Dijkstra's Algorithm Visualization
Dijkstra's Algorithm finds the shortest path from a source node to all other nodes in a graph with non-negative weights. It uses a priority queue to explore nodes in order of increasing distance. This algorithm works by maintaining a set of visited nodes and, at each step, selecting the unvisited node with the smallest known distance to visit next. This process continues until all nodes have been visited, making it essential for network routing problems. Unlike MST algorithms, Dijkstra’s focuses on optimal paths from a single source.
Pseudocode
dist = map vertex -> Infinity
dist[src] = 0
heap = new MinHeap()
heap.insert(src, 0)
while heap is not empty:
(u, d) = heap.extract()
if u is not visited:
relax(u, d)
function relax(u, d):
mark u as visited
for each neighbor v of u:
alt = d + weight(u, v)
if alt < dist[v]:
dist[v] = alt
heap.insert(v, alt)
AI Summary
(5 credits)
Common Interview Questions
Why does Dijkstra's algorithm fail or produce incorrect results on graphs with negative edge weights?
Dijkstra's algorithm makes a greedy assumption: once a node is popped from the Min-Priority Queue, its shortest distance is finalized and will never decrease. A negative edge encountered later could reveal a shorter path to an already finalized node, invalidating Dijkstra's greedy guarantee.
What is the time complexity of Dijkstra's algorithm using a Min-Heap vs an unsorted array?
Min-Heap implementation: O((V + E) log V). Unsorted Array implementation: O(V²). Array implementation is actually faster for complete/extremely dense graphs where E ≈ V², while Min-Heap excels on sparse graphs.
How is Dijkstra's algorithm adapted in real-world GPS navigation systems (e.g. Google Maps)?
Raw Dijkstra is too slow on worldwide road networks. GPS systems use A* Search with geometric heuristics, Contraction Hierarchies (CH), and Multi-Level Dijkstra to precompute highway routes and answer shortest path queries in milliseconds.
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