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WaveletMatrix.go
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WaveletMatrix.go
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// Usage:
// !Attention: nums[i] >= 0
// Count(start, end, value) – [start, end) 中值为 value 的数的个数.
// CountRange(start, end, lower, upper) – [start, end) 中值在 [lower, upper) 内的数的个数.
// Kth(start, end, k) – [start, end) 中第 k(0-indexed) 小的数.
// KthMax(start, end, k) – [start, end) 中第 k(0-indexed) 大的数.
// Lower(start, end, value) – [start, end) 中值小于 value 的最大值.不存在的话返回 -INF.
// Higher(start, end, value) – [start, end) 中值大于 value 的最小值.不存在的话返回 INF.
// Floor(start, end, value) – [start, end) 中值不超过 value 的最大值.不存在的话返回 -INF.
// Ceiling(start, end, value) – [start, end) 中值不小于 value 的最小值.不存在的话返回 INF.
// Referece:
// https://beet-aizu.github.io/library/datastructure/waveletmatrix.cpp
// https://blog.hamayanhamayan.com/entry/2017/06/13/103352
package main
import (
"bufio"
"fmt"
"math/bits"
"os"
)
func main() {
CF813E()
}
func demo() {
nums := []int{1, 2, 3, 1, 5, 6, 7, 8, 9, 10}
M := NewWaveletMatrix(nums)
fmt.Println(M.Count(0, 1, 1))
fmt.Println(M.CountRange(0, 10, 1, 5))
fmt.Println(M.Index(1, 1))
fmt.Println(M.IndexWithStart(1, 0, 2))
fmt.Println(M.Kth(0, 10, 2))
fmt.Println(M.KthMax(0, 10, 2))
fmt.Println(M.Lower(0, 3, 2))
fmt.Println(M.Floor(0, 3, 2))
fmt.Println(M.Higher(0, 10, 1))
fmt.Println(M.Ceiling(0, 10, 1))
}
const INF int = 1e18
// Army Creation
// https://www.luogu.com.cn/problem/CF813E
// 有n个数,q次询问,每次询问[l,r]中最多可以选多少个数使得相同的数最多有k个,强制在线.
// !k不变,可以预处理出每个数前面的第k个相同的数在哪.
// !对于询问区间[l,r],若前面的第k个数的位置小于l,则可以选择这个数
func CF813E() {
in := bufio.NewReader(os.Stdin)
out := bufio.NewWriter(os.Stdout)
defer out.Flush()
var n, k int
fmt.Fscan(in, &n, &k)
nums := make([]int, n)
for i := 0; i < n; i++ {
fmt.Fscan(in, &nums[i])
}
preRes := 0
parse := func(v int) int {
return (v+preRes)%n + 1
}
mp := make(map[int][]int)
for i, v := range nums {
mp[v] = append(mp[v], i)
}
maxStart := make([]int, n) // 选择下标为i的数时,区间的左端点需要严格小于maxStart[i]
for _, pos := range mp {
for i := k; i < len(pos); i++ {
maxStart[pos[i]] = pos[i-k] + 1
}
}
wm := NewWaveletMatrix(maxStart)
query := func(start, end int) int {
return wm.CountRange(start, end, 0, start+1)
}
var q int
fmt.Fscan(in, &q)
for i := 0; i < q; i++ {
var start, end int
fmt.Fscan(in, &start, &end)
start = parse(start)
end = parse(end)
if start > end {
start, end = end, start
}
start--
preRes = query(start, end)
fmt.Fprintln(out, preRes)
}
}
// 给你一个长度为 n 下标从 0 开始的整数数组 nums ,它包含 1 到 n 的所有数字,请你返回上升四元组的数目。
// 如果一个四元组 (i, j, k, l) 满足以下条件,我们称它是上升的:
// 0 <= i < j < k < l < n 且
// nums[i] < nums[k] < nums[j] < nums[l] 。
// 4 <= nums.length <= 4000, nums 中所有数字 互不相同 ,nums 是一个排列。
// https://leetcode.cn/problems/count-increasing-quadruplets/
func countQuadruplets(nums []int) int64 {
W := NewWaveletMatrix(nums)
res := 0
for j := 1; j < len(nums)-2; j++ {
for k := j + 1; k < len(nums)-1; k++ {
if !(nums[k] < nums[j]) {
continue
}
left := W.CountRange(0, j, 0, nums[k])
right := W.CountRange(k+1, len(nums), nums[j]+1, INF)
res += (left * right)
}
}
return int64(res)
}
// 给定非负整数数组 nums 构建一个 WaveletMatrix.
func NewWaveletMatrix(data []int) *WaveletMatrix {
dataCopy := make([]int, len(data))
max_ := 0
for i, v := range data {
if v > max_ {
max_ = v
}
dataCopy[i] = v
}
maxLog := bits.Len(uint(max_)) + 1
n := len(dataCopy)
mat := make([]*BitVector, maxLog)
zs := make([]int, maxLog)
buff1 := make([]int, maxLog)
buff2 := make([]int, maxLog)
ls, rs := make([]int, n), make([]int, n)
for dep := 0; dep < maxLog; dep++ {
mat[dep] = NewBitVector(n + 1)
p, q := 0, 0
for i := 0; i < n; i++ {
k := (dataCopy[i] >> uint(maxLog-dep-1)) & 1
if k == 1 {
rs[q] = dataCopy[i]
mat[dep].Set(i)
q++
} else {
ls[p] = dataCopy[i]
p++
}
}
zs[dep] = p
mat[dep].Build()
ls = dataCopy
for i := 0; i < q; i++ {
dataCopy[p+i] = rs[i]
}
}
return &WaveletMatrix{
n: n,
maxLog: maxLog,
mat: mat,
zs: zs,
buff1: buff1,
buff2: buff2,
}
}
type WaveletMatrix struct {
n int
maxLog int
mat []*BitVector
zs []int
buff1, buff2 []int
}
// [start, end) 内的 value 的個数.
func (w *WaveletMatrix) Count(start, end, value int) int {
return w.count(value, end) - w.count(value, start)
}
// [start, end) 内 [lower, upper) 的个数.
func (w *WaveletMatrix) CountRange(start, end, lower, upper int) int {
return w.freqDfs(0, start, end, 0, lower, upper)
}
// 第k(0-indexed)个value的位置.
func (w *WaveletMatrix) Index(value, k int) int {
w.count(value, w.n)
for dep := w.maxLog - 1; dep >= 0; dep-- {
bit := (value >> uint(w.maxLog-dep-1)) & 1
k = w.mat[dep].IndexWithStart(bit, k, w.buff1[dep])
if k < 0 || k >= w.buff2[dep] {
return -1
}
k -= w.buff1[dep]
}
return k
}
func (w *WaveletMatrix) IndexWithStart(value, k, start int) int {
return w.Index(value, k+w.count(value, start))
}
// [start, end) 内第k(0-indexed)小的数(kth).
func (w *WaveletMatrix) Kth(start, end, k int) int {
return w.KthMax(start, end, end-start-k-1)
}
// [start, end) 内第k(0-indexed)大的数.
func (w *WaveletMatrix) KthMax(start, end, k int) int {
if k < 0 || k >= end-start {
return -1
}
res := 0
for dep := 0; dep < w.maxLog; dep++ {
p, q := w.mat[dep].Count(1, start), w.mat[dep].Count(1, end)
if k < q-p {
start = w.zs[dep] + p
end = w.zs[dep] + q
res |= 1 << uint(w.maxLog-dep-1)
} else {
k -= q - p
start -= p
end -= q
}
}
return res
}
// [start, end) 内第k(0-indexed)小的数(kth).
func (w *WaveletMatrix) KthMin(start, end, k int) int {
return w.KthMax(start, end, end-start-k-1)
}
// [start, end) 中比 value 严格小的数, 不存在返回 -INF.
func (w *WaveletMatrix) Lower(start, end, value int) int {
k := w.lt(start, end, value)
if k != 0 {
return w.KthMin(start, end, k-1)
}
return -INF
}
// [start, end) 中比 value 严格大的数, 不存在返回 INF.
func (w *WaveletMatrix) Higher(start, end, value int) int {
k := w.le(start, end, value)
if k == end-start {
return INF
}
return w.KthMin(start, end, k)
}
// [start, end) 中不超过 value 的最大值, 不存在返回 -INF.
func (w *WaveletMatrix) Floor(start, end, value int) int {
count := w.Count(start, end, value)
if count > 0 {
return value
}
return w.Lower(start, end, value)
}
// [start, end) 中不小于 value 的最小值, 不存在返回 INF.
func (w *WaveletMatrix) Ceiling(start, end, value int) int {
count := w.Count(start, end, value)
if count > 0 {
return value
}
return w.Higher(start, end, value)
}
func (w *WaveletMatrix) access(k int) int {
res := 0
for dep := 0; dep < w.maxLog; dep++ {
bit := w.mat[dep].Get(k)
res = (res << 1) | bit
k = w.mat[dep].Count(bit, k) + w.zs[dep]*dep
}
return res
}
func (w *WaveletMatrix) count(value, end int) int {
left, right := 0, end
for dep := 0; dep < w.maxLog; dep++ {
w.buff1[dep] = left
w.buff2[dep] = right
bit := (value >> uint(w.maxLog-dep-1)) & 1
left = w.mat[dep].Count(bit, left) + w.zs[dep]*bit
right = w.mat[dep].Count(bit, right) + w.zs[dep]*bit
}
return right - left
}
func (w *WaveletMatrix) freqDfs(d, left, right, val, a, b int) int {
if left == right {
return 0
}
if d == w.maxLog {
if a <= val && val < b {
return right - left
}
return 0
}
nv := (1 << uint(w.maxLog-d-1)) | val
nnv := ((1 << uint(w.maxLog-d-1)) - 1) | nv
if nnv < a || b <= val {
return 0
}
if a <= val && nnv < b {
return right - left
}
lc, rc := w.mat[d].Count(1, left), w.mat[d].Count(1, right)
return w.freqDfs(d+1, left-lc, right-rc, val, a, b) + w.freqDfs(d+1, lc+w.zs[d], rc+w.zs[d], nv, a, b)
}
func (w *WaveletMatrix) ll(left, right, v int) (int, int) {
res := 0
for dep := 0; dep < w.maxLog; dep++ {
w.buff1[dep] = left
w.buff2[dep] = right
bit := v >> uint(w.maxLog-dep-1) & 1
if bit == 1 {
res += right - left + w.mat[dep].Count(1, left) - w.mat[dep].Count(1, right)
}
left = w.mat[dep].Count(bit, left) + w.zs[dep]*bit
right = w.mat[dep].Count(bit, right) + w.zs[dep]*bit
}
return res, right - left
}
func (w *WaveletMatrix) lt(left, right, v int) int {
a, _ := w.ll(left, right, v)
return a
}
func (w *WaveletMatrix) le(left, right, v int) int {
a, b := w.ll(left, right, v)
return a + b
}
type BitVector struct {
n int
block []int
sum []int
}
func NewBitVector(n int) *BitVector {
blockCount := (n + 63) >> 6
return &BitVector{
n: n,
block: make([]int, blockCount+1),
sum: make([]int, blockCount+1),
}
}
func (f *BitVector) Set(i int) {
f.block[i>>6] |= 1 << uint(i&63)
}
func (f *BitVector) Build() {
for i := 0; i < len(f.block)-1; i++ {
f.sum[i+1] = f.sum[i] + bits.OnesCount(uint(f.block[i]))
}
}
func (f *BitVector) Get(i int) int {
return (f.block[i>>6] >> uint(i&63)) & 1
}
func (f *BitVector) Count(value, end int) int {
mask := (1 << uint(end&63)) - 1
res := f.sum[end>>6] + bits.OnesCount(uint(f.block[end>>6]&mask))
if value == 1 {
return res
}
return end - res
}
func (f *BitVector) Index(value, k int) int {
if k < 0 || f.Count(value, f.n) <= k {
return -1
}
left, right := 0, f.n
for right-left > 1 {
mid := (left + right) >> 1
if f.Count(value, mid) >= k+1 {
right = mid
} else {
left = mid
}
}
return right - 1
}
func (f *BitVector) IndexWithStart(value, k, start int) int {
return f.Index(value, k+f.Count(value, start))
}