import sys readline=sys.stdin.readline from collections import deque,defaultdict import math from bisect import bisect_left, bisect_right from typing import Generic, Iterable, Iterator, TypeVar, Optional, List T = TypeVar('T') class AVL_Node_dict: """ノード Attributes: key (any): ノードのキー。比較可能なものであれば良い。(1, 4)などタプルも可。 val (any): ノードの値。 left (Node): 左の子ノード。 right (Node): 右の子ノード。 bias (int): 平衡度。(左部分木の高さ)-(右部分木の高さ)。 size (int): 自分を根とする部分木の大きさ """ def __init__(self,parent,key,value): self.parent=parent self.key=key self.value=value self.left=None self.right=None self.bias=0 self.size=1 class AVLTree_dict: def __init__(self): self.root=None def Rotate_Left(self,node): node_right=node.right node_right.size=node.size node.size-=1 if node_right.right!=None: node.size-=node_right.right.size if node_right.bias==-1: node_right.bias=0 node.bias=0 else: #assert node_right.bias==0 node_right.bias=1 node.bias=-1 node.right=node_right.left node_right.left=node return node_right def Rotate_Right(self,node): node_left=node.left node_left.size=node.size node.size-=1 if node_left.left!=None: node.size-=node_left.left.size if node_left.bias==1: node_left.bias=0 node.bias=0 else: #assert node_left.bias==0 node_left.bias=-1 node.bias=1 node.left=node_left.right node_left.right=node return node_left def Rotate_Left_Right(self,node): node_left=node.left node_left_right=node_left.right #assert node.bias==2 #assert node_left.bias==-1 #assert node_left_right.bias in (-1,0,1) node_left_right.size=node.size node.size-=node_left.size if node_left_right.right!=None: node.size+=node_left_right.right.size node_left.size-=1 if node_left_right.right!=None: node_left.size-=node_left_right.right.size node_left.right=node_left_right.left node_left_right.left=node_left node.left=node_left_right.right node_left_right.right=node self.Update_Bias_Double(node_left_right) return node_left_right def Rotate_Right_Left(self,node): node_right=node.right node_right_left=node_right.left #assert node.bias==-2 #assert node_right.bias==1 #assert node_right_left.bias in (-1,0,1) node_right_left.size=node.size node.size-=node_right.size if node_right_left.left!=None: node.size+=node_right_left.left.size node_right.size-=1 if node_right_left.left!=None: node_right.size-=node_right_left.left.size node_right.left=node_right_left.right node_right_left.right=node_right node.right=node_right_left.left node_right_left.left=node self.Update_Bias_Double(node_right_left) return node_right_left def Update_Bias_Double(self,node): #assert node.right.bias*node.left.bias==-2 #assert node.right.bias>0 if node.bias==1: node.right.bias=-1 node.left.bias=0 elif node.bias==-1: node.right.bias=0 node.left.bias=1 else: node.right.bias=0 node.left.bias=0 node.bias=0 def __getitem__(self,key): v=self.root while v!=None: if keykey: if retu==None or retu[0]>v.key: retu=(v.key,v.value) v=v.left else: v=v.right return retu def Bisect_Left(self,key): retu=None v=self.root while v!=None: if v.key None: "Evenly divide `a` into buckets." if a is None: a = list(self) size = self.size = len(a) bucket_size = int(math.ceil(math.sqrt(size / self.BUCKET_RATIO))) self.a = [a[size * i // bucket_size : size * (i + 1) // bucket_size] for i in range(bucket_size)] def __init__(self, a: Iterable[T] = []) -> None: "Make a new SortedSet from iterable. / O(N) if sorted and unique / O(N log N)" a = list(a) if not all(a[i] < a[i + 1] for i in range(len(a) - 1)): a = sorted(set(a)) self._build(a) def __iter__(self) -> Iterator[T]: for i in self.a: for j in i: yield j def __reversed__(self) -> Iterator[T]: for i in reversed(self.a): for j in reversed(i): yield j def __len__(self) -> int: return self.size def __repr__(self) -> str: return "SortedSet" + str(self.a) def __str__(self) -> str: s = str(list(self)) return "{" + s[1 : len(s) - 1] + "}" def _find_bucket(self, x: T) -> List[T]: "Find the bucket which should contain x. self must not be empty." for a in self.a: if x <= a[-1]: return a return a def __contains__(self, x: T) -> bool: if self.size == 0: return False a = self._find_bucket(x) i = bisect_left(a, x) return i != len(a) and a[i] == x def add(self, x: T) -> bool: "Add an element and return True if added. / O(√N)" if self.size == 0: self.a = [[x]] self.size = 1 return True a = self._find_bucket(x) i = bisect_left(a, x) if i != len(a) and a[i] == x: return False a.insert(i, x) self.size += 1 if len(a) > len(self.a) * self.REBUILD_RATIO: self._build() return True def discard(self, x: T) -> bool: "Remove an element and return True if removed. / O(√N)" if self.size == 0: return False a = self._find_bucket(x) i = bisect_left(a, x) if i == len(a) or a[i] != x: return False a.pop(i) self.size -= 1 if len(a) == 0: self._build() return True def lt(self, x: T) -> Optional[T]: "Find the largest element < x, or None if it doesn't exist." for a in reversed(self.a): if a[0] < x: return a[bisect_left(a, x) - 1] def le(self, x: T) -> Optional[T]: "Find the largest element <= x, or None if it doesn't exist." for a in reversed(self.a): if a[0] <= x: return a[bisect_right(a, x) - 1] def gt(self, x: T) -> Optional[T]: "Find the smallest element > x, or None if it doesn't exist." for a in self.a: if a[-1] > x: return a[bisect_right(a, x)] def ge(self, x: T) -> Optional[T]: "Find the smallest element >= x, or None if it doesn't exist." for a in self.a: if a[-1] >= x: return a[bisect_left(a, x)] def __getitem__(self, x: int) -> T: "Return the x-th element, or IndexError if it doesn't exist." if x < 0: x += self.size if x < 0: raise IndexError for a in self.a: if x < len(a): return a[x] x -= len(a) raise IndexError def index(self, x: T) -> int: "Count the number of elements < x." ans = 0 for a in self.a: if a[-1] >= x: return ans + bisect_left(a, x) ans += len(a) return ans def index_right(self, x: T) -> int: "Count the number of elements <= x." ans = 0 for a in self.a: if a[-1] > x: return ans + bisect_right(a, x) ans += len(a) return ans class Convex_Hull_Trick: def __init__(self,avl=False): self.avl=avl if self.avl: self.lines=AVLTree_dict() else: self.lines_A=SortedSet() self.lines_B={} def is_removed(self,line0,line1,line2): if line0==None: return False if line2==None: return False a0,b0=line0 a1,b1=line1 a2,b2=line2 return (a1-a0)*(b2-b1)<=(b1-b0)*(a2-a1) def add_line(self,a,b): if self.avl: bb=self.lines[a] else: if a in self.lines_B: bb=self.lines_B[a] else: bb=None if bb!=None and bb<=b: return if self.avl: line0=self.lines.Bisect_Left(a) else: aa=self.lines_A.lt(a) if aa==None: line0=None else: line0=(aa,self.lines_B[aa]) line1=(a,b) if self.avl: line2=self.lines.Bisect_Right(a) else: aa=self.lines_A.gt(a) if aa==None: line2=None else: line2=(aa,self.lines_B[aa]) if self.is_removed(line0,line1,line2): return if self.avl: self.lines[a]=b else: self.lines_A.add(a) self.lines_B[a]=b line2=(a,b) if self.avl: line1=self.lines.Bisect_Left(line2[0]) line0=None if line1==None else self.lines.Bisect_Left(line1[0]) else: aa=self.lines_A.lt(line2[0]) if aa==None: line1=None else: line1=(aa,self.lines_B[aa]) if line1==None: line0=None else: aa=self.lines_A.lt(line1[0]) if aa==None: line0=None else: line0=(aa,self.lines_B[aa]) while self.is_removed(line0,line1,line2): if self.avl: del self.lines[line1[0]] else: assert self.lines_A.discard(line1[0]) del self.lines_B[line1[0]] line1=line0 if self.avl: line0=None if line1==None else self.lines.Bisect_Left(line1[0]) else: aa=self.lines_A.lt(line1[0]) if aa==None: line0=None else: line0=(aa,self.lines_B[aa]) line0=(a,b) if self.avl: line1=self.lines.Bisect_Right(line0[0]) line2=None if line1==None else self.lines.Bisect_Right(line1[0]) else: aa=self.lines_A.gt(line0[0]) if aa==None: line1=None else: line1=(aa,self.lines_B[aa]) if line1==None: line2=None else: aa=self.lines_A.gt(line1[0]) if aa==None: line2=None else: line2=(aa,self.lines_B[aa]) while self.is_removed(line0,line1,line2): if self.avl: del self.lines[line1[0]] else: assert self.lines_A.discard(line1[0]) del self.lines_B[line1[0]] line1=line2 if self.avl: line2=None if line1==None else self.lines.Bisect_Right(line1[0]) else: aa=self.lines_A.gt(line1[0]) if aa==None: line2=None else: line2=(aa,self.lines_B[aa]) def __call__(self,x): if self.avl: size=len(self.lines) else: size=len(self.lines_A) if not size: return None ok,ng=-1,size-1 while ng-ok>1: mid=(ok+ng)//2 if self.avl: a0,b0=self.lines.Find_Kth_Element(mid) a1,b1=self.lines.Find_Kth_Element(mid+1) else: a0=self.lines_A[mid] b0=self.lines_B[a0] a1=self.lines_A[mid+1] b1=self.lines_B[a1] if a0*x+b0>a1*x+b1: ok=mid else: ng=mid if self.avl: a,b=self.lines.Find_Kth_Element(ok+1) else: a=self.lines_A[ok+1] b=self.lines_B[a] return a*x+b def __getitem__(self,a): if self.avl: return self.lines[a] else: if a in self.lines_A: return self.lines_B[a] else: return None def __setitem__(self,a,b): self.add_line(a,b) class Convex_Hull_Trick_deque: """ f_i = a_ix + b_i とする。f_i の追加および、min_i f(x) の取得ができるデータ構造。 ただし、傾き a_i は降順に追加されなければならない。 また、クエリ x も昇順に実行されなければならない。 """ def __init__(self): self.lines=deque() def add_line(self,a,b): lines=self.lines while len(lines) >= 2: a1,b1=lines[-2] a2,b2=lines[-1] if (a2-a1)*(b-b2)<(b2-b1)*(a-a2): break lines.pop() lines.append((a, b)) def __call__(self, x): lines=self.lines a,b=lines[0] y=a*x+b while len(lines) >= 2: a2, b2 = lines[1] y2 = a2 * x + b2 if y < y2: break y = y2 lines.popleft() return y SS=SortedSet N=int(readline()) A=list(map(int,readline().split())) X=list(map(int,readline().split())) Y=list(map(int,readline().split())) dp=Convex_Hull_Trick() cost=0 for i in range(N): dp[-2*X[i]]=cost+X[i]**2+Y[i]**2 cost=dp(A[i])+A[i]**2 ans=cost print(ans) for x in X: dp[-2*x]