Check if the Rectangle Corner Is Reachable
<p>You are given two positive integers <code>xCorner</code> and <code>yCorner</code>, and a 2D array <code>circles</code>, where <code>circles[i] = [x<sub>i</sub>, y<sub>i</sub>, r<sub>i</sub>]</code> denotes a circle with center at <code>(x<sub>i</sub>, y<sub>i</sub>)</code> and radius <code>r<sub>i</sub></code>.</p>

<p>There is a rectangle in the coordinate plane with its bottom left corner at the origin and top right corner at the coordinate <code>(xCorner, yCorner)</code>. You need to check whether there is a path from the bottom left corner to the top right corner such that the <strong>entire path</strong> lies inside the rectangle, <strong>does not</strong> touch or lie inside <strong>any</strong> circle, and touches the rectangle <strong>only</strong> at the two corners.</p>

<p>Return <code>true</code> if such a path exists, and <code>false</code> otherwise.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 4, circles = [[2,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">true</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3200-3299/3235.Check%20if%20the%20Rectangle%20Corner%20Is%20Reachable/images/example2circle1.png" style="width: 346px; height: 264px;" /></p>

<p>The black curve shows a possible path between <code>(0, 0)</code> and <code>(3, 4)</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 3, circles = [[1,1,2]]</span></p>

<p><strong>Output:</strong> <span class="example-io">false</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3200-3299/3235.Check%20if%20the%20Rectangle%20Corner%20Is%20Reachable/images/example1circle.png" style="width: 346px; height: 264px;" /></p>

<p>No path exists from <code>(0, 0)</code> to <code>(3, 3)</code>.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 3, circles = [[2,1,1],[1,2,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">false</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3200-3299/3235.Check%20if%20the%20Rectangle%20Corner%20Is%20Reachable/images/example0circle.png" style="width: 346px; height: 264px;" /></p>

<p>No path exists from <code>(0, 0)</code> to <code>(3, 3)</code>.</p>
</div>

<p><strong class="example">Example 4:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 4, yCorner = 4, circles = [[5,5,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">true</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3200-3299/3235.Check%20if%20the%20Rectangle%20Corner%20Is%20Reachable/images/rectangles.png" style="width: 346px; height: 264px;" /></p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>3 &lt;= xCorner, yCorner &lt;= 10<sup>9</sup></code></li>
	<li><code>1 &lt;= circles.length &lt;= 1000</code></li>
	<li><code>circles[i].length == 3</code></li>
	<li><code>1 &lt;= x<sub>i</sub>, y<sub>i</sub>, r<sub>i</sub> &lt;= 10<sup>9</sup></code></li>
</ul>
Depth-First Search; Breadth-First Search; Union Find; Geometry; Array; Math
class Solution {
public:
    bool canReachCorner(int xCorner, int yCorner, vector<vector<int>>& circles) {
        using ll = long long;
        auto inCircle = [&](ll x, ll y, ll cx, ll cy, ll r) {
            return (x - cx) * (x - cx) + (y - cy) * (y - cy) <= r * r;
        };
        auto crossLeftTop = [&](ll cx, ll cy, ll r) {
            bool a = abs(cx) <= r && (cy >= 0 && cy <= yCorner);
            bool b = abs(cy - yCorner) <= r && (cx >= 0 && cx <= xCorner);
            return a || b;
        };
        auto crossRightBottom = [&](ll cx, ll cy, ll r) {
            bool a = abs(cx - xCorner) <= r && (cy >= 0 && cy <= yCorner);
            bool b = abs(cy) <= r && (cx >= 0 && cx <= xCorner);
            return a || b;
        };

        int n = circles.size();
        vector<bool> vis(n);
        auto dfs = [&](this auto&& dfs, int i) -> bool {
            auto c = circles[i];
            ll x1 = c[0], y1 = c[1], r1 = c[2];
            if (crossRightBottom(x1, y1, r1)) {
                return true;
            }
            vis[i] = true;
            for (int j = 0; j < n; ++j) {
                if (vis[j]) {
                    continue;
                }
                auto c2 = circles[j];
                ll x2 = c2[0], y2 = c2[1], r2 = c2[2];
                if ((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2) > (r1 + r2) * (r1 + r2)) {
                    continue;
                }
                if (x1 * r2 + x2 * r1 < (r1 + r2) * xCorner && y1 * r2 + y2 * r1 < (r1 + r2) * yCorner
                    && dfs(j)) {
                    return true;
                }
            }
            return false;
        };

        for (int i = 0; i < n; ++i) {
            auto c = circles[i];
            ll x = c[0], y = c[1], r = c[2];
            if (inCircle(0, 0, x, y, r) || inCircle(xCorner, yCorner, x, y, r)) {
                return false;
            }
            if (!vis[i] && crossLeftTop(x, y, r) && dfs(i)) {
                return false;
            }
        }
        return true;
    }
};
