Swim in Rising Water

Swim in Rising Water

What is this problem about?

The Swim in Rising Water coding problem is a challenging grid-based pathfinding question. You are given an n x n grid where each cell represents an elevation. At time t, you can swim to any adjacent cell (up, down, left, right) as long as the elevation of both the current cell and the adjacent cell is less than or equal to t. You start at the top-left corner (0, 0) and want to reach the bottom-right corner (n-1, n-1). Your goal is to find the minimum time t such that there is a path from the start to the end.

Why is this asked in interviews?

This "Hard" problem is frequently asked by top-tier companies like Uber, Microsoft, and Google because it can be solved using several different advanced algorithmic techniques. It tests your ability to adapt shortest-path algorithms or use binary search in combination with graph traversal. Companies use the Swim in Rising Water interview question to evaluate a candidate's mastery of graph theory and their ability to choose the most efficient data structure for a given set of constraints.

Algorithmic pattern used

There are three main algorithmic patterns that can be applied to this problem:

1. Dijkstra’s Algorithm with a Min-Heap: Instead of the standard "sum of weights," you use the "maximum elevation encountered so far" as the distance. A priority queue helps you always explore the path with the smallest possible maximum elevation.
2. Binary Search on Answer with BFS/DFS: Since the time t is monotonic (if you can reach the end at time t, you can also reach it at any time > t), you can binary search for the minimum t and use BFS or DFS to check for connectivity.
3. Union Find: Sort all cells by elevation and add them to a Union Find structure one by one until the start and end cells are in the same component.

Example explanation

Consider a 2x2 grid: 0 2 1 3

• At t=0: You are at (0,0). Neighbors are 1 and 2. Both are > 0. You can't move.
• At t=1: You can move to (1,0) because its elevation is 1. Now you are at (1,0). Neighbor is (1,1) with elevation 3. Still can't reach the end.
• At t=2: From (0,0), you could move to (0,1). But (0,1) doesn't help you reach (1,1) yet.
• At t=3: You can move from (1,0) or (0,1) to (1,1). The minimum time is 3.

Common mistakes candidates make

A common error is trying to use standard BFS without a priority queue or binary search, which doesn't account for the "rising water" aspect. Another mistake is forgetting that you must consider the elevation of the starting cell as well (the minimum time must be at least grid[0][0]). Some candidates also overcomplicate the Union Find approach by not sorting the edges or cells correctly.

Interview preparation tip

When preparing for the Swim in Rising Water coding problem, practice Dijkstra's algorithm and understand how it can be modified for "Min-Max" path problems. Also, get comfortable with the Binary Search on Answer interview pattern, as it is a very powerful and versatile technique for finding the minimum or maximum possible value in a valid range.