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This approach leverages the concept of "Manhattan distance" coupled with the "parity check" to determine if the cell can be reached in exactly t steps.
The minimum number of steps required to reach from (sx, sy) to (fx, fy) in a grid is the maximum of the horizontal and vertical distances between these points, known as Chebyshev distance. If this minimal distance is greater than t, the function should return false instantly.
Moreover, if the difference between t and this minimum distance is even, the extra steps required can effectively be taken by oscillating on the grid or taking additional allowable steps. If not, reaching in t steps becomes impossible.
Time Complexity: O(1), as it involves simple arithmetic operations.
Space Complexity: O(1), as no extra space is used.
1def is_cell_reachable(sx, sy, fx, fy, t):
2 min_steps = max(abs(fx - sx), abs(fy - sy))
3 returnThe Python function computes the required minimum steps and parity check using simple built-in functions and returns True or False based on whether it can reach in exactly t steps.
This approach models the grid and implements a Breadth-First Search (BFS) to simulate traversing from the starting point to check if reaching precisely at t seconds is possible.
Instead of directly computing the distance, this method carries out a brute-force simulation of the movement, exploring all potential paths using a FIFO queue structure inherent to BFS algorithms. The goal is to maintain a count of steps and analyze pathways that may uniquely cover the grid in t steps.
Time Complexity: Highly complex, typically O(n^2).
Space Complexity: Also complex, with decisions based on array allocations with a theoretical upper limit.
1#include <stdbool.h>
2#include <stdlib.h>
3#include <stdio.h>
4
5int dir[8][2] = {{-1, 0}, {1, 0}, {0, -1}, {0, 1}, {-1, -1}, {-1, 1}, {1, -1}, {1, 1}};
6
7bool bfs(int sx, int sy, int fx, int fy, int t) {
8 // Assume this is initialized with a queue and relevant BFS logic.
9 // Left incomplete for basic simulation purposes and brevity.
10 return false;
11}
12
13bool isCellReachable(int sx, int sy, int fx, int fy, int t) {
14 return bfs(sx, sy, fx, fy, t);
15}This approach is conceptual as implementing full BFS is impractical for infinite grids or large bounds directly within C without optimized constraints.
The BFS is understood as a simulation iterating every possible step sequence up to t.