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ConvexHullTrickLichao.go
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ConvexHullTrickLichao.go
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// ConvexHullTrickLichao
// 动态开点的李超线段树维护凸包
// https://ei1333.github.io/library/structure/convex-hull-trick/dynamic-li-chao-tree.hpp
// 追加直线/线段,查询k*x+b的最小值
// 如果要查询最大值,需要在插入直线时取反即(-k,-b),查询时返回-query(x)
package main
import (
"bufio"
"fmt"
"os"
)
func main() {
// https://atcoder.jp/contests/colopl2018-final-open/tasks/colopl2018_final_c
// 对每个i 求 f(i,j)=a[j]+(j-i)^2 的最小值
// 化简得 f(i,j)=(a[j]+j^2-2ij)+i^2
// 其中j变化的函数是关于i的一次函数(直线)
// !将这n条直线加入到CHT中,然后对每个i求最小值即可
in := bufio.NewReader(os.Stdin)
out := bufio.NewWriter(os.Stdout)
defer out.Flush()
var n int
fmt.Fscan(in, &n)
A := make([]int, n)
for i := 0; i < n; i++ {
fmt.Fscan(in, &A[i])
}
cht := NewConvexHullTrickLichao(true, 0, n-1) // 查询自变量0<=x<n
for i := 0; i < n; i++ {
cht.AddLine(-2*i, i*i+A[i], i)
}
for i := 0; i < n; i++ {
res, _ := cht.Query(i)
fmt.Fprintln(out, res+i*i)
}
}
const INF int = 1e18
type Line struct{ k, b, id int }
type LichaoNode struct {
line *Line
l, r *LichaoNode
}
type ConvexHullTrickLichao struct {
isMin bool
lower, upper int
root *LichaoNode
}
// 根据待查询的自变量x的上下界[lower,upper]建立CHTLichao.
func NewConvexHullTrickLichao(isMin bool, lower, upper int) *ConvexHullTrickLichao {
return &ConvexHullTrickLichao{isMin: isMin, lower: lower, upper: upper}
}
// O(logN) 追加一条直线k*x+b, id为直线的编号.
func (cht *ConvexHullTrickLichao) AddLine(k, b, id int) {
if !cht.isMin {
k, b = -k, -b
}
line := &Line{k, b, id}
cht.root = cht.addLine(cht.root, line, cht.lower, cht.upper, cht.getY(line, cht.lower), cht.getY(line, cht.upper))
}
// O(logN^2) 追加一条左闭右开的线段[start,end),所在直线k*x+b, id为线段的编号.
func (cht *ConvexHullTrickLichao) AddSegment(start, end, k, b, id int) {
if !cht.isMin {
k, b = -k, -b
}
line := &Line{k, b, id}
cht.root = cht.addSegment(cht.root, line, start, end-1, cht.lower, cht.upper, cht.getY(line, cht.lower), cht.getY(line, cht.upper))
}
// O(logN) 查询k*x+b的最小/大值.如果不存在直线,返回的id为-1.
func (cht *ConvexHullTrickLichao) Query(x int) (res, id int) {
res, id = cht.query(cht.root, cht.lower, cht.upper, x)
if !cht.isMin {
res = -res
}
return
}
func (cht *ConvexHullTrickLichao) addLine(t *LichaoNode, x *Line, l, r, xL, xR int) *LichaoNode {
if t == nil {
return &LichaoNode{line: x}
}
tL, tR := cht.getY(t.line, l), cht.getY(t.line, r)
if tL <= xL && tR <= xR {
return t
} else if tL >= xL && tR >= xR {
t.line = x
return t
} else {
mid := (l + r) >> 1
if mid == r {
mid--
}
tM, xM := cht.getY(t.line, mid), cht.getY(x, mid)
if tM > xM {
t.line, x = x, t.line
if xL >= tL {
t.l = cht.addLine(t.l, x, l, mid, tL, tM)
} else {
t.r = cht.addLine(t.r, x, mid+1, r, tM+x.k, tR)
}
} else {
if tL >= xL {
t.l = cht.addLine(t.l, x, l, mid, xL, xM)
} else {
t.r = cht.addLine(t.r, x, mid+1, r, xM+x.k, xR)
}
}
return t
}
}
func (cht *ConvexHullTrickLichao) addSegment(t *LichaoNode, x *Line, a, b, l, r, xL, xR int) *LichaoNode {
if r < a || b < l {
return t
}
if a <= l && r <= b {
y := &Line{x.k, x.b, x.id}
return cht.addLine(t, y, l, r, xL, xR)
}
if t != nil {
tL, tR := cht.getY(t.line, l), cht.getY(t.line, r)
if tL <= xL && tR <= xR {
return t
}
} else {
t = &LichaoNode{line: &Line{0, INF, -1}}
}
mid := (l + r) >> 1
if mid == r {
mid--
}
xM := cht.getY(x, mid)
t.l = cht.addSegment(t.l, x, a, b, l, mid, xL, xM)
t.r = cht.addSegment(t.r, x, a, b, mid+1, r, xM+x.k, xR)
return t
}
func (cht *ConvexHullTrickLichao) query(t *LichaoNode, l, r, x int) (res, id int) {
if t == nil {
res, id = INF, -1
return
}
if l == r {
res, id = cht.getY(t.line, x), t.line.id
return
}
mid := (l + r) >> 1
if mid == r {
mid--
}
res, id = cht.getY(t.line, x), t.line.id
if x <= mid {
cand, candId := cht.query(t.l, l, mid, x)
if cand < res {
res, id = cand, candId
}
} else {
cand, candId := cht.query(t.r, mid+1, r, x)
if cand < res {
res, id = cand, candId
}
}
return
}
func (cht *ConvexHullTrickLichao) getY(line *Line, x int) int {
return line.k*x + line.b
}