Finding the shortest path between two points in a grid is one of the most visual problems in computer science. It appears everywhere: from video game AI to GPS navigation, from robot motion planning to network routing.

But not all search algorithms are created equal. Some find the shortest path, others just find a path. Some explore everything, others are guided by intuition. The best way to understand the difference is to watch them work.

Below you can experiment with four classic pathfinding algorithms on a grid with random obstacles. Click to draw or erase walls, move the start and end points, pick an algorithm, and hit Run to watch it explore.

The Algorithms

BFS (Breadth-First Search)

BFS explores in all directions equally, layer by layer, like ripples in a pond. It always finds the shortest path in an unweighted grid because it visits all cells at distance $d$ before any cell at distance $d+1$.

DFS (Depth-First Search)

DFS dives deep into one direction before backtracking. It can find a path quickly, but that path is often far from optimal — it meanders through the grid like someone exploring a maze by always turning left.

A* (A-Star)

A* is BFS with a brain. It uses a heuristic (Manhattan distance in our case) to estimate how far each cell is from the goal. Cells that look more promising are explored first. With an admissible heuristic, A* guarantees the shortest path while typically exploring far fewer cells than BFS.

Where $g(n)$ is the cost so far and $h(n)$ is the estimated remaining cost.

IDA* (Iterative Deepening A*)

IDA* combines the optimality of A* with the memory efficiency of DFS. Instead of keeping a priority queue, it performs a series of depth-limited DFS searches, increasing the depth limit each time. The limit is not a simple depth counter but the $f$-value threshold from A*.

What to Try

Try changing the wall density and see how each algorithm behaves. Place the start and goal in opposite corners, then swap them and compare the result. You can also try nearby cells, or cells separated by a wall. The differences become especially clear with algorithms like DFS, which explore in a fixed order.

After each run, the panel below the visualization shows the number of explored cells and the length of the path found, so you can compare which algorithms explore more and whether they actually find the optimal path.