0、基本模板

0.1单点修改

class Node:def __init__(self,l,r,s):self.l=lself.r=rself.s=s
def pushup(u,l,r):u.s = l.s + r.s
def build(u,l,r):if l==r:tr[u]=Node(l,r,a[l])else:mid = l+r>>1tr[u]=Node(l,r,0)build(u<<1,l,mid); build(u<<1|1,mid+1,r)pushup(tr[u],tr[u<<1],tr[u<<1|1])
def modify(u,x,v):if tr[u].l==tr[u].r==x:tr[u].v=velse:mid = tr[u].l+tr[u].r>>1if x<=mid:modify(u<<1,l,r)else:modify(u<<1|1,l,r)pushup(tr[u],tr[u<<1],tr[u<<1|1])
def query(u,l,r):if l<=tr[u].l<=tr[u].r<=r:return tr[u]else:mid = tr[u].l+tr[u].r>>1if mid>=r:return query(u<<1,l,r)elif mid<l:return query(u<<1|1,l,r)else:res = Node(0,0,0)left = query(u<<1,l,r)right = query(u<<1|1,l,r)pushup(res,left,right)return res
tr = [None]*(N*4)
build(1,1,n)

• 维护的属性可以是：$sum、min、max、gcd、lcm$、区间积、区间异或和、区间异或和、区间最大值个数、区间最小值个数等

class Node:def __init__(self, l, r, s, add, min_val, max_val, gcd_val, lcm_val, product_val, xor_val, max_count, min_count):self.l = lself.r = rself.s = sself.add = addself.min_val = min_valself.max_val = max_valself.gcd_val = gcd_valself.lcm_val = lcm_valself.product_val = product_valself.xor_val = xor_valself.max_count = max_countself.min_count = min_count
def pushup(u, l, r):u.s = l.s + r.su.min_val = min(l.min_val, r.min_val)u.max_val = max(l.max_val, r.max_val)u.gcd_val = gcd(l.gcd_val, r.gcd_val)u.lcm_val = lcm(l.lcm_val, r.lcm_val)u.product_val = l.product_val * r.product_valu.xor_val = l.xor_val ^ r.xor_valif l.max_val == r.max_val:u.max_count = l.max_count + r.max_countelse:u.max_count = l.max_count if l.max_val > r.max_val else r.max_countif l.min_val == r.min_val:u.min_count = l.min_count + r.min_countelse:u.min_count = l.min_count if l.min_val < r.min_val else r.min_count

0.2区间修改

class Node:def __init__(self,l,r,s,add):self.l=lself.r=rself.s=sself.add=add
def pushup(u,l,r):u.s = l.s+r.s
def pushdown(u,l,r):if u.add:l.add += u.add; l.s += (l.r-l.l+1)*u.addr.add += u.add; r.s += (r.r-r.l+1)*u.addu.add=0
def build(u,l,r):if l==r:tr[u]=Node(l,r,a[l],0)else:mid = l+r>>1tr[u]=Node(l,r,0,0)build(u<<1,l,mid); build(u<<1|1,mid+1,r)pushup(tr[u],tr[u<<1],tr[u<<1|1])
def modify(u,l,r,x):if l<=tr[u].l<=tr[u].r<=r:tr[u].s += (tr[u].r-tr[u].l+1)*xtr[u].add += xelse:pushdown(tr[u],tr[u<<1],tr[u<<1|1])mid = tr[u].l + tr[u].r>>1if mid>=l:modify(u<<1,l,r,x)if mid<r:modify(u<<1|1,l,r,x)pushup(tr[u],tr[u<<1],tr[u<<1|1])
def query(u,l,r):if l<=tr[u].l<=tr[u].r<=r:return tr[u]else:pushdown(tr[u],tr[u<<1],tr[u<<1|1])mid = tr[u].l+tr[u].r>>1if mid>=r:return query(u<<1,l,r)elif mid<l:return query(u<<1|1,l,r)else:res = Node(0,0,0,0)left = query(u<<1,l,r)right = query(u<<1|1,l,r)pushup(res,left,right)return res
tr = [None]*(N*4)
build(1,1,n)        

• 维护各种属性

class Node:def __init__(self, l, r, s, add, min_val, max_val, gcd_val, lcm_val, product_val, xor_val, max_count, min_count):self.l = lself.r = rself.s = sself.add = addself.min_val = min_valself.max_val = max_valself.gcd_val = gcd_valself.lcm_val = lcm_valself.product_val = product_valself.xor_val = xor_valself.max_count = max_countself.min_count = min_count
def pushup(u,l,r):u.s = l.s + r.su.min_val = min(l.min_val, r.min_val)u.max_val = max(l.max_val, r.max_val)u.gcd_val = gcd(l.gcd_val, r.gcd_val)u.lcm_val = lcm(l.lcm_val, r.lcm_val)u.product_val = l.product_val * r.product_valu.xor_val = l.xor_val ^ r.xor_valif l.max_val == r.max_val:u.max_count = l.max_count + r.max_countelse:u.max_count = l.max_count if l.max_val > r.max_val else r.max_countif l.min_val == r.min_val:u.min_count = l.min_count + r.min_countelse:u.min_count = l.min_count if l.min_val < r.min_val else r.min_count
def modify(u, l, r, x):if l <= tr[u].l <= tr[u].r <= r:tr[u].s += (tr[u].r - tr[u].l + 1) * xtr[u].min_val += xtr[u].max_val += xtr[u].gcd_val = x  # 假设为区间赋值tr[u].lcm_val = x  # 假设为区间赋值tr[u].product_val *= xtr[u].xor_val ^= xif x > 0 and tr[u].max_val == tr[u].max_val + x:tr[u].max_count += 1if x < 0 and tr[u].min_val == tr[u].min_val + x:tr[u].min_count += 1tr[u].add += xelse:pushdown(tr[u],tr[u<<1],tr[u<<1|1])mid = tr[u].l + tr[u].r>>1if mid>=l:modify(u<<1,l,r,x)if mid<r:modify(u<<1|1,l,r,x)pushup(tr[u],tr[u<<1],tr[u<<1|1])