]) { const weight = graph[currentNode][neighbor]; const distanceThroughCurrent = distances[currentNode] + weight; if (distanceThroughCurrent < distances[neighbor]) { distances[neighbor] = distanceThroughCurrent; // Optionally, you might also store the 'previous' node to reconstruct the path // previous[neighbor] = currentNode; } }

graph TD A[Start Node] --> B{Select unvisited node with min distance}; B --> C{Process Neighbors}; C --> D{Update distances if shorter path found}; D --> B; B -- No unvisited nodes or destination reached --> E[Algorithm Ends];