Swimming Solution

Process cells in order of increasing elevation. At each step, union the cell with its neighbors (if they're already processed):

function swimInWater(grid):
    n = grid.length
    uf = new UnionFind(n * n)

    // Sort cells by elevation
    cells = []
    for i from 0 to n-1:
        for j from 0 to n-1:
            cells.add((grid[i][j], i, j))
    sort(cells)

    function idx(r, c):
        return r * n + c

    processed = n x n grid of false

    for each (elevation, r, c) in cells:
        processed[r][c] = true
        for each (dr, dc) in [(0,1),(0,-1),(1,0),(-1,0)]:
            nr = r + dr
            nc = c + dc
            if 0 <= nr < n and 0 <= nc < n and processed[nr][nc]:
                uf.union(idx(r, c), idx(nr, nc))

        if uf.connected(idx(0, 0), idx(n-1, n-1)):
            return elevation

    return -1 // should never reach

Time: $O(n^2 \log n)$ for sorting. Space: $O(n^2)$.