Dijkstra's Algorithm: Step-by-Step Guide
Dijkstra's Algorithm is a famous algorithm used to find the shortest path
from a starting node to all other nodes in a graph. It is widely applied in routing and navigation systems,
network optimization, and many other fields.
This visualization will walk you through each step of the algorithm and help you understand how it works,
starting from the initialization to the final path discovery.
How It Works
- Initialization: All nodes are initially assigned a tentative distance
of
infinity, except for the start node, which has a distance of 0. This means
all nodes are "far away" initially.
- Choosing the Next Node: At each step, the algorithm selects the unvisited node with the
smallest tentative distance. This is efficiently achieved using a priority queue, such
as a min-heap. If there are multiple nodes with the same smallest distance, one is chosen arbitrarily,
as it doesn't impact the result.
- Relaxation (Updating Distances): For the selected node, the algorithm checks its
neighbors. If traveling from the current node to a neighbor results in a shorter path than previously
known, the neighbor’s distance is updated.
- Marking Nodes as Visited: After considering all neighboring nodes, the current node is
marked as visited (colored green), and its shortest distance is finalized. The algorithm will never
revisit a visited node because the shortest path to it has already been determined.
- Repetition: The algorithm repeats the process of selecting the node with the smallest
distance, updating its neighbors, and marking it as visited until all nodes are visited or the
destination is reached.
- Destination Node: Once the destination node is visited, it is marked in red to show
that the shortest path has been found.
- The Golden Path: Finally, the shortest path (golden path) from the start node to the
destination node is highlighted, showing the optimal route.
Key Concepts:
- Tentative Distance: The current known shortest distance from the start node to each
node.
- Greedy Approach: Dijkstra's algorithm always chooses the closest node to expand first,
gradually discovering the shortest path.
- Arbitrary Node Selection: When multiple unvisited nodes have the same tentative
distance, one is arbitrarily chosen. This doesn’t affect the algorithm's correctness because all choices
are equivalent at that point.
- Graph Requirements: Dijkstra's algorithm assumes that edge weights are
non-negative. Negative edge weights can create cycles that continually decrease the
path length, causing the algorithm to fail in finding the shortest path. If negative weights are
present, algorithms like Bellman-Ford are needed
instead.
Why It Works
Dijkstra’s Algorithm guarantees finding the shortest path because it processes nodes in
order of their known shortest distance. By always expanding the closest node, it ensures that the shortest
path to each node is found as the algorithm progresses.
Beyond Binary Heaps
An exotic data structure known as a Fibonacci Heap can be used to implement the priority
queue for better theoretical time complexity, reducing the algorithm's running time from O((V + E)
log V) to O(E + V log V). However, Fibonacci Heaps are complex and have large
constant
factors, which can make them less practical for real-world applications.