// 维护区间贡献的 Wavelet Matrix // !注意查询区间贡献时, 异或无效 // CountRange(start, end, a, b, xor) - 区间 [start, end) 中值在 [a, b) 之间的数的个数和这些数的和. // CountPrefix(start, end, x, xor) - 区间 [start, end) 中值在 [0, x) 之间的数的个数和这些数的和. // Kth(start, end, k, xor) - 区间 [start, end) 中第 k 小的数(0-indexed) 和前 k 小的数的和(不包括这个数). // Floor(start, end, x, xor) - 区间 [start, end) 中值小于等于 x 的最大值 // Ceiling(start, end, x, xor) - 区间 [start, end) 中值大于等于 x 的最小值 // MaxRightValue(start, end, xor, check) - 返回使得 check(prefixSum) 为 true 的最大value, 其中prefixSum为[0,val)内的数的和. // MaxRightCount(start, end, xor, check) - 返回使得 check(prefixSum) 为 true 的区间前缀个数的最大值. package main import ( "bufio" "fmt" "math/bits" "os" "sort" "time" ) func main() { // demo() // CF1771F() 区间最短距离和() } // Hossam and Range Minimum Query // https://www.luogu.com.cn/problem/CF1771F // 在线查询区间出现次数为奇数的数的最小值. // // 异或哈希+Wavelet Matrix二分: 最大值 max_ { max_ = v } } if log == -1 { log = bits.Len(uint(max_)) } res := &WaveletMatrixSum{e: e, op: op, inv: inv} res.unit = res.e() n := len(numsCopy) mid := make([]int, log) bv := make([]*BitVector, log) for i := 0; i < log; i++ { bv[i] = NewBitVector(n) } preSum := make([][]E, log+1) for i := range preSum { preSum[i] = make([]E, n+1) for j := range preSum[i] { preSum[i][j] = res.unit } } a0, a1 := make([]E, n), make([]E, n) for d := log - 1; d >= -1; d-- { p0, p1 := 0, 0 for i := 0; i < n; i++ { tmp := numsCopy[i] if f != nil { tmp = f(tmp) } preSum[d+1][i+1] = res.op(preSum[d+1][i], tmp) } if d == -1 { break } for i := 0; i < n; i++ { f := (numsCopy[i] >> d) & 1 if f == 0 { a0[p0] = numsCopy[i] p0++ } else { bv[d].Set(i) a1[p1] = numsCopy[i] p1++ } } mid[d] = p0 bv[d].Build() numsCopy, a0 = a0, numsCopy for i := 0; i < p1; i++ { numsCopy[p0+i] = a1[i] } } res.n, res.log = n, log res.mid, res.bv, res.preSum = mid, bv, preSum return res } // 返回区间 [left, right) 中 范围在 [a, b) 中的 (元素的个数, op 的结果) func (wm *WaveletMatrixSum) CountRange(left, right, a, b, xor int) (int, E) { c1, s1 := wm.CountPrefix(left, right, a, xor) c2, s2 := wm.CountPrefix(left, right, b, xor) return c2 - c1, wm.op(wm.inv(s1), s2) } // 返回区间 [left, right) 中 范围在 [0, x) 中的 (元素的个数, op 的结果) func (wm *WaveletMatrixSum) CountPrefix(left, right, x, xor int) (int, E) { if x >= 1<= 0; d-- { add := (x >> d) & 1 f := (xor >> d) & 1 l0, r0 := wm.bv[d].Rank(left, 0), wm.bv[d].Rank(right, 0) kf := f*(right-left-r0+l0) + (f^1)*(r0-l0) if add == 1 { count += kf if f == 1 { sum = wm.op(sum, wm.get(d, left+wm.mid[d]-l0, right+wm.mid[d]-r0)) left, right = l0, r0 } else { sum = wm.op(sum, wm.get(d, l0, r0)) left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0 } } else if f == 0 { left, right = l0, r0 } else { left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0 } } return count, sum } // 返回区间 [left, right) 中的 (第k小的元素, 前k个元素(不包括第k小的元素) 的 op 的结果) // // 如果k < 0, 返回 (-1, 0); 如果k >= right-left, 返回 (-1, 区间 op 的结果) func (wm *WaveletMatrixSum) Kth(left, right, k, xor int) (int, E) { if k < 0 { return -1, 0 } if right-left <= k { return -1, wm.get(wm.log, left, right) } res, sum := 0, wm.unit for d := wm.log - 1; d >= 0; d-- { f := (xor >> d) & 1 l0, r0 := wm.bv[d].Rank(left, 0), wm.bv[d].Rank(right, 0) kf := f*(right-left-r0+l0) + (f^1)*(r0-l0) if k < kf { if f == 0 { left, right = l0, r0 } else { left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0 } } else { k -= kf res |= 1 << d if f == 1 { sum = wm.op(sum, wm.get(d, left+wm.mid[d]-l0, right+wm.mid[d]-r0)) left, right = l0, r0 } else { sum = wm.op(sum, wm.get(d, l0, r0)) left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0 } } } if k != 0 { sum = wm.op(sum, wm.get(0, left, left+k)) } return res, sum } // 返回使得 check(prefixSum) 为 true 的最大值 val. // // !(即区间内小于 val 的数的聚合值 prefixSum 满足 check函数, 找到这样的最大的 val) // 如果整个区间都满足, 返回 INF. // eg: val = 5 => 即区间内值域在 [0,5) 中的数的聚合值满足 check 函数. func (wm *WaveletMatrixSum) MaxRightValue(left, right, xor int, check func(preSum E) bool) E { if check(wm.get(wm.log, left, right)) { return INF } res := 0 sum := wm.unit for d := wm.log - 1; d >= 0; d-- { f := (xor >> d) & 1 l0, r0 := wm.bv[d].Rank(left, 0), wm.bv[d].Rank(right, 0) var loSum E if f == 0 { loSum = wm.get(d, l0, r0) } else { loSum = wm.get(d, left+wm.mid[d]-l0, right+wm.mid[d]-r0) } if check(wm.op(sum, loSum)) { sum = wm.op(sum, loSum) res |= 1 << d if f == 1 { left, right = l0, r0 } else { left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0 } } else if f == 0 { left, right = l0, r0 } else { left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0 } } return res } // 返回使得 check(prefixSum) 为 true 的区间前缀个数的最大值. // // eg: count = 4 => 即区间内的数排序后, 前4个数的和满足 check 函数. func (wm *WaveletMatrixSum) MaxRightCount(left, right, xor int, check func(preSum E) bool) int { if check(wm.get(wm.log, left, right)) { return right - left } res := 0 sum := wm.unit for d := wm.log - 1; d >= 0; d-- { f := (xor >> d) & 1 l0, r0 := wm.bv[d].Rank(left, 0), wm.bv[d].Rank(right, 0) var kf int var loSum E if f == 0 { kf = r0 - l0 loSum = wm.get(d, l0, r0) } else { kf = (right - left) - (r0 - l0) loSum = wm.get(d, left+wm.mid[d]-l0, right+wm.mid[d]-r0) } if check(wm.op(sum, loSum)) { sum = wm.op(sum, loSum) res += kf if f == 1 { left, right = l0, r0 } else { left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0 } } else if f == 0 { left, right = l0, r0 } else { left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0 } } res += wm.binarySearch(func(k int) bool { return check(wm.op(sum, wm.get(0, left, left+k))) }, 0, right-left) return res } // [left, right) 中小于等于 x 的数中最大的数 // // 如果不存在则返回-INF func (w *WaveletMatrixSum) Floor(start, end, value, xor int) int { less, _ := w.CountPrefix(start, end, value, xor) if less == 0 { return -INF } res, _ := w.Kth(start, end, less-1, xor) return res } // [left, right) 中大于等于 x 的数中最小的数 // // 如果不存在则返回INF func (w *WaveletMatrixSum) Ceiling(start, end, value, xor int) int { less, _ := w.CountPrefix(start, end, value, xor) if less == end-start { return INF } res, _ := w.Kth(start, end, less, xor) return res } func (wm *WaveletMatrixSum) binarySearch(f func(E) bool, ok, ng int) int { for abs(ok-ng) > 1 { x := (ok + ng) >> 1 if f(x) { ok = x } else { ng = x } } return ok } func (wm *WaveletMatrixSum) get(d, l, r int) E { return wm.op(wm.inv(wm.preSum[d][l]), wm.preSum[d][r]) } func abs(a int) int { if a < 0 { return -a } return a } type BitVector struct { bits []uint64 preSum []int32 } func NewBitVector(n int) *BitVector { return &BitVector{bits: make([]uint64, n>>6+1), preSum: make([]int32, n>>6+1)} } func (bv *BitVector) Set(i int) { bv.bits[i>>6] |= 1 << (i & 63) } func (bv *BitVector) Build() { for i := 0; i < len(bv.bits)-1; i++ { bv.preSum[i+1] = bv.preSum[i] + int32(bits.OnesCount64(bv.bits[i])) } } func (bv *BitVector) Rank(k int, f int) int { m, s := bv.bits[k>>6], bv.preSum[k>>6] res := int(s) + bits.OnesCount64(m&((1<<(k&63))-1)) if f == 1 { return res } return k - res } // (紧)离散化. // // offset: 离散化的起始值偏移量. // // getRank: 给定一个数,返回它的排名`(offset ~ offset + count)`. // count: 离散化(去重)后的元素个数. func DiscretizeCompressed(nums []int, offset int) (getRank func(value int) int, getValue func(rank int) int, count int) { set := make(map[int]struct{}, len(nums)) for _, v := range nums { set[v] = struct{}{} } count = len(set) rank := make([]int, 0, count) for v := range set { rank = append(rank, v) } sort.Ints(rank) mp := make(map[int]int, count) for i, v := range rank { mp[v] = i + offset } getRank = func(v int) int { return mp[v] } getValue = func(r int) int { return rank[r-offset] } count = len(nums) return } type Random struct { seed uint64 hashBase uint64 } func NewRandom() *Random { return &Random{seed: uint64(time.Now().UnixNano()/2 + 1)} } func NewRandomWithSeed(seed int) *Random { return &Random{seed: uint64(seed)} } func (r *Random) Rng() uint64 { r.seed ^= r.seed << 7 r.seed ^= r.seed >> 9 return r.seed }