grid47 Exploring patterns and algorithms
Sep 16, 2024
6 min read
Solution to LeetCode 519: Random Flip Matrix Problem
You are given a binary grid of size m x n, where each cell is initially set to 0. The task is to design an algorithm that can randomly pick a cell that is still 0, flip it to 1, and ensure that all available cells are equally likely to be selected.
Problem
Approach
Steps
Complexity
Input: The input consists of a pair of integers m and n representing the size of the grid, followed by a sequence of operations.
Example: [3, 1], [], [], [], []
Constraints:
• 1 <= m, n <= 10^4
• There will always be at least one cell with value 0 for every flip operation.
• No more than 1000 flip and reset operations will be performed.
Output: The output should be an array representing the randomly selected index [i, j] for the flip operation, and null for other operations.
• The flip operation should return the index of a randomly selected cell that is still 0 and flips it to 1.
Goal: To randomly select an index where the value is 0, flip it to 1, and ensure equal likelihood for all available indices.
Steps:
• 1. Maintain a list or map to track available cells with value 0.
• 2. For each flip, randomly pick an index from the list of available cells.
• 3. After picking, flip the cell to 1 and update the list accordingly.
• 4. Reset the grid when the reset operation is called, restoring all values to 0.
Goal: The constraints for the input and output of the problem.
Steps:
• 1 <= m, n <= 10^4
• At least one free cell will be available for each flip call.
• No more than 1000 flip and reset operations.
Assumptions:
• The grid is initially filled with 0s.
• Each flip picks a valid available index and updates the grid.
• Input: [3, 1], [], [], [], []
• Explanation: A grid of size 3x1 is initialized with all zeros. Each flip returns a random index of a free cell, and after resetting, the grid is restored to its initial state.
Approach: The solution involves maintaining a list or map of available cells and efficiently selecting a random index while minimizing calls to the random function.
Observations:
• We can maintain a map or list that keeps track of the available cells to pick from.
• When selecting a cell, it should be done efficiently to minimize random function calls.
• The reset operation should restore the grid efficiently.
Steps:
• 1. Initialize the grid and maintain a list of available cells (cells with value 0).
• 2. When a flip occurs, randomly select an index from the list of available cells.
• 3. After each flip, update the grid and the list of available cells.
• 4. The reset operation restores the grid to its initial state and updates the available cells list.
Empty Inputs:
• If the grid is of size 1x1, there is only one cell to flip.
Large Inputs:
• The algorithm must handle large grid sizes up to 10^4 efficiently.
Special Values:
• The grid should handle reset and flip operations correctly when all cells are flipped to 1 and then reset.
Constraints:
• Ensure that no flip operation is called when there are no available cells.
classSolution {
map<int, int> mp;
int row, col, lmt;
public:Solution(int m, int n) {
row = m;
col = n;
reset();
}
vector<int> flip() {
int idx = rand()%lmt;
lmt--;
int x = mp.count(idx)? mp[idx] : idx;
mp[idx] = mp.count(lmt)? mp[lmt]: lmt;
return vector<int>{x/col, x%col};
}
voidreset() {
mp.clear();
lmt = row*col;
}
};
/**
* Your Solution object will be instantiated and called as such:
* Solution* obj = new Solution(m, n);
* vector<int> param_1 = obj->flip();
* obj->reset();
1 : Class Definition
classSolution {
This defines the `Solution` class, which will contain methods for flipping and resetting the matrix simulation.
2 : Data Members
map<int, int> mp;
This is a map `mp` used to store the indices of flipped cells, mapping the index to a new location.
3 : Data Members
int row, col, lmt;
These integer variables represent the number of rows (`row`), columns (`col`), and the limit (`lmt`) for random index generation.
4 : Constructor
public:
This marks the start of the public section of the class where constructors and methods will be defined.
5 : Constructor
Solution(int m, int n) {
Constructor for the `Solution` class that takes the number of rows `m` and columns `n` as parameters and initializes the object.
6 : Constructor Initialization
row = m;
Assign the value of `m` to the `row` data member.
7 : Constructor Initialization
col = n;
Assign the value of `n` to the `col` data member.
8 : Constructor Method Call
reset();
Call the `reset()` method to initialize or reset the state of the object.
9 : Method Definition
vector<int> flip() {
This method `flip` returns a random 2D index (row and column) based on a random number and the current state.
10 : Randomization
int idx = rand()%lmt;
Generate a random index `idx` between 0 and `lmt-1`.
11 : Update Limit
lmt--;
Decrease the limit `lmt` after every flip to reduce the number of possible indices.
12 : Mapping Logic
int x = mp.count(idx)? mp[idx] : idx;
Check if the random index `idx` is already mapped; if yes, use the mapped value, otherwise, use the index itself.
13 : Mapping Update
mp[idx] = mp.count(lmt)? mp[lmt]: lmt;
Map the current `idx` to the new limit `lmt`. This operation swaps the locations of the random index and the last available index.
14 : Return Statement
return vector<int>{x/col, x%col};
Return the row and column of the flipped index as a vector, where `x/col` is the row and `x%col` is the column.
15 : Method Definition
voidreset() {
This method `reset` is used to clear the map `mp` and reset the limit `lmt` to the total number of cells.
16 : Reset Logic
mp.clear();
Clear the map `mp` to reset all the flipped cell mappings.
17 : Reset Logic
lmt = row*col;
Reset the limit `lmt` to the total number of cells (rows * columns).
Best Case: O(1) for reset operation.
Average Case: O(1) for flip operation when using optimized data structures.
Worst Case: O(1) for both flip and reset operations.
Description: Both flip and reset operations are optimized to O(1) time using efficient data structures.
Best Case: O(1) when grid is empty (only when grid is reset).
Worst Case: O(m * n) to store the grid and available cells.
Description: The space complexity is O(m * n) due to the need to track the available cells in the grid.
