package main import ( "bufio" "fmt" "math/bits" "os" ) func main() { in := bufio.NewReader(os.Stdin) out := bufio.NewWriter(os.Stdout) defer out.Flush() var n int fmt.Fscan(in, &n) xs := make([]int, n) for i := 0; i < n; i++ { fmt.Fscan(in, &xs[i]) xs[i]-- } wm := NewWaveletMatrixSum(xs, -1, nil) res := n * (n - 1) for i := 0; i < n; i++ { res -= wm.CountRange(0, i, 0, xs[i], 0) // 左侧不相撞的 res -= wm.CountRange(i+1, n, xs[i]+1, 2*n, 0) // 右侧不相撞的 } fmt.Fprintln(out, res/2) } func demo() { nums := []int{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} wm := NewWaveletMatrixSum(nums, -1, nums) fmt.Println(wm.CountRange(0, 10, 3, 7, 0)) fmt.Println(wm.Kth(0, 10, 9, 0)) fmt.Println(wm.KthValueAndSum(0, 10, 10, 0)) fmt.Println(wm.Sum(0, 10, 1, 3, 0)) fmt.Println(wm.Median(false, 0, 10, 0)) fmt.Println(wm.CountRangeSegments([][2]int{{0, 1}, {5, 10}}, 3, 7, 0)) fmt.Println(wm.KthSegments([][2]int{{0, 1}, {5, 10}}, 5, 0)) fmt.Println(wm.KthValueAndSumSegments([][2]int{{0, 1}, {5, 10}}, 5, 0)) fmt.Println(wm.SumAllSegments([][2]int{{0, 1}, {5, 10}})) fmt.Println(wm.MedianSegments(true, [][2]int{{0, 1}, {5, 10}}, 0)) } const INF int = 1e18 type E = int func (*WaveletMatrixForTree) e() E { return 0 } func (*WaveletMatrixForTree) op(a, b E) E { return a + b } func (*WaveletMatrixForTree) inv(a E) E { return -a } type WaveletMatrixForTree struct { n, log int mid []int bv []*BitVector preSum [][]int unit E } // // log:如果要支持异或,则需要按照异或的值来决定值域 // 设为-1时表示不使用异或 // sumData:如果要支持区间和,则需要传入前缀和数组 // 设为nil时表示不使用区间和 func NewWaveletMatrixSum(nums []E, log int, sumData []E) *WaveletMatrixForTree { res := &WaveletMatrixForTree{} res.build(nums, log, sumData) return res } // 返回区间 [left, right) 中 范围在 [a, b) 中的 元素的个数. func (wm *WaveletMatrixForTree) CountRange(left, right, a, b, xor int) int { return wm.prefixCount(left, right, b, xor) - wm.prefixCount(left, right, a, xor) } func (wm *WaveletMatrixForTree) CountRangeSegments(segments [][2]int, a, b, xor int) int { res := 0 for _, seg := range segments { res += wm.CountRange(seg[0], seg[1], a, b, xor) } return res } // 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 *WaveletMatrixForTree) KthValueAndSum(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 count := 0 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 count+kf > k { if f == 0 { left, right = l0, r0 } else { left, right = wm.mid[d]-l0+left, wm.mid[d]-r0+right } } else { var s E if f == 0 { s = wm.get(d, l0, r0) } else { s = wm.get(d, wm.mid[d]-l0+left, wm.mid[d]-r0+right) } count += kf res |= 1 << d sum = wm.op(sum, s) if f == 0 { left, right = wm.mid[d]-l0+left, wm.mid[d]-r0+right } else { left, right = l0, r0 } } } sum = wm.op(sum, wm.get(0, left, left+k-count)) return res, sum } // 如果k < 0, 返回 (-1, 0); 如果k >= segments总长, 返回 (-1, 区间 op 的结果) func (wm *WaveletMatrixForTree) KthValueAndSumSegments(segments [][2]int, k, xor int) (int, E) { if k < 0 { return -1, 0 } totalLen := 0 for _, seg := range segments { totalLen += seg[1] - seg[0] } if k >= totalLen { return -1, wm.SumAllSegments(segments) } count := 0 sum := wm.unit res := 0 for d := wm.log - 1; d >= 0; d-- { f := (xor >> d) & 1 c := 0 for _, seg := range segments { L, R := seg[0], seg[1] l0, r0 := wm.bv[d].Rank(L, 0), wm.bv[d].Rank(R, 0) c += f*(R-L-r0+l0) + (f^1)*(r0-l0) } if count+c > k { for _, seg := range segments { L, R := seg[0], seg[1] l0, r0 := wm.bv[d].Rank(L, 0), wm.bv[d].Rank(R, 0) if f == 0 { seg[0], seg[1] = L, R } else { seg[0], seg[1] = wm.mid[d]-l0+L, wm.mid[d]-r0+R } } } else { count += c res |= 1 << d for _, seg := range segments { L, R := seg[0], seg[1] l0, r0 := wm.bv[d].Rank(L, 0), wm.bv[d].Rank(R, 0) var s E if f == 0 { s = wm.get(d, l0, r0) } else { s = wm.get(d, wm.mid[d]-l0+L, wm.mid[d]-r0+R) } sum = wm.op(sum, s) if f == 0 { seg[0], seg[1] = wm.mid[d]-l0+L, wm.mid[d]-r0+R } else { seg[0], seg[1] = L, R } } } } for _, seg := range segments { L, R := seg[0], seg[1] t := min(R-L, k-count) sum = wm.op(sum, wm.get(0, L, L+t)) count += t } return res, sum } // 如果不存在,返回-1. func (wm *WaveletMatrixForTree) Kth(left, right, k, xor int) E { if k < 0 || k >= right-left { return -1 } count := 0 res := 0 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) c := f*(right-left-r0+l0) + (f^1)*(r0-l0) if count+c > k { if f == 0 { left, right = l0, r0 } else { left, right = wm.mid[d]-l0+left, wm.mid[d]-r0+right } } else { count += c res |= 1 << d if f == 0 { left, right = wm.mid[d]-l0+left, wm.mid[d]-r0+right } else { left, right = l0, r0 } } } return res } // 如果不存在,返回-1. func (wm *WaveletMatrixForTree) KthSegments(segments [][2]int, k, xor int) E { if k < 0 { return -1 } totalLen := 0 for _, seg := range segments { totalLen += seg[1] - seg[0] } if k >= totalLen { return -1 } count := 0 res := 0 for d := wm.log - 1; d >= 0; d-- { f := (xor >> d) & 1 c := 0 for _, seg := range segments { L, R := seg[0], seg[1] l0, r0 := wm.bv[d].Rank(L, 0), wm.bv[d].Rank(R, 0) c += f*(R-L-r0+l0) + (f^1)*(r0-l0) } if count+c > k { for _, seg := range segments { L, R := seg[0], seg[1] l0, r0 := wm.bv[d].Rank(L, 0), wm.bv[d].Rank(R, 0) if f == 0 { seg[0], seg[1] = l0, r0 } else { seg[0], seg[1] = wm.mid[d]-l0+L, wm.mid[d]-r0+R } } } else { count += c res |= 1 << d for _, seg := range segments { L, R := seg[0], seg[1] l0, r0 := wm.bv[d].Rank(L, 0), wm.bv[d].Rank(R, 0) if f == 0 { seg[0], seg[1] = wm.mid[d]-l0+L, wm.mid[d]-r0+R } else { seg[0], seg[1] = l0, r0 } } } } return res } // 区间中位数. // upper: true表示上中位数, false表示下中位数. func (wm *WaveletMatrixForTree) Median(upper bool, left, right, xor int) E { n := right - left var k int if upper { k = n / 2 } else { k = (n - 1) / 2 } return wm.Kth(left, right, k, xor) } func (wm *WaveletMatrixForTree) MedianSegments(upper bool, segments [][2]int, xor int) E { n := 0 for _, seg := range segments { n += seg[1] - seg[0] } var k int if upper { k = n / 2 } else { k = (n - 1) / 2 } return wm.KthSegments(segments, k, xor) } func (wm *WaveletMatrixForTree) Sum(left, right, k1, k2, xor int) E { return wm.prefixSum(left, right, k2, xor) - wm.prefixSum(left, right, k1, xor) } func (wm *WaveletMatrixForTree) SumAll(left, right int) E { return wm.get(wm.log, left, right) } func (wm *WaveletMatrixForTree) SumAllSegments(segments [][2]int) E { res := wm.unit for _, seg := range segments { res = wm.op(res, wm.get(wm.log, seg[0], seg[1])) } return res } // 返回使得 check(count,prefixSum) 为 true 的最大 (count, prefixSum) 对. // !(即区间内小于 val 的数的个数count和 和 prefixSum 满足 check函数, 找到这样的最大的 (count, prefixSum). // eg: val = 5 => 即区间内值域在 [0,5) 中的数的和满足 check 函数. func (wm *WaveletMatrixForTree) MaxRight(left, right, xor int, check func(count int, preSum E) bool) (int, E) { if tmp := wm.get(wm.log, left, right); check(right-left, tmp) { return right - left, tmp } count := 0 res := 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) c := (f << d) * (right - left - (r0 - l0)) s := wm.get(d, left+(wm.mid[d]-l0)*f, right+(wm.mid[d]-r0)*f) if check(count+c, wm.op(res, s)) { count += c res = wm.op(res, s) if f == 0 { left, right = l0, r0 } } else { if f == 0 { left, right = l0, r0 } } } k := wm.binarySearch(func(k int) bool { return check(count+k, wm.op(res, wm.get(0, left, left+k))) }, 0, right-left) count += k res = wm.op(res, wm.get(0, left, left+k)) return count, res } func (w *WaveletMatrixForTree) build(nums []E, log int, sumData []E) { numsCopy := make([]E, len(nums)) max_ := 1 for i, v := range nums { numsCopy[i] = v if v > max_ { max_ = v } } if log == -1 { log = bits.Len(uint(max_)) } makeSum := sumData != nil sumData = append(sumData[:0:0], sumData...) w.unit = w.e() n := len(numsCopy) mid := make([]int, log) bv := make([]*BitVector, log) for i := 0; i < log; i++ { bv[i] = NewBitVector(n) } var preSum [][]E if makeSum { preSum = make([][]E, log+1) for i := range preSum { preSum[i] = make([]E, n+1) for j := range preSum[i] { preSum[i][j] = w.unit } } } a0, a1 := make([]E, n), make([]E, n) s0, s1 := make([]E, n), make([]E, n) for d := log - 1; d >= -1; d-- { p0, p1 := 0, 0 if makeSum { for i := 0; i < n; i++ { preSum[d+1][i+1] = w.op(preSum[d+1][i], sumData[i]) } } if d == -1 { break } for i := 0; i < n; i++ { f := (numsCopy[i] >> d) & 1 if f == 0 { if makeSum { s0[p0] = sumData[i] } a0[p0] = numsCopy[i] p0++ } else { if makeSum { s1[p1] = sumData[i] } bv[d].Set(i) a1[p1] = numsCopy[i] p1++ } } mid[d] = p0 bv[d].Build() numsCopy, a0 = a0, numsCopy sumData, s0 = s0, sumData for i := 0; i < p1; i++ { numsCopy[p0+i] = a1[i] if makeSum { sumData[p0+i] = s1[i] } } } w.n, w.log = n, log w.mid, w.bv, w.preSum = mid, bv, preSum } func (wm *WaveletMatrixForTree) 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 } // 返回区间 [left, right) 中 范围在 [0, x) 中的 元素的个数. func (wm *WaveletMatrixForTree) prefixCount(left, right, x, xor int) int { if x == 0 { return 0 } 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 { 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 count } func (wm *WaveletMatrixForTree) prefixSum(left, right, k, xor int) E { _, res := wm.KthValueAndSum(left, right, k, xor) return res } func (wm *WaveletMatrixForTree) prefixSumSegments(segments [][2]int, k, xor int) E { _, res := wm.KthValueAndSumSegments(segments, k, xor) return res } func (wm *WaveletMatrixForTree) 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 } func min(a, b int) int { if a > b { return b } return a } type BitVector struct { data [][2]int } func NewBitVector(n int) *BitVector { return &BitVector{data: make([][2]int, (n+63)>>5)} } func (bv *BitVector) Set(i int) { bv.data[i>>5][0] |= 1 << (i & 31) } func (bv *BitVector) Build() { for i := 0; i < len(bv.data)-1; i++ { bv.data[i+1][1] = bv.data[i][1] + bits.OnesCount(uint(bv.data[i][0])) } } func (bv *BitVector) Rank(k int, f int) int { a, b := bv.data[k>>5][0], bv.data[k>>5][1] ret := b + bits.OnesCount(uint(a&((1<<(k&31))-1))) if f == 1 { return ret } return k - ret }