请求讲解BFS、DFS、LRTA*算法在带权网格寻路中的每步决策逻辑

Got it, let's walk through exactly how BFS, DFS, and LRTA* make step-by-step decisions for your grid-based pathfinding project—including handling walls, 1-cost land, and 2-cost grass. I'll keep this practical since you're building this for a course assignment.

BFS (Breadth-First Search) #

BFS is all about exploring nodes level by level—think of it as ripples spreading out from the start. But a quick heads-up: standard BFS prioritizes number of steps, not total movement cost. So it'll find the shortest path in terms of how many tiles you step on, but not necessarily the cheapest one (if grass is in the way). Here's its decision process:

• Initialization:
  • Add the start node to a queue, mark it as visited, and track two things: the total cost to reach it (starts at 0) and its parent node (to reconstruct the path later).
• Loop until queue is empty or end is found:
  • Pull the front node from the queue (this is the oldest node we added, keeping the level order).
  • If this node is the end, stop and backtrack through parent nodes to get the path.
  • Check all 4 adjacent nodes (up, down, left, right—adjust if you allow 8-directional movement):
    • Skip any node that's a wall (can't move there).
    • If the node hasn't been visited yet:
      • Calculate the total cost to reach it: current node's cost + 1 (land) or 2 (grass).
      • Mark it as visited, set its parent to the current node, and add it to the end of the queue.
• Key quirk: BFS will explore every tile at step 1 before moving to step 2, so it guarantees the shortest path in steps—but if a step-2 path has all grass (total cost 4) and a step-3 path has all land (total cost 3), BFS will pick the step-2 path even though it's more expensive.

DFS (Depth-First Search) #

DFS is the "go all in on one direction" algorithm. It uses a stack (or recursion) to dive as deep as possible into one path before backtracking. It's great for checking if a path exists, but almost never finds the shortest (step or cost) path. Here's how it decides where to go:

• Initialization:
  • Push the start node onto a stack, mark it as visited, and track its parent node.
• Loop until stack is empty or end is found:
  • Pop the top node from the stack (this is the most recent node we added, so we keep going deep).
  • If this node is the end, stop and backtrack the path.
  • Check all adjacent nodes (you can choose the order—e.g., up first, then right, etc.—this affects which path it takes):
    • Skip walls.
    • If the node isn't visited:
      • Mark it as visited, set its parent to the current node, push it onto the stack, and immediately start exploring this new node (no waiting for other adjacent nodes).
  • If all adjacent nodes are either walls or visited, pop the current node from the stack (backtrack) and go back to the previous node to try other directions.
• Key quirk: DFS might get stuck exploring a long dead-end path before ever reaching the end, but it uses way less memory than BFS since it only tracks the current path, not all nodes at a given level.

LRTA* (Learning Real-Time A*) #

LRTA* is a heuristic, real-time algorithm—perfect if you don't want to precompute the entire map, or if the map might change dynamically. It learns as it moves, gradually getting better at finding the cheapest path. Here's its decision flow, tailored to your cost grid:

• Initialization:
  • For every node, set an initial heuristic value h(n): this is your best guess of the cost from n to the end. Manhattan distance (absolute difference in x + absolute difference in y) works great for grids—multiply by 1 (the lowest cost) to get a conservative estimate.
  • Set the current node to the start, and track g(current) (the actual cost to reach the start, which is 0).
• Loop until current node is the end:
  • Look at all reachable adjacent nodes (non-walls):
    • For each adjacent node n, calculate the estimated total cost to reach the end through n: g(current) + cost(current→n) + h(n). Here, cost(current→n) is 1 if n is land, 2 if it's grass.
  • Pick the adjacent node with the smallest estimated total cost—this is your next move.
  • Update the heuristic for the current node: Set h(current) to the minimum value of cost(current→n) + h(n) across all reachable n. This is LRTA* learning: it's updating the guess for how much it costs to get from the current node to the end, based on what it just saw.
  • Move to the selected adjacent node, update g(new_node) to g(current) + cost(current→new_node).
• Key quirk: The first time you run LRTA*, it might take a non-optimal path, but every subsequent run (or every time it revisits nodes) will get closer to the cheapest path. It doesn't need to pre-explore the entire map, which makes it great for real-time scenarios.

内容的提问来源于stack exchange，提问作者johN