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Graph Algorithms Visualization
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Explore fundamental graph algorithms through interactive visualizations
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<h2 class="text-xl font-bold text-white mb-4">Graph Algorithms</h2>
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<h3 class="font-medium text-white">Breadth-First Search</h3>
<p class="text-sm text-gray-300 mt-1">Level-order traversal algorithm</p>
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<h3 class="font-medium text-white">Depth-First Search</h3>
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<h3 class="font-medium text-white">Dijkstra's Algorithm</h3>
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Time: O(V + E)
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Space: O(V)
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<text x="100" y="155" text-anchor="middle" class="text-white font-bold">A</text>
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<text x="300" y="155" text-anchor="middle" class="text-white font-bold">C</text>
<text x="400" y="105" text-anchor="middle" class="text-white font-bold">D</text>
<text x="200" y="205" text-anchor="middle" class="text-white font-bold">E</text>
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<h3 class="font-bold text-white mb-2">Current Step</h3>
<p class="text-gray-300">Visiting node C. Adding its unvisited neighbors D and F to the queue.</p>
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<h3 class="font-bold text-white mb-2">BFS Queue</h3>
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<div class="bg-blue-500 px-3 py-1 rounded">D</div>
<div class="bg-blue-500 px-3 py-1 rounded">F</div>
<div class="bg-gray-700 px-3 py-1 rounded text-gray-400">Empty</div>
<div class="bg-gray-700 px-3 py-1 rounded text-gray-400">Empty</div>
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<h3 class="font-bold text-white mb-2">Pseudocode</h3>
<pre class="text-green-400 text-sm">
BFS(graph, start):
queue = [start]
visited = set([start])
while queue:
node = queue.pop(0)
process(node)
for neighbor in graph[node]:
if neighbor not in visited:
visited.add(neighbor)
queue.append(neighbor)</pre>
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<h3 class="font-bold text-white mb-3">Mathematical Analysis</h3>
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\( \text{Time Complexity: } O(V + E) \)
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BFS visits each vertex exactly once and examines each edge exactly twice
(once from each endpoint in undirected graphs). Therefore, the time complexity
is proportional to the sum of vertices and edges.
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\( \text{Space Complexity: } O(V) \)
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The space complexity is determined by the maximum size of the queue, which
can contain at most \( V-1 \) vertices in the worst case (when all but one
vertex are at the same level of the BFS tree).
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