Pacific Atlantic Water Flow
Given an m x n matrix of heights, water flows from a cell to adjacent cells with height <= current height. The Pacific ocean touches the left and top edges. The Atlantic ocean touches the right and bottom edges.
Return cells where water can flow to both oceans. Output each cell as "row col" on a separate line, sorted by row then column.
Example:
1,2,2,3,5 3,2,3,4,4 2,4,5,3,1 6,7,1,4,5 5,1,1,2,4
0 4 1 3 1 4 2 2 3 0 3 1 4 0
- We start by identifying the Pacific and Atlantic ocean boundaries: the Pacific touches the left and top edges, and the Atlantic touches the right and bottom edges.
- We then perform a depth-first search (DFS) from each ocean boundary to find the cells that can flow to each ocean, considering the height constraint: a cell can flow to an adjacent cell if the adjacent cell's height is hadj​≤hcurrent​.
- The DFS from the Pacific ocean boundary marks the cells that can flow to the Pacific, and the DFS from the Atlantic ocean boundary marks the cells that can flow to the Atlantic.
- We find the intersection of the cells that can flow to both oceans and output them in the required format, sorted by row then column.
Constraints:
- 1 <= m, n <= 200
- 0 <= heights[i][j] <= 10^5
Background Knowledge
The problem "Pacific Atlantic Water Flow" involves understanding the concept of graph traversal and flow in a grid-based system. The grid represents a matrix of heights, where water can flow from a cell to its adjacent cells if the height of the adjacent cell is less than or equal to the current cell's height. This is similar to the concept of flow networks, where flow can occur between nodes (cells) based on certain conditions. In this case, the flow is driven by the height differences between cells.
The problem also touches on the idea of reachability, where we need to find cells that can reach both the Pacific and Atlantic oceans. This involves understanding the concept of connected components in a graph, where a connected component is a subgraph in which there is a path between any two vertices. In this problem, we can think of the Pacific and Atlantic oceans as two separate connected components, and we need to find the cells that belong to both components.
To solve this problem, we need to have a good understanding of graph traversal algorithms, such as depth-first search (DFS) and breadth-first search (BFS). These algorithms allow us to explore the grid and find the cells that can reach both oceans. We also need to understand how to represent the grid as a graph, where each cell is a node, and the edges represent the flow between cells.
Algorithm/Approach
The general approach to solve this type of problem involves using a graph traversal algorithm to explore the grid and find the cells that can reach both oceans. We can start by identifying the cells that are directly connected to the Pacific and Atlantic oceans, and then use a traversal algorithm to find the cells that can reach these oceans. We can use DFS or BFS to traverse the grid, and keep track of the cells that can reach both oceans.
Step-by-Step Strategy
To implement the solution, we can follow these steps:
- Identify the cells that are directly connected to the Pacific and Atlantic oceans.
- Use a graph traversal algorithm (such as DFS or BFS) to traverse the grid, starting from the cells that are directly connected to the oceans.
- Keep track of the cells that can reach both oceans, using a data structure such as a set or matrix.
- Once we have traversed the entire grid, we can output the cells that can reach both oceans, sorted by row and then column.
Common Pitfalls
Some common pitfalls to watch out for when implementing the solution include:
- Not properly handling the boundary cases, such as cells that are on the edge of the grid.
- Not correctly implementing the graph traversal algorithm, such as not properly marking visited cells.
- Not efficiently keeping track of the cells that can reach both oceans, such as using a data structure that has high overhead.
Time & Space Complexity
The expected time complexity of the solution is O(mâ‹…n), where m and n are the dimensions of the grid. This is because we need to traverse the entire grid to find the cells that can reach both oceans. The expected space complexity is also O(mâ‹…n), as we need to keep track of the cells that can reach both oceans. However, the actual space complexity may be lower if we use a more efficient data structure to keep track of the cells.