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expl_ore_ation_chapter_2.py
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expl_ore_ation_chapter_2.py
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# Copyright (c) 2021 kamyu. All rights reserved.
#
# Facebook Hacker Cup 2021 Round 3 - Problem D. Expl-ore-ation Chapter 2
# https://www.facebook.com/codingcompetitions/hacker-cup/2021/round-3/problems/D2
#
# Time: O((R * C + S + K) * log(R * C + S + K) + (R * C + S + K) * alpha(R * C) + ((R * C) * log(R * C) + S + K) * logK) = O((R * C + S + K) * log(R * C + S + K) + ((R * C) * log(R * C) + S + K) * logK) = O((R * C + K) * log(R * C + K) + ((R * C) * log(R * C) + K) * logK), since O(S) <= O(R * C)
# Space: O(R * C + K)
#
# Template:
# https://github.com/cheran-senthil/PyRival/blob/master/pyrival/data_structures/SortedList.py
class SortedList(object):
def __init__(self, iterable=[], _load=200):
"""Initialize sorted list instance."""
values = sorted(iterable)
self._len = _len = len(values)
self._load = _load
self._lists = _lists = [values[i:i + _load] for i in xrange(0, _len, _load)]
self._list_lens = [len(_list) for _list in _lists]
self._mins = [_list[0] for _list in _lists]
self._fen_tree = []
self._rebuild = True
def _fen_build(self):
"""Build a fenwick tree instance."""
self._fen_tree[:] = self._list_lens
_fen_tree = self._fen_tree
for i in xrange(len(_fen_tree)):
if i | i + 1 < len(_fen_tree):
_fen_tree[i | i + 1] += _fen_tree[i]
self._rebuild = False
def _fen_update(self, index, value):
"""Update `fen_tree[index] += value`."""
if not self._rebuild:
_fen_tree = self._fen_tree
while index < len(_fen_tree):
_fen_tree[index] += value
index |= index + 1
def _fen_query(self, end):
"""Return `sum(_fen_tree[:end])`."""
if self._rebuild:
self._fen_build()
_fen_tree = self._fen_tree
x = 0
while end:
x += _fen_tree[end - 1]
end &= end - 1
return x
def _fen_findkth(self, k):
"""Return a pair of (the largest `idx` such that `sum(_fen_tree[:idx]) <= k`, `k - sum(_fen_tree[:idx])`)."""
_list_lens = self._list_lens
if k < _list_lens[0]:
return 0, k
if k >= self._len - _list_lens[-1]:
return len(_list_lens) - 1, k + _list_lens[-1] - self._len
if self._rebuild:
self._fen_build()
_fen_tree = self._fen_tree
idx = -1
for d in reversed(xrange(len(_fen_tree).bit_length())):
right_idx = idx + (1 << d)
if right_idx < len(_fen_tree) and k >= _fen_tree[right_idx]:
idx = right_idx
k -= _fen_tree[idx]
return idx + 1, k
def _delete(self, pos, idx):
"""Delete value at the given `(pos, idx)`."""
_lists = self._lists
_mins = self._mins
_list_lens = self._list_lens
self._len -= 1
self._fen_update(pos, -1)
del _lists[pos][idx]
_list_lens[pos] -= 1
if _list_lens[pos]:
_mins[pos] = _lists[pos][0]
else:
del _lists[pos]
del _list_lens[pos]
del _mins[pos]
self._rebuild = True
def _loc_left(self, value):
"""Return an index pair that corresponds to the first position of `value` in the sorted list."""
if not self._len:
return 0, 0
_lists = self._lists
_mins = self._mins
lo, pos = -1, len(_lists) - 1
while lo + 1 < pos:
mi = (lo + pos) >> 1
if value <= _mins[mi]:
pos = mi
else:
lo = mi
if pos and value <= _lists[pos - 1][-1]:
pos -= 1
_list = _lists[pos]
lo, idx = -1, len(_list)
while lo + 1 < idx:
mi = (lo + idx) >> 1
if value <= _list[mi]:
idx = mi
else:
lo = mi
return pos, idx
def _loc_right(self, value):
"""Return an index pair that corresponds to the last position of `value` in the sorted list."""
if not self._len:
return 0, 0
_lists = self._lists
_mins = self._mins
pos, hi = 0, len(_lists)
while pos + 1 < hi:
mi = (pos + hi) >> 1
if value < _mins[mi]:
hi = mi
else:
pos = mi
_list = _lists[pos]
lo, idx = -1, len(_list)
while lo + 1 < idx:
mi = (lo + idx) >> 1
if value < _list[mi]:
idx = mi
else:
lo = mi
return pos, idx
def add(self, value):
"""Add `value` to sorted list."""
_load = self._load
_lists = self._lists
_mins = self._mins
_list_lens = self._list_lens
self._len += 1
if _lists:
pos, idx = self._loc_right(value)
self._fen_update(pos, 1)
_list = _lists[pos]
_list.insert(idx, value)
_list_lens[pos] += 1
_mins[pos] = _list[0]
if _load + _load < len(_list):
_lists.insert(pos + 1, _list[_load:])
_list_lens.insert(pos + 1, len(_list) - _load)
_mins.insert(pos + 1, _list[_load])
_list_lens[pos] = _load
del _list[_load:]
self._rebuild = True
else:
_lists.append([value])
_mins.append(value)
_list_lens.append(1)
self._rebuild = True
def discard(self, value):
"""Remove `value` from sorted list if it is a member."""
_lists = self._lists
if _lists:
pos, idx = self._loc_right(value)
if idx and _lists[pos][idx - 1] == value:
self._delete(pos, idx - 1)
def remove(self, value):
"""Remove `value` from sorted list; `value` must be a member."""
_len = self._len
self.discard(value)
if _len == self._len:
raise ValueError('{0!r} not in list'.format(value))
def pop(self, index=-1):
"""Remove and return value at `index` in sorted list."""
pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
value = self._lists[pos][idx]
self._delete(pos, idx)
return value
def bisect_left(self, value):
"""Return the first index to insert `value` in the sorted list."""
pos, idx = self._loc_left(value)
return self._fen_query(pos) + idx
def bisect_right(self, value):
"""Return the last index to insert `value` in the sorted list."""
pos, idx = self._loc_right(value)
return self._fen_query(pos) + idx
def count(self, value):
"""Return number of occurrences of `value` in the sorted list."""
return self.bisect_right(value) - self.bisect_left(value)
def __len__(self):
"""Return the size of the sorted list."""
return self._len
def __getitem__(self, index):
"""Lookup value at `index` in sorted list."""
pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
return self._lists[pos][idx]
def __delitem__(self, index):
"""Remove value at `index` from sorted list."""
pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
self._delete(pos, idx)
def __contains__(self, value):
"""Return true if `value` is an element of the sorted list."""
_lists = self._lists
if _lists:
pos, idx = self._loc_left(value)
return idx < len(_lists[pos]) and _lists[pos][idx] == value
return False
def __iter__(self):
"""Return an iterator over the sorted list."""
return (value for _list in self._lists for value in _list)
def __reversed__(self):
"""Return a reverse iterator over the sorted list."""
return (value for _list in reversed(self._lists) for value in reversed(_list))
def __repr__(self):
"""Return string representation of sorted list."""
return 'SortedList({0})'.format(list(self))
class UnionFind(object): # Time: O(n * alpha(n)), Space: O(n)
def __init__(self, n):
self.set = range(n)
self.rank = [0]*n
self.intervals = [SortedList([(-INF, -INF), (INF, INF)]) for _ in xrange(n)] # modified
def find_set(self, x):
stk = []
while self.set[x] != x: # path compression
stk.append(x)
x = self.set[x]
while stk:
self.set[stk.pop()] = x
return x
def union_set(self, x, y):
x, y = self.find_set(x), self.find_set(y)
if x == y:
return False
if self.rank[x] > self.rank[y]: # union by rank
x, y = y, x
self.set[x] = self.set[y]
if self.rank[x] == self.rank[y]:
self.rank[y] += 1
# belows are all added
if len(self.intervals[x]) > len(self.intervals[y]):
self.intervals[x], self.intervals[y] = self.intervals[y], self.intervals[x]
for a, b in self.intervals[x]: # merge smaller self.intervals[x] into larger self.intervals[y]
if a in (-INF, INF):
continue
self.__add_interval(a, b, self.intervals[y]) # O(R * C * log(R * C)) times in total
self.intervals[x] = SortedList() # clear
return True
def __add_interval(self, a, b, interval): # Total Time: O(((R * C) * log(R * C) + (S + K)) * logK)
i = interval.bisect_left((a+1,))-1
c, d = interval[i]
if d < a-1:
i += 1
c, d = interval[i]
overlapped = 0
while c <= b+1:
a, b = min(a, c), max(b, d)
del interval[i]
overlapped += d-c+1
c, d = interval[i] # at most O(K) times in total, since each time at least increases 1 unit
interval.add((a, b)) # O((R * C) * log(R * C) + (S + K)) times in total, each costs Time: O(logK)
return (b-a+1)-overlapped
def add_robot(self, x, a, b): # added
return self.__add_interval(a, b, self.intervals[self.find_set(x)])
def expl_ore_ation_chapter_2():
R, C = map(int, raw_input().strip().split())
H = []
for _ in xrange(R):
H.extend(map(int, raw_input().strip().split()))
S = []
for _ in xrange(R):
S.extend(map(int, raw_input().strip().split()))
K = input()
changes = [map(int, raw_input().strip().split()) for _ in xrange(K)]
intervals = [[] for _ in xrange(R*C)]
for i, s in enumerate(S):
intervals[i].append((0, s))
for idx, (A, B, U) in enumerate(changes):
A -= 1
B -= 1
i = A*C+B
if not idx:
intervals[i][:] = []
intervals[i].append((idx, U))
events = []
for i in xrange(R*C):
if i-C >= 0: # up
events.append((min(H[i], H[i-C]), -1, i, i-C))
if i%C-1 >= 0: # left
events.append((min(H[i], H[i-1]), -1, i, i-1))
for i, interval in enumerate(intervals):
for j, (a, u) in enumerate(interval):
b = interval[j+1][0]-1 if j+1 < len(interval) else K-1
events.append((u, i, a, b))
events.sort(reverse=True) # Time: O((R * C + S + K) * log(R * C + S + K)), Space: O(R * C + S + K)
uf = UnionFind(R*C)
result = total = 0
for h, i, a, b in events:
if i == -1:
uf.union_set(a, b)
continue
if H[i] <= h:
continue
result += b-a+1
total += uf.add_robot(i, a, b)
return "%s %s" % (result, total)
INF = float("inf")
for case in xrange(input()):
print 'Case #%d: %s' % (case+1, expl_ore_ation_chapter_2())