Cave

Solution Approach

Use binary search to determine each switch-door mapping:

1. Process doors in order $0, 1, \ldots, N-1$.

2. For door $d$, we know the correct settings for switches controlling doors $0, \ldots, d-1$ (so those doors are open). The remaining switches are ``unknown.''

3. Binary search on the set of unknown switches to find which one controls door $d$:

   • Set the known switches to their correct values (doors 0..d-1 open).

   • For unknown switches: set the first half to 0 and the rest to 1 (or vice versa).

   • Query. If door $d$ is open, the controlling switch is in one half; if closed, it's in the other half.

   • Recurse on the correct half until one switch remains.

   • Then determine the correct position for that switch by toggling it. enumerate

     Complexity

     • Queries: $O(N \log N)$ --- for each of $N$ doors, $O(\log N)$ binary search queries.

     • For $N = 5000$: $5000 \times 13 \approx 65000 < 70000$. Fits within budget.

     C++ Solution

     #include <bits/stdc++.h>
     using namespace std;
     
     // Grader functions:
     // int tryCombination(int S[]) - returns first closed door, or -1 if all open
     // void answer(int S[], int D[]) - report solution
     // S[i] = correct position (0/1) for switch i
     // D[i] = which door switch i controls
     
     int tryCombination(int S[]);
     void answer(int S[], int D[]);
     
     void exploreCave(int N){
         vector<int> switchPos(N, -1);  // correct position for each switch
         vector<int> switchDoor(N, -1); // which door each switch controls
         vector<bool> determined(N, false); // whether switch i is determined
     
         for(int door = 0; door < N; door++){
             // Collect undetermined switches
             vector<int> unknown;
             for(int i = 0; i < N; i++){
                 if(!determined[i]) unknown.push_back(i);
             }
     
             // First, determine the "test value" for unknown switches
             // that makes door 'door' close (so we can detect it).
             // Try setting all unknown switches to 0:
             int *S = new int[N];
             for(int i = 0; i < N; i++){
                 if(determined[i]){
                     S[i] = switchPos[i]; // known correct value
                 } else {
                     S[i] = 0;
                 }
             }
             int result = tryCombination(S);
             int testVal; // the value for unknown switches that closes door 'door'
             if(result == door){
                 // All unknowns at 0 -> door is closed
                 // So the controlling switch's correct value is 1 (currently at 0 = wrong)
                 testVal = 0; // setting unknown to 0 closes the door
             } else {
                 // Door is open with all unknowns at 0
                 // So correct value is 0 for the controlling switch
                 // Set unknowns to 1 to close it
                 testVal = 1;
             }
     
             // Binary search on unknown switches
             int lo = 0, hi = (int)unknown.size() - 1;
             while(lo < hi){
                 int mid = (lo + hi) / 2;
                 // Set switches unknown[lo..mid] to testVal, rest to 1-testVal
                 for(int i = 0; i < N; i++){
                     if(determined[i]){
                         S[i] = switchPos[i];
                     } else {
                         S[i] = 1 - testVal; // default: opposite of testVal
                     }
                 }
                 for(int i = lo; i <= mid; i++){
                     S[unknown[i]] = testVal;
                 }
     
                 result = tryCombination(S);
                 if(result == door){
                     // The controlling switch is in [lo..mid] (with testVal, door closes)
                     hi = mid;
                 } else {
                     // Door is open -> controlling switch NOT in [lo..mid]
                     lo = mid + 1;
                 }
             }
     
             // Switch unknown[lo] controls door 'door'
             int sw = unknown[lo];
             switchDoor[sw] = door;
             switchPos[sw] = 1 - testVal; // testVal closes it, so correct = opposite
             determined[sw] = true;
     
             delete[] S;
         }
     
         int *S = new int[N], *D = new int[N];
         for(int i = 0; i < N; i++){
             S[i] = switchPos[i];
             D[i] = switchDoor[i];
         }
         answer(S, D);
         delete[] S; delete[] D;
     }
     
     int main(){
         int N;
         cin >> N;
         exploreCave(N);
         return 0;
     }
     
     int tryCombination(int S[]){ return -1; }
     void answer(int S[], int D[]){}