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subarrays-distinct-element-sum-of-squares-ii.py
164 lines (149 loc) · 6.1 KB
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subarrays-distinct-element-sum-of-squares-ii.py
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# Time: O(nlogn)
# Space: O(n)
import collections
from sortedcontainers import SortedList
# bit, fenwick tree, sorted list, math
class Solution(object):
def sumCounts(self, nums):
"""
:type nums: List[int]
:rtype: int
"""
MOD = 10**9+7
class BIT(object): # 0-indexed.
def __init__(self, n):
self.__bit = [0]*(n+1) # Extra one for dummy node.
def add(self, i, val):
i += 1 # Extra one for dummy node.
while i < len(self.__bit):
self.__bit[i] = (self.__bit[i]+val) % MOD
i += (i & -i)
def query(self, i):
i += 1 # Extra one for dummy node.
ret = 0
while i > 0:
ret = (ret+self.__bit[i]) % MOD
i -= (i & -i)
return ret
def update(accu, d):
i = sl.bisect_left(idxs[x][-1])
accu = (accu + d*(len(nums)*(2*len(sl)-1) - (2*i+1)*idxs[x][-1] - 2*(bit.query(len(nums)-1)-bit.query(idxs[x][-1])))) % MOD
bit.add(idxs[x][-1], d*idxs[x][-1])
return accu
idxs = collections.defaultdict(list)
for i in reversed(xrange(len(nums))):
idxs[nums[i]].append(i)
result = 0
sl = SortedList(idxs[x][-1] for x in idxs)
accu = (len(nums)*len(sl)**2) % MOD
for i, x in enumerate(sl):
accu = (accu-(2*i+1)*x) % MOD
bit = BIT(len(nums))
for x in sl:
bit.add(x, x)
for x in nums:
result = (result+accu) % MOD # accu = sum(count(i, k) for k in range(i, len(nums)))
accu = update(accu, -1)
del sl[0]
idxs[x].pop()
if not idxs[x]:
continue
sl.add(idxs[x][-1])
accu = update(accu, +1)
assert(accu == 0)
return result
# Time: O(nlogn)
# Space: O(n)
# dp, segment tree, math
class Solution2(object):
def sumCounts(self, nums):
"""
:type nums: List[int]
:rtype: int
"""
MOD = 10**9+7
# Template:
# https://github.com/kamyu104/LeetCode-Solutions/blob/master/Python/longest-substring-of-one-repeating-character.py
class SegmentTree(object):
def __init__(self, N,
build_fn=None,
query_fn=lambda x, y: y if x is None else x if y is None else (x+y)%MOD,
update_fn=lambda x, y: y if x is None else (x+y)%MOD):
self.tree = [None]*(1<<((N-1).bit_length()+1))
self.base = len(self.tree)>>1
self.lazy = [None]*self.base
self.query_fn = query_fn
self.update_fn = update_fn
if build_fn is not None:
for i in xrange(self.base, self.base+N):
self.tree[i] = build_fn(i-self.base)
for i in reversed(xrange(1, self.base)):
self.tree[i] = query_fn(self.tree[i<<1], self.tree[(i<<1)+1])
self.count = [1]*len(self.tree) # added
for i in reversed(xrange(1, self.base)): # added
self.count[i] = self.count[i<<1] + self.count[(i<<1)+1]
def __apply(self, x, val):
self.tree[x] = self.update_fn(self.tree[x], val*self.count[x]) # modified
if x < self.base:
self.lazy[x] = self.update_fn(self.lazy[x], val)
def __push(self, x):
for h in reversed(xrange(1, x.bit_length())):
y = x>>h
if self.lazy[y] is not None:
self.__apply(y<<1, self.lazy[y])
self.__apply((y<<1)+1, self.lazy[y])
self.lazy[y] = None
def update(self, L, R, h): # Time: O(logN), Space: O(N)
def pull(x):
while x > 1:
x >>= 1
self.tree[x] = self.query_fn(self.tree[x<<1], self.tree[(x<<1)+1])
if self.lazy[x] is not None:
self.tree[x] = self.update_fn(self.tree[x], self.lazy[x]*self.count[x]) # modified
L += self.base
R += self.base
# self.__push(L) # enable if range assignment
# self.__push(R) # enable if range assignment
L0, R0 = L, R
while L <= R:
if L & 1: # is right child
self.__apply(L, h)
L += 1
if R & 1 == 0: # is left child
self.__apply(R, h)
R -= 1
L >>= 1
R >>= 1
pull(L0)
pull(R0)
def query(self, L, R):
if L > R:
return None
L += self.base
R += self.base
self.__push(L)
self.__push(R)
left = right = None
while L <= R:
if L & 1:
left = self.query_fn(left, self.tree[L])
L += 1
if R & 1 == 0:
right = self.query_fn(self.tree[R], right)
R -= 1
L >>= 1
R >>= 1
return self.query_fn(left, right)
result = accu = 0
sl = {}
st = SegmentTree(len(nums))
for i in xrange(len(nums)):
j = sl[nums[i]] if nums[i] in sl else -1
# sum(count(k, i)^2 for k in range(i+1)) - sum(count(k, i-1)^2 for k in range(i))
# = sum(2*count(k, i-1)+1 for k in range(j+1, i+1))
# = (i-j) + sum(2*count(k, i-1) for k in range(j+1, i+1))
accu = (accu+((i-j)+2*max(st.query(j+1, i), 0)))%MOD
result = (result+accu)%MOD
st.update(j+1, i, 1) # count(k, i) = count(k, i-1)+(1 if k >= j+1 else 0) for k in range(i+1)
sl[nums[i]] = i
return result