885. Spiral Matrix III

Problem Description

In this LeetCode problem, you are given the dimensions of a grid (rows x cols) and a starting cell within that grid, defined by its row (rStart) and column (cStart). Your task is to simulate a walk from the starting cell in a clockwise spiral pattern until you have visited every cell in the grid. This spiral pattern means you move east, then south, then west, and finally north, increasing the distance you move in a specific direction before turning by one cell each time you complete a full cycle (east, south, west, and north). As you perform this walk, even if you move outside the grid's boundaries, you keep the pattern, and you may return to the grid's boundary later. The goal is to visit all cells on the grid and return a list of their coordinates in the order they were visited.

Intuition

The problem can be solved by simulating the walk in a spiral pattern as described. The intuition behind this solution is to track movement direction and counter changes in direction while traversing the grid. We should keep in mind that it is necessary to handle cases where the walk takes us outside the grid boundary.

We initialize the solution by adding the starting position to our result set. If the grid is of size 1x1, the problem is already solved with this step. For larger grids, we have to begin walking in a spiral. We know that for a spiral walk, the number of steps we take in the east or west directions (horizontal steps) will always be one more than the number of steps we took in the previous vertical movement (north or south). Conversely, the number of steps we take in the north or south direction will be equal to the number of steps we will take in the next horizontal movement.

To implement this pattern, we maintain a variable k that represents how many steps we should move in the current direction. We start with k = 1, as the first move after the starting position is just one step to the right (east). We then increase k by 2 after a full cycle of directions (east > south > west > north) to account for the changing step count in the spiral path.

For each direction, we move k steps, updating our current position and conditionally add the new position to the result set if it lies within the grid's boundaries. We keep doing so until we have visited all rows * cols cells.

Solution Approach

The implementation of the solution follows a pattern-based approach by simulating movements on the grid. Here's how it's done step by step:

1. We define a list ans to keep track of the cells visited, starting with the initial cell [rStart, cStart].
2. We check if the grid size is 1x1. If so, we immediately return ans because the single starting cell is the only cell to visit.
3. We then set k = 1, which will determine the number of steps we take in a given direction before turning. This k will be increased as necessary to simulate the spiral pattern.
4. A while True loop begins, which will run until we've visited all rows * cols cells of the grid.
5. Inside the loop, we iterate over a list of direction increments and the corresponding step counts. The list has tuples in the format [dr, dc, dk], where dr is the row increment (0 for east or west, 1 for south, -1 for north), dc is the column increment (1 for east, -1 for west, 0 for north or south), and dk being the number of steps to move in that direction.
6. For each direction, we run another loop for dk steps:
   • For each step, we update rStart and cStart with dr and dc respectively.
   • We then check if the new position is within the grid's boundaries by verifying 0 <= rStart < rows and 0 <= cStart < cols. If yes, we append the position to ans.
   • We check if we have visited rows * cols cells. If we have, we return ans as the complete list of visited cells.
7. After completing the movement in all four directions, we increment k by 2 to maintain the spiral pattern for the next cycle of directions.

The algorithm utilizes simple iteration and directional increments to traverse the grid in a predictable, spiral pattern. No complex data structures are necessary beyond a list to hold the visited cell coordinates. The pattern's consistency allows us to increment k strategically, ensuring we expand our traversal in a spiraling outward fashion. This method provides full coverage of the grid while recording our path.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach. Consider a grid of size 3x3, where our starting cell is at rStart = 1 and cStart = 1 (the center of the grid).

1. We initialize ans with the starting cell, so ans = [[1, 1]].
2. Since the grid size is not 1x1, we proceed with the solution.
3. We set k = 1 as our initial step length for the eastward move.
4. Since the grid has more than one cell, we enter the while True loop to start the spiral traversal.
5. The first direction from our starting cell is east. We only need to move one step east (dc = 1, dr = 0). We move to cell [1, 2], which is within the grid. We add this to ans, resulting in ans = [[1, 1], [1, 2]].
6. Next is to move one step south (dc = 0, dr = 1). We move to [2, 2]. Again, it's within the grid, so ans becomes ans = [[1, 1], [1, 2], [2, 2]].
7. Following is westward movement, but since k is 1, we'd move only one step west (dc = -1, dr = 0) to [2, 1]. We add this to ans, making it ans = [[1, 1], [1, 2], [2, 2], [2, 1]].
8. The last direction in this cycle is north (dc = 0, dr = -1). We move one step north to [1, 1], but since this cell has already been visited, we don't add it again to ans`.
9. We've completed a full cycle (east > south > west > north), so we increment k by 2, making k = 3.
10. For the next eastward movement, we will move three steps, but after one step, we're already at the edge. The next position would be [1, 3], within the grid, so it's added to ans - ans = [[1, 1], [1, 2], [2, 2], [2, 1], [1, 3]].
11. Continuing the cycle, we go south three steps, adding positions [2, 3] and [3, 3] to ans.
12. Going west for three steps, we add [3, 2] and [3, 1] to ans.
13. Finally, moving north, we add [2, 1] and the last remaining cell [1, 1] is ignored as it's already visited.
14. All cells of the 3x3 grid have been added to ans, and the traversal is complete.

The final ans, with the cells visited in the order of the spiral from the center, would be:

1[[1, 1], [1, 2], [2, 2], [2, 1], [1, 3], [2, 3], [3, 3], [3, 2], [3, 1], [2, 1]]

This example demonstrated the simulated spiral movement through a grid following the solution approach outlined.

Solution Implementation

Time and Space Complexity

The given Python code generates all the coordinates in a matrix in a spiral order starting from a given cell (rStart, cStart). Considering each coordinate in a rows x cols grid needs to be visited exactly once, let's analyze both the time complexity and space complexity of the code.

Time Complexity:

The time complexity of this function can be analyzed by understanding the number of steps taken to complete the spiral. The number of steps increases as the spiral grows. The spiral growth is implemented by incrementing the step size k after every two directions (right-up and left-down). Initially, k starts at 1, and after each full cycle (right -> down -> left -> up), k is incremented by 2. This means that for a matrix that requires n cycles to fill, the total number of steps S would be the sum of an arithmetic series:

1S = 1 + 2 + 3 + 4 + ... + (2n - 1) + 2n

This sum is equal to n(2n + 1), which is O(n^2).

Since n is proportional to the larger dimension of the matrix (rows or cols), and the number of iterations required to cover all cells of the matrix will be bound by the total number of cells, we can conclude that the time complexity is O(max(rows, cols)^2).

Space Complexity:

The space complexity of the algorithm is determined by the amount of space needed to store the output. Since the output ans is a list of all coordinates in the matrix, it needs to store exactly rows * cols elements.

Thus, the space complexity is O(rows * cols).

Learn more about how to find time and space complexity quickly using problem constraints.