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.
1using System;
2using System.Collections.Generic;
3
4public class SafestPathFinder {
5 private static int[][] directions = new int[4][] {
6 new int[] { 0, 1 },
7 new int[] { 1, 0 },
8 new int[] { 0, -1 },
9 new int[] { -1, 0 }
10 };
11
12 public int MaxSafenessFactor(int[][] grid) {
13 int n = grid.Length;
14 int[][] distance = new int[n][];
15 for (int i = 0; i < n; i++)
16 distance[i] = new int[n];
17
18 for (int i = 0; i < n; i++)
19 for (int j = 0; j < n; j++)
20 distance[i][j] = int.MaxValue;
21
22 Queue<(int, int)> queue = new Queue<(int, int)>();
23
24 for (int r = 0; r < n; r++) {
25 for (int c = 0; c < n; c++) {
26 if (grid[r][c] == 1) {
27 queue.Enqueue((r, c));
28 distance[r][c] = 0;
29 }
30 }
31 }
32
33 while (queue.Count > 0) {
34 var (r, c) = queue.Dequeue();
35 foreach (var dir in directions) {
36 int nr = r + dir[0];
37 int nc = c + dir[1];
38 if (nr >= 0 && nr < n && nc >= 0 && nc < n && distance[nr][nc] == int.MaxValue) {
39 distance[nr][nc] = distance[r][c] + 1;
40 queue.Enqueue((nr, nc));
41 }
42 }
43 }
44
45 PriorityQueue<(int, int, int)> pq = new();
46 pq.Enqueue((0, 0, distance[0][0]));
47 bool[,] visited = new bool[n, n];
48
49 while (pq.Count > 0) {
50 var (r, c, safeness) = pq.Dequeue();
51 if (r == n - 1 && c == n - 1) return safeness;
52 if (visited[r, c]) continue;
53 visited[r, c] = true;
54
55 foreach (var dir in directions) {
56 int nr = r + dir[0];
57 int nc = c + dir[1];
58 if (nr >= 0 && nr < n && nc >= 0 && nc < n && !visited[nr, nc]) {
59 int newSafeness = Math.Min(safeness, distance[nr][nc]);
60 pq.Enqueue((nr, nc, newSafeness));
61 }
62 }
63 }
64
65 return 0;
66 }
67
68 public static void Main() {
69 SafestPathFinder spf = new SafestPathFinder();
70 int[][] grid = new int[][] {
71 new int[] { 0, 0, 1 },
72 new int[] { 0, 0, 0 },
73 new int[] { 0, 0, 0 }
74 };
75 Console.WriteLine(spf.MaxSafenessFactor(grid)); // Output: 2
76 }
77}
78
79public class PriorityQueue<T> where T : IComparable<T> {
80 private List<T> data = new List<T>();
81
82 public int Count => data.Count;
83
84 public void Enqueue(T item) {
85 data.Add(item);
86 int ci = data.Count - 1;
87 while (ci > 0) {
88 int pi = (ci - 1) / 2;
89 if (data[ci].CompareTo(data[pi]) <= 0) break;
90 var tmp = data[ci];
91 data[ci] = data[pi];
92 data[pi] = tmp;
93 ci = pi;
94 }
95 }
96
97 public T Dequeue() {
98 var ret = data[0];
99 var li = data.Count - 1;
100 data[0] = data[li];
101 data.RemoveAt(li);
102 --li;
103 int pi = 0;
104 while (true) {
105 int ci = pi * 2 + 1;
106 if (ci > li) break;
107 int rc = ci + 1;
108 if (rc <= li && data[rc].CompareTo(data[ci]) > 0) ci = rc;
109 if (data[pi].CompareTo(data[ci]) >= 0) break;
110 var tmp = data[pi];
111 data[pi] = data[ci];
112 data[ci] = tmp;
113 pi = ci;
114 }
115 return ret;
116 }
117}
118The C# solution begins by performing BFS to calculate the distance from every cell to the nearest thief. After that, it implements a version of Dijkstra’s algorithm to efficiently find the path maximizing safeness from the start to the endpoint.
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.