| Mineral Deposits (interactive) | |
| You handle signal processing for an extra-terrestrial mining company. Your vessel is approaching an asteroid. | |
| Preliminary scans show the presence of k mineral deposits on the asteroid, but their precise locations are unknown. | |
| The surface of the asteroid is modeled as a grid of integer coordinates. Each mineral deposit i is at unknown integer | |
| coordinates (x_i, y_i) with βb β€ x_i β€ b and βb β€ y_i β€ b, for some integer b corresponding to the size of your initial scan. | |
| You may send probes to the surface in waves. | |
| If you send a wave of d probes at coordinates (s_j, t_j) for j = 1..d, then when a probe arrives, it measures the Manhattan | |
| distance to each of the k deposits. All data packets arrive together and are indistinguishable across probes. | |
| Thus, one wave returns the kΒ·d integer distances: | |
| |x_i β s_j| + |y_i β t_j| for all i in {1..k} and j in {1..d}. | |
| The list of returned distances is in non-decreasing order. | |
| Goal | |
| β Minimize the number of probe waves needed to determine all deposit locations. | |
| Interaction | |
| At the start, read a single line containing three integers: b, k, w β the scan boundary, the number of deposits, | |
| and the maximum number of waves you may send. | |
| You may then make at most w queries, each representing one wave. A query is printed as: | |
| ? d s1 t1 s2 t2 ... sd td | |
| with 1 β€ d β€ 2000. Each probe coordinate must satisfy β10^8 β€ s_j, t_j β€ 10^8. | |
| The judge replies with one line containing kΒ·d integers in non-decreasing order: the multiset of all Manhattan | |
| distances between the deposits and the d probe coordinates. | |
| The total number of probes across all waves must not exceed 2Β·10^4. | |
| To finish, print one line: | |
| ! x1 y1 x2 y2 ... xk yk | |
| containing the coordinates of all k deposits in any order. This must be your last line of output. | |
| Base Constraints | |
| 1 β€ b β€ 10^8, 1 β€ k β€ 20, and 2 β€ w β€ 10^4. | |
| Scoring | |
| For each test case, your score is: | |
| (# of mineral deposits found) / k | |
| Your overall score is the average over all test cases. There are no point-based subtasks in this version. | |
| Example | |
| If k = 2 deposits are at (1, 2) and (β3, β2), and you send d = 3 probes to (β4, β3), (β1, 0), and (2, β1), | |
| you must print: | |
| ? 3 -4 -3 -1 0 2 -1 | |
| and the response would be the six distances: | |
| 2 4 4 4 6 10 | |
| If the next wave has d = 2 probes at (1, 2) and (0, β2), you must print: | |
| ? 2 1 2 0 -2 | |
| and the response would be: | |
| 0 3 5 8 | |
| Finally you might answer: | |
| ! 1 2 β3 β2 | |
| Implementation notes: | |
| You may not ask more than w queries. Once you ask w queries and do the respective calculations, you should just print out the locations of the mineral deposits. |