<p>Winter is coming! During the contest, your first job is to design a standard heater with a fixed warm radius to warm all the houses.</p>

<p>Every house can be warmed, as long as the house is within the heater&#39;s warm radius range.&nbsp;</p>

<p>Given the positions of <code>houses</code> and <code>heaters</code> on a horizontal line, return <em>the minimum radius standard of heaters&nbsp;so that those heaters could cover all houses.</em></p>

<p><strong>Notice</strong> that&nbsp;all the <code>heaters</code> follow your radius standard, and the warm radius will the same.</p>

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

<pre>
<strong>Input:</strong> houses = [1,2,3], heaters = [2]
<strong>Output:</strong> 1
<strong>Explanation:</strong> The only heater was placed in the position 2, and if we use the radius 1 standard, then all the houses can be warmed.
</pre>

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

<pre>
<strong>Input:</strong> houses = [1,2,3,4], heaters = [1,4]
<strong>Output:</strong> 1
<strong>Explanation:</strong> The two heaters were placed at positions 1 and 4. We need to use a radius 1 standard, then all the houses can be warmed.
</pre>

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

<pre>
<strong>Input:</strong> houses = [1,5], heaters = [2]
<strong>Output:</strong> 3
</pre>

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

<ul>
	<li><code>1 &lt;= houses.length, heaters.length &lt;= 3 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= houses[i], heaters[i] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution {
    bool check(int mid, vector<int>& houses, vector<int>& heaters) {
        int i = 0, j = 0;
        while (i < houses.size()) {
            // Check if the current house is within the radius 'mid' of the current heater
            if (j < heaters.size() && abs(heaters[j] - houses[i]) <= mid) {
                // Move to the next house
                i++;
            } else if (j < heaters.size() - 1 ) {
                // Move to the next heater
                j++;
            } else {
                // If neither condition is met, the house can't be heated
                return false;
            }
        }
        return true;
    }

public:
    int findRadius(vector<int>& houses, vector<int>& heaters) {
        // Sort the houses and heaters to ensure correct comparison
        sort(houses.begin(), houses.end());
        sort(heaters.begin(), heaters.end());

        int low = 0;
        int high = max(houses.back(), heaters.back()) - min(houses.front(), heaters.front());
        int ans = high;  // Initialize answer to the maximum possible radius

        while (low <= high) {
            int mid = low + (high - low) / 2;
            if (check(mid, houses, heaters)) {
                ans = mid;  // Store the smallest radius
                high = mid - 1;
            } else {
                low = mid + 1;
            }
        }
        return ans;
    }
};

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