1810. Minimum Path Cost in a Hidden Grid
Problem Description
In this problem, you are tasked with finding the minimum total cost for a robot to move from its starting cell to a target cell in an unknown and potentially partially blocked grid. The grid is represented by a two-dimensional array with m rows and n columns. Each cell in the grid has a cost associated with moving to it, except for the starting cell which does not incur a cost before the robot moves.
The grid is hidden and inaccessible directly. The only interaction with the grid is allowed through a provided GridMaster class, which contains methods that let you check if the robot can move in a certain direction, move the robot in a direction (returning the cost of that cell), and check if the current cell is the target cell.
Here are the key points:
- You don't know the size of the grid or the location of the starting and target cells.
- The robot should move to the target cell with the minimum total cost.
- You may only interact with the grid using the provided
GridMasterclass functions. - You must determine if there is a path to the target, and if so, calculate its cost. If no path exists, return
-1.
Intuition
To solve this problem, we need a two-pronged approach. First, we must map out the grid as much as we can to identify the target cell and the costs of moving through empty cells. Second, once we have mapped the grid, we need to find the shortest path from the start cell to the target cell in terms of cost, not distance.
Since we can only interact with the grid through the GridMaster, we can use depth-first search (DFS) to explore the grid. We need to keep track of the cells we have visited to avoid infinite loops and excessive querying of the GridMaster. Essentially, the process would involve these steps:
- Start from the initial cell (where the robot is), mark it as visited, and explore all four directions (
U,D,L,R- up, down, left, right). - Use the
canMovefunction to check if the robot can move in a specific direction. If it can, use themovefunction to move the robot, mark the new cell as visited, record the cost, and recursively continue the process from the new cell. - If the
isTargetfunction returnstrue, we've found the target cell, so we record its location. - After mapping out the accessible grid (or as far as we can go without hitting blocked cells or grid boundaries), we then apply a shortest-path algorithm, like Dijkstra’s algorithm, to find the path with the minimal cost from the start cell to the target cell. This part of the problem is where we use a priority queue to explore the least costly paths first.
The grid is large, so it makes sense to offset the initial cell to the middle of our grid representation (e.g., if we use a 2D array of size 200x200, we start at position (100, 100)). This choice is because the grid size is unknown, and we need room to explore in all directions. In the shortest-path stage, we use a priority queue to ensure we're always exploring the path with the minimal cost so far, making the process efficient.
To sum up, the intuition is to map out the grid using DFS and GridMaster and then use Dijkstra's algorithm to calculate the minimum total cost to reach the target cell if possible. If the target cell was not located during the mapping phase, or if there is no valid path, we return -1 as specified by the problem.
Solution Approach
The solution consists of two primary components – mapping the grid and then finding the shortest path to the target.
Mapping the Grid
To map out the grid, we initialize a two-dimensional list g of size N x N, with N being large enough to ensure that the robot can explore the grid in all directions from the initial position, which we offset to the center ((100, 100) in this case). Initially, all values in g are set to -1, which denotes unexplored cells.
We define a recursive function dfs(i, j) that accepts the current coordinates (i, j) of the robot in the grid. For each direction (U, D, L, R), the function performs the following steps:
- It checks if the robot can move in the current direction using
master.canMove(dir). - If the move is possible, it moves the robot with
master.move(dir), records the cost of the new cell ing, and callsdfs(x, y)recursively with the updated coordinates. - If the
master.isTarget()returnstrue, the coordinates of the target cell are stored in thetargetvariable.
To prevent moving outside the initialized grid, we check that the new coordinates are within bounds (0 <= x < N and 0 <= y < N) and that the cell has not been visited before (g[x][y] == -1).
After exploring in one direction and completing the depth-first search, we move the robot back to its previous position using master.move(ndir) to backtrack, where ndir is the opposite direction.
Finding the Shortest Path
Once the grid is mapped, we proceed to the shortest-path part of the algorithm. We use Dijkstra's algorithm for this purpose:
- We initialize a priority queue
qand add the start cell(cost, i, j)with the initial cost set to0. - We also initialize a
distlist to keep track of the minimum cost required to reach each cell, setting the initial cell's cost to0and the rest to infinity (INF). - While the priority queue
qis not empty, extract the cell with the minimum cost (w, i, j). - If
(i, j)is the target cell, return the minimum costw. - Otherwise, for each adjacent cell
(x, y), calculate the potential new costw + g[x][y]. If this cost is less thandist[x][y], updatedist[x][y]with the new cost and push the updated cost and coordinates back into the priority queue.
The use of the priority queue ensures that the cell with the currently known lowest cost is always processed next. This prioritization speeds up the search for the shortest path, as paths with a high cost are not processed until necessary.
If the target cell is not found during the initial DFS exploration, or if there is no path from the starting cell to the target cell, the function returns -1. Otherwise, the function returns the cost of the cheapest path.
The key algorithms, data structures, and patterns used in the solution include:
- Depth-First Search (DFS): To explore the grid and map out the accessible cells.
- Dijkstra's Algorithm: To find the shortest path to the target in terms of cost.
- Priority Queue (Min Heap): To efficiently retrieve the next cell to explore based on the lowest known cost (used in conjunction with Dijkstra's algorithm).
- Backtracking: To ensure that after exploring a path, the robot returns to its previous position.
This approach ensures a thorough exploration of the grid and an efficient calculation of the minimum total cost to reach the target cell.
Example Walkthrough
Let's use a small example to illustrate the solution approach. Assume we have a 5x5 grid with the following conditions (actual grid values are not known to the robot, they are only shown here for illustration):
1| S | 1 | - | - | - | S: Start (0,0) 2| - | 1 | 1 | - | T | T: Target 3| - | - | 1 | 1 | 1 | 1: Cost to move 4| - | 1 | - | - | - | -: Blocked cell 5| - | 1 | 1 | 1 | - |
The robot starts at the top-left corner S and the target is at position T. The costs to move to any cell are all 1s and the blocked cells are represented by -.
Mapping the Grid
- The robot starts DFS from
(0,0), marking this cell visited in ourgarray, initially(100, 100). - The robot can move right to
(0,1)and marks it in the grid with a cost of1. - It cannot move up or down as it is the top edge of the grid. The robot attempts to move left but the cell is blocked, as per the assumed grid structure.
- Now, the robot tries to move right from
(0,1)to(0,2)but it's blocked so it will backtrack to(0,1). - Next, the robot moves down from
(0,1)to(1,1)and marks the cell with a cost of1. - From
(1,1), the robot continues to move right, marking cost1at(1,2), and moves down to(2,2), then to(3,2), and finally to(4,2), marking each with cost1. - The robot now can move right from
(4,2)to(4,3)and then up to(1,3), marking all those cells with the cost1. At(1,3), it moves right and discovers the targetTat(1,4).
By the end of the DFS, the robot has the following map:
1| 0 | 1 | X | X | X | 2| X | 1 | 1 | 1 | T | 3| X | X | 1 | 1 | 1 | 4| X | 1 | X | X | X | 5| X | 1 | 1 | 1 | X |
0 denotes the start cell, X is unvisited or blocked, T is the target, and 1 denotes visited cells with associated costs.
Finding the Shortest Path
- We initialize the priority queue with the start cell
(0, 100, 100)cost and position. - The priority queue pops the start cell, and we check its neighbors. Since they are all either
Xor have a cost of1, we add those with cost1to the priority queue with their cumulative costs. - The process continues, with the priority queue ensuring cells with the lowest cumulative cost are explored first.
- Once the target cell's coordinates are popped from the queue, the algorithm stops and the cost associated with the target cell is the minimum cost.
Following the described approach with the specified grid, the robot would find a path with a total minimum cost of 3 to move from S to T through (1, 1), (1, 2), (1, 3), and reaching the target at (1, 4).
Python Solution
1from heapq import heappop, heappush
2
3class Solution(object):
4 def findShortestPath(self, master: 'GridMaster') -> int:
5 # DFS to build the grid and locate the target.
6 def dfs(i, j):
7 nonlocal target
8 # Check if the current position is the target.
9 if master.isTarget():
10 target = (i, j)
11 # Explore all possible directions.
12 for direction, (delta_i, delta_j, opposite) in directions.items():
13 new_i, new_j = i + delta_i, j + delta_j
14 if (0 <= new_i < N) and (0 <= new_j < N) and master.canMove(direction) and grid[new_i][new_j] == -1:
15 grid[new_i][new_j] = master.move(direction)
16 dfs(new_i, new_j)
17 master.move(opposite)
18
19 # Initialize variables.
20 target = (-1, -1)
21 N = 200
22 INF = float('inf')
23 grid = [[-1] * N for _ in range(N)]
24 directions = {
25 'U': (-1, 0, 'D'),
26 'D': (1, 0, 'U'),
27 'L': (0, -1, 'R'),
28 'R': (0, 1, 'L'),
29 }
30
31 # Start DFS from the center of the grid.
32 dfs(100, 100)
33
34 # If the target is not found, return -1.
35 if target == (-1, -1):
36 return -1
37
38 # Dijkstra's algorithm to find the shortest path to the target.
39 priority_queue = [(0, 100, 100)] # Weight, i, j
40 distances = [[INF] * N for _ in range(N)]
41 distances[100][100] = 0 # Start with distance 0 from the starting point.
42
43 # Process the nodes in the queue.
44 while priority_queue:
45 weight, i, j = heappop(priority_queue)
46 # Check if we've reached the target.
47 if (i, j) == target:
48 return weight
49 # Explore all possible directions.
50 for delta_i, delta_j, _ in directions.values():
51 new_i, new_j = i + delta_i, j + delta_j
52 # Update the distance if a shorter path is found.
53 if (0 <= new_i < N) and (0 <= new_j < N) and grid[new_i][new_j] != -1 and distances[new_i][new_j] > weight + grid[new_i][new_j]:
54 distances[new_i][new_j] = weight + grid[new_i][new_j]
55 heappush(priority_queue, (distances[new_i][new_j], new_i, new_j))
56
57 # If the target was not reached, return 0.
58 return 0
59
Java Solution
1import java.util.Arrays;
2import java.util.PriorityQueue;
3import java.util.Comparator;
4
5class Solution {
6 // Direction mappings
7 private static final char[] DIRS = {'U', 'R', 'D', 'L'};
8 private static final char[] OPPOSITE_DIRS = {'D', 'L', 'U', 'R'};
9 private static final int[] DIRS_DELTA = {-1, 0, 1, 0, -1};
10
11 // Constants for grid size and infinity equivalent for distances
12 private static final int GRID_SIZE = 200;
13 private static final int INF = Integer.MAX_VALUE / 2; // To avoid overflow
14
15 // Grid representation and distances
16 private static int[][] grid = new int[GRID_SIZE][GRID_SIZE];
17 private static int[][] distances = new int[GRID_SIZE][GRID_SIZE];
18
19 // Variable to store target position
20 private int[] targetPosition;
21
22 /**
23 * Method to find the shortest path to the target using Dijkstra's algorithm.
24 * @param master The GridMaster instance to interact with the grid.
25 * @return The shortest distance to the target or -1 if the target is unreachable.
26 */
27 public int findShortestPath(GridMaster master) {
28 targetPosition = new int[] {-1, -1};
29
30 // Initialize grid values to -1 and distances to infinity (INF)
31 for (int i = 0; i < GRID_SIZE; ++i) {
32 Arrays.fill(grid[i], -1);
33 Arrays.fill(distances[i], INF);
34 }
35
36 // Fill in the available paths and target position with a DFS
37 dfs(100, 100, master);
38
39 // If target is not found, return -1
40 if (targetPosition[0] == -1) {
41 return -1;
42 }
43
44 // Implement Dijkstra's algorithm to find the shortest path
45 PriorityQueue<int[]> priorityQueue = new PriorityQueue<>(Comparator.comparingInt(a -> a[0]));
46
47 // Starting point
48 priorityQueue.offer(new int[] {0, 100, 100});
49 distances[100][100] = 0;
50
51 // Process nodes
52 while (!priorityQueue.isEmpty()) {
53 int[] point = priorityQueue.poll();
54 int weight = point[0], row = point[1], col = point[2];
55
56 // If target position is reached, return the path weight
57 if (row == targetPosition[0] && col == targetPosition[1]) {
58 return weight;
59 }
60
61 // Explore all possible directions from the current node
62 for (int k = 0; k < 4; ++k) {
63 int newRow = row + DIRS_DELTA[k], newCol = col + DIRS_DELTA[k + 1];
64
65 // Check bounds and if the new cell is accessible
66 if (newRow >= 0 && newRow < GRID_SIZE && newCol >= 0 && newCol < GRID_SIZE
67 && grid[newRow][newCol] != -1 && distances[newRow][newCol] > weight + grid[newRow][newCol]) {
68 distances[newRow][newCol] = weight + grid[newRow][newCol];
69 priorityQueue.offer(new int[] {distances[newRow][newCol], newRow, newCol});
70 }
71 }
72 }
73 // The target was not reachable
74 return -1;
75 }
76
77 /**
78 * Depth-first search to map out the reachable cells in the grid.
79 * @param row The starting row coordinate.
80 * @param col The starting column coordinate.
81 * @param master The GridMaster instance to interact with the grid.
82 */
83 private void dfs(int row, int col, GridMaster master) {
84 // Check if the current position is the target
85 if (master.isTarget()) {
86 targetPosition = new int[] {row, col};
87 }
88
89 // Explore all possible directions
90 for (int k = 0; k < 4; ++k) {
91 char moveDirection = DIRS[k], oppositeDirection = OPPOSITE_DIRS[k];
92 int newRow = row + DIRS_DELTA[k], newCol = col + DIRS_DELTA[k + 1];
93
94 // Validate new positions and ensure we haven't visited it before
95 if (newRow >= 0 && newRow < GRID_SIZE && newCol >= 0 && newCol < GRID_SIZE &&
96 master.canMove(moveDirection) && grid[newRow][newCol] == -1) {
97 grid[newRow][newCol] = master.move(moveDirection);
98 dfs(newRow, newCol, master);
99 master.move(oppositeDirection); // Move back
100 }
101 }
102 }
103}
104
C++ Solution
1#include <vector>
2#include <queue>
3#include <climits>
4#include <algorithm>
5
6class Solution {
7private:
8 // Direction mappings
9 const char DIRS[4] = {'U', 'R', 'D', 'L'};
10 const char OPPOSITE_DIRS[4] = {'D', 'L', 'U', 'R'};
11 const int DIRS_DELTA[5] = {-1, 0, 1, 0, -1};
12
13 // Constants for grid size and infinity equivalent for distances
14 static constexpr int GRID_SIZE = 200;
15 static constexpr int INF = INT_MAX / 2; // To avoid overflow
16
17 // Grid representation and distances
18 int grid[GRID_SIZE][GRID_SIZE];
19 int distances[GRID_SIZE][GRID_SIZE];
20
21 // Variable to store target position
22 std::pair<int, int> targetPosition;
23
24public:
25 Solution() {
26 targetPosition = {-1, -1};
27 std::fill(&grid[0][0], &grid[0][0] + GRID_SIZE * GRID_SIZE, -1);
28 std::fill(&distances[0][0], &distances[0][0] + GRID_SIZE * GRID_SIZE, INF);
29 }
30
31 /**
32 * Method to find the shortest path to the target using Dijkstra's algorithm.
33 * @param master The GridMaster instance to interact with the grid.
34 * @return The shortest distance to the target or -1 if the target is unreachable.
35 */
36 int findShortestPath(GridMaster &master) {
37 // Fill in the available paths and target position with a DFS
38 dfs(100, 100, master);
39
40 // If target is not found, return -1
41 if (targetPosition.first == -1) {
42 return -1;
43 }
44
45 // Implement Dijkstra's algorithm to find the shortest path
46 auto cmp = [](const std::vector<int> &a, const std::vector<int> &b) {
47 return a[0] > b[0];
48 };
49 std::priority_queue<std::vector<int>, std::vector<std::vector<int>>, decltype(cmp)> priorityQueue(cmp);
50
51 // Starting point
52 priorityQueue.push({0, 100, 100});
53 distances[100][100] = 0;
54
55 // Process nodes
56 while (!priorityQueue.empty()) {
57 std::vector<int> point = priorityQueue.top();
58 priorityQueue.pop();
59 int weight = point[0], row = point[1], col = point[2];
60
61 // If target position is reached, return the path weight
62 if (row == targetPosition.first && col == targetPosition.second) {
63 return weight;
64 }
65
66 // Explore all possible directions from the current node
67 for (int k = 0; k < 4; ++k) {
68 int newRow = row + DIRS_DELTA[k], newCol = col + DIRS_DELTA[k + 1];
69
70 // Check bounds and if the new cell is accessible
71 if (newRow >= 0 && newRow < GRID_SIZE && newCol >= 0 && newCol < GRID_SIZE
72 && grid[newRow][newCol] != -1 && distances[newRow][newCol] > weight + grid[newRow][newCol]) {
73 distances[newRow][newCol] = weight + grid[newRow][newCol];
74 priorityQueue.push({distances[newRow][newCol], newRow, newCol});
75 }
76 }
77 }
78
79 // The target was not reachable
80 return -1;
81 }
82
83private:
84 /**
85 * Depth-first search to map out the reachable cells in the grid.
86 * @param row The starting row coordinate.
87 * @param col The starting column coordinate.
88 * @param master The GridMaster instance to interact with the grid.
89 */
90 void dfs(int row, int col, GridMaster &master) {
91 // Check if the current position is the target
92 if (master.isTarget()) {
93 targetPosition = {row, col};
94 }
95
96 // Explore all possible directions
97 for (int k = 0; k < 4; ++k) {
98 char moveDirection = DIRS[k], oppositeDirection = OPPOSITE_DIRS[k];
99 int newRow = row + DIRS_DELTA[k], newCol = col + DIRS_DELTA[k + 1];
100
101 // Validate new positions and ensure we haven't visited it before
102 if (newRow >= 0 && newRow < GRID_SIZE && newCol >= 0 && newCol < GRID_SIZE &&
103 master.canMove(moveDirection) && grid[newRow][newCol] == -1) {
104 grid[newRow][newCol] = master.move(moveDirection);
105 dfs(newRow, newCol, master);
106 master.move(oppositeDirection); // Move back
107 }
108 }
109 }
110};
111
112/**
113 * The GridMaster class will be provided by the problem in LeetCode and is used here
114 * just for the code to be syntactically correct. This provides the methods used to query
115 * the state of the grid and move within it.
116 */
117class GridMaster {
118public:
119 bool canMove(char direction);
120 int move(char direction);
121 bool isTarget();
122};
123
Typescript Solution
1// Utility types for direction mappings
2const DIRS: char[] = ['U', 'R', 'D', 'L'];
3const OPPOSITE_DIRS: char[] = ['D', 'L', 'U', 'R'];
4const DIRS_DELTA: number[] = [-1, 0, 1, 0, -1];
5
6// Constants for grid size and infinity equivalent for distances
7const GRID_SIZE: number = 200;
8const INF: number = Number.MAX_SAFE_INTEGER / 2; // To prevent overflow
9
10// Grid representation and distances
11let grid: number[][] = Array.from({ length: GRID_SIZE }, () => new Array(GRID_SIZE).fill(-1));
12let distances: number[][] = Array.from({ length: GRID_SIZE }, () => new Array(GRID_SIZE).fill(INF));
13
14// Variable to store the target position
15let targetPosition: number[] = [-1, -1];
16
17// GridMaster interface as a placeholder for actual GridMaster interactions
18interface GridMaster {
19 canMove(direction: char): boolean;
20 move(direction: char): number;
21 isTarget(): boolean;
22}
23
24// Method to find the shortest path to target using Dijkstra's algorithm
25function findShortestPath(master: GridMaster): number {
26 targetPosition = [-1, -1];
27
28 // Fill in the available paths and target position with a DFS
29 dfs(100, 100, master);
30
31 // If target is not found, return -1
32 if (targetPosition[0] === -1) {
33 return -1;
34 }
35
36 // PriorityQueue setup with a comparator for shortest distance at the top
37 const priorityQueue: [number, number, number][] = [];
38
39 // Starting point
40 enqueue(priorityQueue, 0, 100, 100);
41 distances[100][100] = 0;
42
43 // Process nodes
44 while (priorityQueue.length > 0) {
45 const [weight, row, col] = dequeue(priorityQueue);
46
47 // If target position is reached, return the path weight
48 if (row === targetPosition[0] && col === targetPosition[1]) {
49 return weight;
50 }
51
52 // Explore all possible directions from the current node
53 for (let k = 0; k < 4; ++k) {
54 const newRow = row + DIRS_DELTA[k];
55 const newCol = col + DIRS_DELTA[k + 1];
56
57 // Check bounds and if the new cell is accessible
58 if (newRow >= 0 && newRow < GRID_SIZE && newCol >= 0 && newCol < GRID_SIZE
59 && grid[newRow][newCol] !== -1 && distances[newRow][newCol] > weight + grid[newRow][newCol]) {
60 distances[newRow][newCol] = weight + grid[newRow][newCol];
61 enqueue(priorityQueue, distances[newRow][newCol], newRow, newCol);
62 }
63 }
64 }
65 // The target was not reachable
66 return -1;
67}
68
69function dfs(row: number, col: number, master: GridMaster): void {
70 // Check if the current position is the target
71 if (master.isTarget()) {
72 targetPosition = [row, col];
73 }
74
75 // Explore all possible directions
76 for (let k = 0; k < 4; k++) {
77 const moveDirection = DIRS[k];
78 const oppositeDirection = OPPOSITE_DIRS[k];
79 const newRow = row + DIRS_DELTA[k];
80 const newCol = col + DIRS_DELTA[k + 1];
81
82 // Validate new positions and ensure we haven't visited it before
83 if (newRow >= 0 && newRow < GRID_SIZE && newCol >= 0 && newCol < GRID_SIZE &&
84 master.canMove(moveDirection) && grid[newRow][newCol] === -1) {
85 grid[newRow][newCol] = master.move(moveDirection);
86 dfs(newRow, newCol, master);
87 master.move(oppositeDirection); // Move back
88 }
89 }
90}
91
92function enqueue(queue: [number, number, number][], weight: number, row: number, col: number): void {
93 queue.push([weight, row, col]);
94 queue.sort((a, b) => a[0] - b[0]);
95}
96
97function dequeue(queue: [number, number, number][]): [number, number, number] {
98 return queue.shift() as [number, number, number];
99}
100
Time and Space Complexity
Time Complexity
The time complexity of the solution is determined by the depth-first search (DFS) used to explore the grid and the subsequent use of Dijkstra's algorithm to find the shortest path to the target.
-
DFS: The DFS function explores potentially every cell once. In the worst case, we could think of this as a recursive call for each accessible cell in the grid. The grid size is bounded by
N, so DFS could potentially beO(N^2)in the worst case. -
Dijkstra's Algorithm: After DFS execution, the code uses Dijkstra's algorithm with a priority queue (min-heap) for finding the shortest path in the weighted grid. Each cell, if accessible, can be pushed and popped from the priority queue at most once. The
heappushandheappopoperations work inO(log V), whereVis the number of vertices in the queue. In this case,Vis at mostN^2, so each heap operation isO(log N^2) = O(2 log N) = O(log N). Since we may process every cell once, the time complexity for this part isO(N^2 log N).
Overall, the time complexity combines the complexities of DFS and Dijkstra's algorithm, which yields O(N^2 + N^2 log N). However, the dominant term here is O(N^2 log N), so the time complexity of the entire function is O(N^2 log N).
Space Complexity
The space complexity includes the space used for the recursion call stack in DFS, the grid g, the priority queue in Dijkstra's algorithm, and the dist array.
-
DFS Stack Space: In the worst case (a completely open grid), the depth of the recursion stack can be
O(N^2)since we might need to traverse the entire grid. -
Grid
g: The gridgis of sizeNxN, and thus requiresO(N^2)space. -
Priority Queue: The min-heap used in Dijkstra's algorithm would at most contain all the cells at one time, so it requires
O(N^2)space in the worst case. -
Distance Array
dist: Similarly, this is also a 2DNxNarray, which requiresO(N^2)space.
Adding all these up, we get a space complexity of O(N^2) as all other factors are also O(N^2).
In conclusion, the provided solution has a time complexity of O(N^2 log N) and a space complexity of O(N^2).
Got a question? Ask the Teaching Assistant anything you don't understand.
Still not clear? Ask in the Forum, Discord or Submit the part you don't understand to our editors.