문제
A* is the classic pathfinding algorithm of games and robotics. It explores cells in order of f = g + h, where g is the cost paid so far and h is a heuristic estimate of the cost still to go.
The grid holds a cost for every cell: entering a cell costs its value, and 0 means a wall that cannot be entered. You start on start (its cost is not paid) and may move up, down, left or right.
Use A* with the Manhattan distance |row - goal_row| + |col - goal_col| as the heuristic. Every open cell costs at least 1, so this heuristic never overestimates and A* returns an optimal answer.
Implement a_star_cost(grid, start, goal).
Input
grid: A matrix of cell costs (0 = wall).
start, goal: [row, col] positions.
Output
Return the minimum total cost to reach goal, or -1 if it cannot be reached (including when start or goal is a wall).
제약 조건
1 ≤ rows, cols ≤ 100
0 ≤ each cost ≤ 9
예시
입력:
goal = [2, 2]
grid = [
[1, 1, 1],
[1, 0, 1],
[1, 1, 1]
]
start = [0, 0]
예상 출력:
expected = 4입력:
std::vector<long long> goal = {2, 2};
std::vector<std::vector<long long>> grid = {
{1, 1, 1},
{1, 0, 1},
{1, 1, 1}
};
std::vector<long long> start = {0, 0};
예상 출력:
long long expected = 4;입력:
long[] goal = new long[] {2, 2};
long[][] grid = new long[][] {
new long[] {1, 1, 1},
new long[] {1, 0, 1},
new long[] {1, 1, 1}
};
long[] start = new long[] {0, 0};
예상 출력:
long expected = 4;설명
The centre is a wall. Going right twice and down twice (or down then right) enters 4 cells of cost 1: total 4.
입력:
goal = [0, 2]
grid = [
[1, 9, 1],
[1, 9, 1],
[1, 1, 1]
]
start = [0, 0]
예상 출력:
expected = 6입력:
std::vector<long long> goal = {0, 2};
std::vector<std::vector<long long>> grid = {
{1, 9, 1},
{1, 9, 1},
{1, 1, 1}
};
std::vector<long long> start = {0, 0};
예상 출력:
long long expected = 6;입력:
long[] goal = new long[] {0, 2};
long[][] grid = new long[][] {
new long[] {1, 9, 1},
new long[] {1, 9, 1},
new long[] {1, 1, 1}
};
long[] start = new long[] {0, 0};
예상 출력:
long expected = 6;설명
Crossing the 9 directly costs 9 + 1 = 10. Going down, along the bottom row and back up enters six cells of cost 1: total 6.
접근 방법
먼저 스스로 풀어 본 뒤 힌트를 하나씩 열어 보세요.
힌트 1
Use a priority queue ordered by traveled cost plus Manhattan distance to the goal.
힌트 2
Track the best cost found so far for each cell, and only expand a cell when this path improves it.
힌트 3
Treat walls and blocked start or goal cells as impossible, and remember entering the start cell is free.
예상 복잡도
O(rows * cols log(rows * cols)) time, O(rows * cols) extra space
참고 자료
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