<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 {
public:
    int findRadius(vector<int>& houses, vector<int>& heaters) {
        sort(houses.begin(), houses.end());
        sort(heaters.begin(), heaters.end());
        int m = houses.size();
        int n = heaters.size();
        map<int, int> mp;
        for (int house : houses) {
            if (mp.empty()) {
                mp[house] = 1;
            } else {
                mp[house] = 1 + (--mp.end())->second;
            }
        }
        int left = 0;
        int right = 1000000000 - 1;
        while (left < right) {
            int mid = (left + right) / 2;
            int last_ended = -1;
            int covers = 0;
            for (int center : heaters) {
                if (center + mid <= last_ended) {
                    continue;
                }
                int x1 = max(last_ended + 1, center - mid);
                int x2 = center + mid;
                last_ended = x2;
                auto begin = mp.lower_bound(x1);
                auto end = mp.upper_bound(x2);
                if (end == mp.begin()) {
                    continue;
                }
                if (begin == mp.begin()) {
                    covers += (--end)->second;
                } else {
                    covers += (--end)->second - (--begin)->second;
                }
            }
            if (covers == houses.size()) {
                right = mid;
            } else {
                left = mid + 1;
            }
        }
        return right;
    }
};
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