This approach involves using Breadth-First Search (BFS) to precompute the minimum Manhattan distance from each cell to any thief in the grid, and then using this information to find the maximum safeness factor for reaching the bottom-right corner.
Time Complexity: O(n2 log n), where n is the grid size, due to the Dijkstra-like process.
Space Complexity: O(n2) for storing distances and safeness factors.
1#include <stdio.h>
2#include <stdlib.h>
3#include <limits.h>
4#include <stdbool.h>
5#include <string.h>
6
7#define MAX 400
8
9int grid[MAX][MAX];
10int n;
11
12int direction[4][2] = {{0,1}, {1,0}, {0,-1}, {-1,0}};
13
14int maxSafenessFactor(int grid[][MAX], int n) {
15 int distance[MAX][MAX];
16 bool visited[MAX][MAX];
17 memset(distance, INT_MAX, sizeof(distance));
18 memset(visited, false, sizeof(visited));
19
20 // BFS for shortest distance from any thief
21 int qsize = n * n;
22 int queue[qsize][2];
23 int front = 0, rear = 0;
24
25 for(int r = 0; r < n; ++r){
26 for(int c = 0; c < n; ++c){
27 if(grid[r][c] == 1) {
28 queue[rear][0] = r;
29 queue[rear][1] = c;
30 rear++;
31 distance[r][c] = 0;
32 visited[r][c] = true;
33 }
34 }
35 }
36
37 while (front < rear) {
38 int r = queue[front][0], c = queue[front][1];
39 front++;
40 for (int d = 0; d < 4; ++d) {
41 int nr = r + direction[d][0], nc = c + direction[d][1];
42 if (nr >= 0 && nr < n && nc >= 0 && nc < n && !visited[nr][nc]) {
43 visited[nr][nc] = true;
44 distance[nr][nc] = distance[r][c] + 1;
45 queue[rear][0] = nr;
46 queue[rear][1] = nc;
47 rear++;
48 }
49 }
50 }
51
52 // Use Dijkstra's rather than simple BFS from (0, 0) due to distance
53 int maxSafeness[MAX][MAX];
54 int pq[qsize][3];
55 int pqSize = 0;
56
57 bool used[MAX][MAX];
58 memset(used, false, sizeof(used));
59
60 // Initialize dijkstra's variables
61 memset(maxSafeness, -1, sizeof(maxSafeness));
62 maxSafeness[0][0] = distance[0][0];
63 pq[pqSize][0] = 0, pq[pqSize][1] = 0, pq[pqSize][2] = distance[0][0];
64 pqSize++;
65
66 while (pqSize > 0) {
67 // Extract the node with maximum safeness factor
68 int maxDistIndex = -1;
69 for (int i = 0; i < pqSize; i++) {
70 if (maxDistIndex == -1 || pq[i][2] > pq[maxDistIndex][2]) {
71 maxDistIndex = i;
72 }
73 }
74
75 int curR = pq[maxDistIndex][0], curC = pq[maxDistIndex][1];
76 int curD = pq[maxDistIndex][2];
77
78 // Remove from queue
79 for (int i = maxDistIndex; i < pqSize - 1; i++) {
80 pq[i][0] = pq[i+1][0];
81 pq[i][1] = pq[i+1][1];
82 pq[i][2] = pq[i+1][2];
83 }
84 pqSize--;
85
86 if (used[curR][curC]) continue;
87 used[curR][curC] = true;
88
89 for (int d = 0; d < 4; ++d) {
90 int nr = curR + direction[d][0], nc = curC + direction[d][1];
91 if (nr >= 0 && nr < n && nc >= 0 && nc < n) {
92 if (min(curD, distance[nr][nc]) > maxSafeness[nr][nc]) {
93 maxSafeness[nr][nc] = min(curD, distance[nr][nc]);
94 pq[pqSize][0] = nr;
95 pq[pqSize][1] = nc;
96 pq[pqSize][2] = maxSafeness[nr][nc];
97 pqSize++;
98 }
99 }
100 }
101 }
102
103 return maxSafeness[n-1][n-1];
104}
105
106int main() {
107 int gridExample[MAX][MAX] = {{0, 0, 1}, {0, 0, 0}, {0, 0, 0}};
108 int n = 3;
109 printf("%d\n", maxSafenessFactor(gridExample, n));
110 return 0;
111}
112This C solution first performs a BFS from all thief positions to calculate the distance from each cell to the nearest thief. Then, it uses Dijkstra-like processing to find the maximum safeness from the top-left corner to the bottom-right corner.
This approach considers processing from both the starting and ending points in a bidirectional BFS style, potentially meeting in the middle for optimized distance calculations and safeness factor determination.
Time Complexity: O(n2) due to simultaneous BFS processing.
Space Complexity: O(n2) as distances from both ends are computed.
1#include <stdio.h>
2#include <stdlib.h>
3#include <limits.h>
4#include <stdbool.h>
5#include <string.h>
6
7#define MAX 400
8
9int grid[MAX][MAX];
10int n;
11
12int direction[4][2] = {{0,1}, {1,0}, {0,-1}, {-1,0}};
13
14int bidirectionalSafenessFactor(int grid[][MAX], int n) {
15 // Distance from start and end
16 int distanceStart[MAX][MAX];
17 int distanceEnd[MAX][MAX];
18
19 memset(distanceStart, INT_MAX, sizeof(distanceStart));
20 memset(distanceEnd, INT_MAX, sizeof(distanceEnd));
21
22 // BFS start queue
23 int queueStart[MAX*MAX][2];
24 int frontStart = 0, rearStart = 0;
25
26 queueStart[rearStart][0] = 0;
27 queueStart[rearStart][1] = 0;
28 rearStart++;
29 distanceStart[0][0] = 0;
30
31 // BFS end queue
32 int queueEnd[MAX*MAX][2];
33 int frontEnd = 0, rearEnd = 0;
34
35 queueEnd[rearEnd][0] = n-1;
36 queueEnd[rearEnd][1] = n-1;
37 rearEnd++;
38 distanceEnd[n-1][n-1] = 0;
39
40 while (frontStart < rearStart && frontEnd < rearEnd) {
41 // Expand from start
42 int r = queueStart[frontStart][0], c = queueStart[frontStart][1];
43 frontStart++;
44
45 for (int d = 0; d < 4; ++d) {
46 int nr = r + direction[d][0], nc = c + direction[d][1];
47 if (nr >= 0 && nr < n && nc >= 0 && nc < n && distanceStart[nr][nc] == INT_MAX) {
48 distanceStart[nr][nc] = distanceStart[r][c] + 1;
49 queueStart[rearStart][0] = nr;
50 queueStart[rearStart][1] = nc;
51 rearStart++;
52 }
53 }
54
55 // Expand from end
56 r = queueEnd[frontEnd][0], c = queueEnd[frontEnd][1];
57 frontEnd++;
58
59 for (int d = 0; d < 4; ++d) {
60 int nr = r + direction[d][0], nc = c + direction[d][1];
61 if (nr >= 0 && nr < n && nc >= 0 && nc < n && distanceEnd[nr][nc] == INT_MAX) {
62 distanceEnd[nr][nc] = distanceEnd[r][c] + 1;
63 queueEnd[rearEnd][0] = nr;
64 queueEnd[rearEnd][1] = nc;
65 rearEnd++;
66 }
67 }
68
69 // Check overlap
70 for (int i = 0; i < n; ++i) {
71 for (int j = 0; j < n; ++j) {
72 if (distanceStart[i][j] != INT_MAX && distanceEnd[i][j] != INT_MAX) {
73 return distanceStart[i][j] + distanceEnd[i][j];
74 }
75 }
76 }
77 }
78
79 return -1;
80}
81
82int main() {
83 int gridExample[MAX][MAX] = {{0, 0, 1}, {0, 0, 0}, {0, 0, 0}};
84 int n = 3;
85 printf("%d\n", bidirectionalSafenessFactor(gridExample, n));
86 return 0;
87}
88The C solution processes the grid using bidirectional BFS starting from both the top-left corner and the bottom-right corner simultaneously, aiming to explore overlapping paths and maximize their safeness factor.