hdu 4578 Transformation [线段树 区间更新]

题意:

一个长度为n的数列a，初始全为0，对这个数列进行m个操作，操作格式如下：

op x y c

op=1: a[i] += c, i=x,x+1,...,y

op=2: a[i] *= c, i=x,x+1,...,y

op=3: a[i] = c, i=x,x+1,...,y

op=4: 输出sum=a[x]+a[x+1]+...+a[y]

思路:

考虑用线段树，对set，mul，add的处理顺序为set->mul->add。

对于一段序列a[x],a[x+1],...,a[y]记

Sigma1(a[i]) = a[x] + a[x+1] +...+ a[y]

Sigma2(a[i]) = a[x]^2 + a[x+1]^2 +...+ a[y]^2

Sigma3(a[i]) = a[x]^3 + a[x+1]^3 +...+ a[y]^3

那么，

Sigma1(n*a[i]+p)

= (n*a[x]+p) + (n*a[x+1]+p) +...+ (n*a[y]+p)

= n*(a[x]+a[x+1]+...+a[y]) + p*(y-x+1)

= n*Sigma1(a[i])

Sigma2(n*a[i]+p)

= (n*a[x]+p)^2 + (n*a[x+1]+p)^2 +...+ (n*a[y]+p)^2

= n*n*(a[x]^2+a[x+1]^2+...+a[y]^2) + 2*n*p*(a[x]+a[x+1]+...+a[y]) + p*p*(y-x+1)

= n*n*Sigma2(a[i]) + 2*n*p*Sigma1(a[i]) + p*p*(y-x+1)

Sigma3(n*a[i]+p)

= (n*a[x]+p) + (n*a[x+1]+p) +...+ (n*a[y]+p)

= n*n*n*(a[x]^3+a[x+1]^3+...+a[y]^3) + 3*n*n*p*(a[x]^2+a[x+1]^2+...+a[y]^2) + 3*n*p*p*(a[x]+a[x+1]+...+a[y]) + p*p*p*(y-x+1)

= n*n*n*Sigma3(a[i]) + 3*n*n*p*Sigma2(a[i]) + 3*n*p*p*Sigma1(a[i]) + p*p*p*(y-x+1)

这样就很容易求解了。

maintain(o):
if setv then cope o with setv
else if R>L then cope o with lo and ro
else cope o as leaf node
cope o with mulv then addv
pushdown(o):
if setv then push setv down
if mulv then push mulv down
if addv then push addv down
update:
if [y1,y2] contains [L,R]
do update directly
else
pushdown
if y1 <= M then update(lo) else maintain(lo)
if y2 > M then update(ro) else maintain(ro)
maintain(o)