Comparing Algorithms

The Big Picture

We have now explored four fundamental pathfinding algorithms: Breadth-First Search, Depth-First Search, Dijkstra's algorithm, and A*. Each operates differently. Each has strengths and weaknesses. Understanding these trade-offs is crucial for choosing the right tool when you face a real pathfinding problem.

No algorithm is universally best. The right choice depends on your specific situation—the structure of your graph, whether edges have weights, whether you know where the goal is, and how much memory you can spare.

Metrics That Matter

When comparing algorithms, we care about several things:

Nodes expanded refers to how many cells the algorithm visits before finding the path (or determining none exists). More nodes means more work, more memory, and more time. An efficient algorithm reaches the goal while expanding as few nodes as possible.

Path length counts the number of steps in the discovered path. In an unweighted graph, this is the primary measure of path quality.

Path cost sums the weights along the path. When edges have different costs, we care about minimizing total cost, not just the number of steps. A longer path might be cheaper if it avoids expensive terrain.

Time complexity describes how runtime grows with input size. All four algorithms we studied run in $O(V + E)$ time in the worst case, where $V$ is the number of vertices and $E$ is the number of edges.

Space complexity describes memory usage. BFS and Dijkstra can require $O(V)$ space to store the frontier. DFS typically uses less memory in practice because it only tracks the current path.

Watch BFS, Dijkstra, and A* race side by side on the same grid. The metrics dashboard shows nodes expanded, path cost, and path length in real time. Notice which algorithm reaches the goal first and which expands the fewest nodes.

When to Use What

Here is a summary of when each algorithm shines:

BFS is your go-to for unweighted graphs. It guarantees the shortest path in terms of number of steps. If all edges cost the same, BFS is simple and optimal.

DFS does not guarantee shortest paths, but it uses less memory and is excellent for tasks like checking connectivity, detecting cycles, or exploring all possible paths. Use it when you need any path, not necessarily the best path.

Dijkstra's algorithm handles weighted graphs. It always finds the path with minimum total cost. Use it when edges have varying weights and you have no additional information about where the goal might be.

A* combines Dijkstra's optimality with heuristic guidance. When you know the goal location and can estimate distance to it, A* typically expands far fewer nodes than Dijkstra while still finding the optimal path. It is the algorithm of choice for most practical pathfinding problems with a known goal.

Scenarios in Action

Different map layouts favor different algorithms. Let us see how each performs across various scenarios:

Try each scenario to see how the algorithms compare:

• Open Field: With no obstacles, A*'s heuristic shines. It heads straight for the goal while BFS and Dijkstra explore in all directions.
• Maze: Dense walls limit the benefit of heuristics. All algorithms must navigate the corridors.
• Weighted Terrain: Only Dijkstra and A* find the truly cheapest path. BFS ignores weights entirely.
• No Path: When the goal is unreachable, all algorithms must explore the entire reachable area before giving up.

Memory and Time Complexity

Let us briefly summarize the theoretical guarantees:

The $\log V$ factor in Dijkstra and A* comes from priority queue operations. In practice, A* with a good heuristic explores far fewer nodes than these worst-case bounds suggest.

DFS can use $O(V)$ space in the worst case (a long chain), but on many graphs it uses much less because it only maintains the current path on the stack.

A Visual Race

For a bit of fun, watch the algorithms race across 3D terrain. The elevated perspective shows how each search strategy expands through space:

Rotate and zoom to watch from different angles. Notice how A*'s search forms a more focused cone toward the goal, while BFS expands uniformly in all directions.

Key Takeaways

• No single algorithm wins everywhere. The right choice depends on your graph structure, whether edges are weighted, and what information you have about the goal.
• BFS is optimal for unweighted graphs—use it when all steps cost the same.
• DFS trades optimality for lower memory usage—use it for exploration, cycle detection, or when any path suffices.
• Dijkstra finds the cheapest path in weighted graphs—use it when costs vary and you have no goal heuristic.
• A* typically expands the fewest nodes when a good heuristic is available—use it for most practical pathfinding with known goal locations.
• Understanding your problem is half the battle. A maze, an open field, a weighted terrain map, and an infinite graph each call for different approaches.