[Verse 1]
Graph sprawled before me like a city maze
Vertices scattered, edges blaze the ways
Starting node selected, distance marked as zero
Every other vertex tagged infinity, my hero
Priority queue holding all the candidates
Extracting minimum distance, algorithm calculates
Neighbors get examined, weights get compared
Relaxation process leaves no stone unspared
[Chorus]
Extract the minimum, relax the edges tight
Update distances when you find a shorter sight
Mark it visited, never check again
Dijkstra's method finds the shortest chain
Greedy choice at every single turn
Optimal substructure helps the lesson burn
Priority queue keeps the order straight
Shortest paths from source, algorithm's fate
[Verse 2]
Heap data structure manages the queue
Binary minimum keeps the runtime true
Logarithmic insertion, extraction clean
Adjacency lists make the graph serene
Relaxation check: current plus the weight
Less than neighbor's distance? Update straight
Decrease key operation in the heap
Maintains the invariant we swore to keep
[Chorus]
Extract the minimum, relax the edges tight
Update distances when you find a shorter sight
Mark it visited, never check again
Dijkstra's method finds the shortest chain
Greedy choice at every single turn
Optimal substructure helps the lesson burn
Priority queue keeps the order straight
Shortest paths from source, algorithm's fate
[Bridge]
No negative weights allowed inside
Single source shortest path, our guide
V log V plus E complexity bound
Fibonacci heap makes it more profound
Predecessor tracking builds the tree
Shortest path reconstruction, guarantee
[Verse 3]
Visited set grows with each extraction
Wavefront spreading, distance satisfaction
Optimal solutions to subproblems found
Greedy property keeps us theorem-bound
Applications vast: routing protocols
Network analysis, GPS controls
Transportation networks, social graphs too
Dijkstra's legacy forever true
[Chorus]
Extract the minimum, relax the edges tight
Update distances when you find a shorter sight
Mark it visited, never check again
Dijkstra's method finds the shortest chain
Greedy choice at every single turn
Optimal substructure helps the lesson burn
Priority queue keeps the order straight
Shortest paths from source, algorithm's fate
[Outro]
Edsger's wisdom echoes through the code
Shortest path algorithm, gold standard mode
From source to every vertex, optimal cost
In the realm of graphs, no distance lost