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Author: MathProgrammer

Contests/Div 2 538/Explanations/Arithmetic Progression Explanation.txt.txt

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+- Use queries of the first kind to find the maximum element using binary search.
+- Find random indices and get the values  and sort them
+    - The difference between any two consecutive elements will be a multiple of $g$
+    - If we choose them randomly enough, there will two that are coprime, for example $3g$ and $5g$ so their GCD will give us $g$
+
+The probability of failing to find 2 coprime differences among a sequence of n elements among k is of the order 10^{-9}
+------
+
+
+mt19937 rng(time(NULL));
+
+int find_maximum(int &queries)
+{
+    int left = 0, right = 1e9 + 5;
+
+    //Always L < Max <= R
+    while(right - left > 1)
+    {
+        int has_higher;
+        int mid = (left + right)/2;
+
+        queries--;
+
+        cout << "> " << mid << "\n"; cout.flush();
+
+        cin >> has_higher;
+
+        if(has_higher)
+            left = mid;
+        else
+            right = mid;
+    }
+
+    return right;
+}
+
+int find_difference(int &queries, int greatest, int no_of_elements)
+{
+    int remaining_range = no_of_elements - 1;
+    vector <int> A;
+
+    vector <int> asked(no_of_elements + 1, false);
+
+    while(queries > 0 && remaining_range > 0)
+    {
+        int question_index = uniform_int_distribution<int>(1, no_of_elements)(rng);
+
+        if(asked[question_index])
+        {
+            continue;
+        }
+
+        cout << "? " << question_index << "\n"; cout.flush();
+
+        asked[question_index] = true;
+
+        int answer;
+        cin >> answer;
+        A.push_back(answer);
+
+        queries--;
+        remaining_range--;
+    }
+
+    sort(all(A));
+
+    if(A[A.size() - 1] != greatest)
+        A.push_back(greatest);
+
+    int difference = A[1] - A[0];
+
+    for(int i = 1; i < A.size(); i++)
+        difference = __gcd(difference, A[i] - A[i - 1]);
+
+    return difference;
+}
+
+int main()
+{
+    int no_of_elements;
+    cin >> no_of_elements;
+
+    const int MAX_QUERIES = 60;
+    int queries = MAX_QUERIES;
+
+    int maximum = find_maximum(queries);
+    int difference = find_difference(queries, maximum, no_of_elements);
+
+    int minimum = maximum - difference*(no_of_elements - 1);
+
+    cout << "! " << minimum << " " << difference << "\n"; cout.flush();
+
+    return 0;
+}