Question

1157. 子数组中占绝大多数的元素

English Version

题目描述

设计一个数据结构，有效地找到给定子数组的 多数元素 。

子数组的 多数元素 是在子数组中出现 threshold 次数或次数以上的元素。

实现 MajorityChecker 类:

• MajorityChecker(int[] arr) 会用给定的数组 arr 对 MajorityChecker 初始化。
• int query(int left, int right, int threshold) 返回子数组中的元素  arr[left...right] 至少出现 threshold 次数，如果不存在这样的元素则返回 -1。

 

示例 1：

输入:
["MajorityChecker", "query", "query", "query"]
[[[1, 1, 2, 2, 1, 1]], [0, 5, 4], [0, 3, 3], [2, 3, 2]]
输出：
[null, 1, -1, 2]

解释：
MajorityChecker majorityChecker = new MajorityChecker([1,1,2,2,1,1]);
majorityChecker.query(0,5,4); // 返回 1
majorityChecker.query(0,3,3); // 返回 -1
majorityChecker.query(2,3,2); // 返回 2

 

提示：

• 1 <= arr.length <= 2 * 104
• 1 <= arr[i] <= 2 * 104
• 0 <= left <= right < arr.length
• threshold <= right - left + 1
• 2 * threshold > right - left + 1
• 调用 query 的次数最多为 104 

解法

方法一：线段树 + 摩尔投票 + 二分查找

我们注意到，题目需要我们找出特定区间内可能的众数，考虑使用线段树来维护每个区间内的候选众数以及其出现的次数。

我们定义线段树的每个节点为 Node，每个节点包含如下属性：

• l：节点的左端点，下标从 $1$ 开始。
• r：节点的右端点，下标从 $1$ 开始。
• x：节点的候选众数。
• cnt：节点的候选众数出现的次数。

线段树主要有以下几个操作：

• build(u, l, r)：建立线段树。
• pushup(u)：用子节点的信息更新父节点的信息。
• query(u, l, r)：查询区间和。

在主函数的初始化方法中，我们先创建一个线段树，并且用哈希表 $d$ 记录每个元素在数组中的所有下标。

在 query(left, right, threshold) 方法中，我们直接调用线段树的 query 方法，得到候选众数 $x$。然后使用二分查找，找到 $x$ 在数组中第一个大于等于 $left$ 的下标 $l$，以及第一个大于 $right$ 的下标 $r$。如果 $r - l \ge threshold$，则返回 $x$，否则返回 $-1$。

时间复杂度方面，初始化方法的时间复杂度为 $O(n)$，查询方法的时间复杂度为 $O(\log n)$。空间复杂度为 $O(n)$。其中 $n$ 为数组的长度。

class Node:
  __slots__ = ("l", "r", "x", "cnt")

  def __init__(self):
    self.l = self.r = 0
    self.x = self.cnt = 0


class SegmentTree:
  def __init__(self, nums):
    self.nums = nums
    n = len(nums)
    self.tr = [Node() for _ in range(n << 2)]
    self.build(1, 1, n)

  def build(self, u, l, r):
    self.tr[u].l, self.tr[u].r = l, r
    if l == r:
      self.tr[u].x = self.nums[l - 1]
      self.tr[u].cnt = 1
      return
    mid = (l + r) >> 1
    self.build(u << 1, l, mid)
    self.build(u << 1 | 1, mid + 1, r)
    self.pushup(u)

  def query(self, u, l, r):
    if self.tr[u].l >= l and self.tr[u].r <= r:
      return self.tr[u].x, self.tr[u].cnt
    mid = (self.tr[u].l + self.tr[u].r) >> 1
    if r <= mid:
      return self.query(u << 1, l, r)
    if l > mid:
      return self.query(u << 1 | 1, l, r)
    x1, cnt1 = self.query(u << 1, l, r)
    x2, cnt2 = self.query(u << 1 | 1, l, r)
    if x1 == x2:
      return x1, cnt1 + cnt2
    if cnt1 >= cnt2:
      return x1, cnt1 - cnt2
    else:
      return x2, cnt2 - cnt1

  def pushup(self, u):
    if self.tr[u << 1].x == self.tr[u << 1 | 1].x:
      self.tr[u].x = self.tr[u << 1].x
      self.tr[u].cnt = self.tr[u << 1].cnt + self.tr[u << 1 | 1].cnt
    elif self.tr[u << 1].cnt >= self.tr[u << 1 | 1].cnt:
      self.tr[u].x = self.tr[u << 1].x
      self.tr[u].cnt = self.tr[u << 1].cnt - self.tr[u << 1 | 1].cnt
    else:
      self.tr[u].x = self.tr[u << 1 | 1].x
      self.tr[u].cnt = self.tr[u << 1 | 1].cnt - self.tr[u << 1].cnt


class MajorityChecker:
  def __init__(self, arr: List[int]):
    self.tree = SegmentTree(arr)
    self.d = defaultdict(list)
    for i, x in enumerate(arr):
      self.d[x].append(i)

  def query(self, left: int, right: int, threshold: int) -> int:
    x, _ = self.tree.query(1, left + 1, right + 1)
    l = bisect_left(self.d[x], left)
    r = bisect_left(self.d[x], right + 1)
    return x if r - l >= threshold else -1


# Your MajorityChecker object will be instantiated and called as such:
# obj = MajorityChecker(arr)
# param_1 = obj.query(left,right,threshold)

class Node {
  int l, r;
  int x, cnt;
}

class SegmentTree {
  private Node[] tr;
  private int[] nums;

  public SegmentTree(int[] nums) {
    int n = nums.length;
    this.nums = nums;
    tr = new Node[n << 2];
    for (int i = 0; i < tr.length; ++i) {
      tr[i] = new Node();
    }
    build(1, 1, n);
  }

  private void build(int u, int l, int r) {
    tr[u].l = l;
    tr[u].r = r;
    if (l == r) {
      tr[u].x = nums[l - 1];
      tr[u].cnt = 1;
      return;
    }
    int mid = (l + r) >> 1;
    build(u << 1, l, mid);
    build(u << 1 | 1, mid + 1, r);
    pushup(u);
  }

  public int[] query(int u, int l, int r) {
    if (tr[u].l >= l && tr[u].r <= r) {
      return new int[] {tr[u].x, tr[u].cnt};
    }
    int mid = (tr[u].l + tr[u].r) >> 1;
    if (r <= mid) {
      return query(u << 1, l, r);
    }
    if (l > mid) {
      return query(u << 1 | 1, l, r);
    }
    var left = query(u << 1, l, r);
    var right = query(u << 1 | 1, l, r);
    if (left[0] == right[0]) {
      left[1] += right[1];
    } else if (left[1] >= right[1]) {
      left[1] -= right[1];
    } else {
      right[1] -= left[1];
      left = right;
    }
    return left;
  }

  private void pushup(int u) {
    if (tr[u << 1].x == tr[u << 1 | 1].x) {
      tr[u].x = tr[u << 1].x;
      tr[u].cnt = tr[u << 1].cnt + tr[u << 1 | 1].cnt;
    } else if (tr[u << 1].cnt >= tr[u << 1 | 1].cnt) {
      tr[u].x = tr[u << 1].x;
      tr[u].cnt = tr[u << 1].cnt - tr[u << 1 | 1].cnt;
    } else {
      tr[u].x = tr[u << 1 | 1].x;
      tr[u].cnt = tr[u << 1 | 1].cnt - tr[u << 1].cnt;
    }
  }
}

class MajorityChecker {
  private SegmentTree tree;
  private Map<Integer, List<Integer>> d = new HashMap<>();

  public MajorityChecker(int[] arr) {
    tree = new SegmentTree(arr);
    for (int i = 0; i < arr.length; ++i) {
      d.computeIfAbsent(arr[i], k -> new ArrayList<>()).add(i);
    }
  }

  public int query(int left, int right, int threshold) {
    int x = tree.query(1, left + 1, right + 1)[0];
    int l = search(d.get(x), left);
    int r = search(d.get(x), right + 1);
    return r - l >= threshold ? x : -1;
  }

  private int search(List<Integer> arr, int x) {
    int left = 0, right = arr.size();
    while (left < right) {
      int mid = (left + right) >> 1;
      if (arr.get(mid) >= x) {
        right = mid;
      } else {
        left = mid + 1;
      }
    }
    return left;
  }
}

/**
 * Your MajorityChecker object will be instantiated and called as such:
 * MajorityChecker obj = new MajorityChecker(arr);
 * int param_1 = obj.query(left,right,threshold);
 */

class Node {
public:
  int l = 0, r = 0;
  int x = 0, cnt = 0;
};

using pii = pair<int, int>;

class SegmentTree {
public:
  SegmentTree(vector<int>& nums) {
    this->nums = nums;
    int n = nums.size();
    tr.resize(n << 2);
    for (int i = 0; i < tr.size(); ++i) {
      tr[i] = new Node();
    }
    build(1, 1, n);
  }

  pii query(int u, int l, int r) {
    if (tr[u]->l >= l && tr[u]->r <= r) {
      return {tr[u]->x, tr[u]->cnt};
    }
    int mid = (tr[u]->l + tr[u]->r) >> 1;
    if (r <= mid) {
      return query(u << 1, l, r);
    }
    if (l > mid) {
      return query(u << 1 | 1, l, r);
    }
    auto left = query(u << 1, l, r);
    auto right = query(u << 1 | 1, l, r);
    if (left.first == right.first) {
      left.second += right.second;
    } else if (left.second >= right.second) {
      left.second -= right.second;
    } else {
      right.second -= left.second;
      left = right;
    }
    return left;
  }

private:
  vector<Node*> tr;
  vector<int> nums;

  void build(int u, int l, int r) {
    tr[u]->l = l;
    tr[u]->r = r;
    if (l == r) {
      tr[u]->x = nums[l - 1];
      tr[u]->cnt = 1;
      return;
    }
    int mid = (l + r) >> 1;
    build(u << 1, l, mid);
    build(u << 1 | 1, mid + 1, r);
    pushup(u);
  }

  void pushup(int u) {
    if (tr[u << 1]->x == tr[u << 1 | 1]->x) {
      tr[u]->x = tr[u << 1]->x;
      tr[u]->cnt = tr[u << 1]->cnt + tr[u << 1 | 1]->cnt;
    } else if (tr[u << 1]->cnt >= tr[u << 1 | 1]->cnt) {
      tr[u]->x = tr[u << 1]->x;
      tr[u]->cnt = tr[u << 1]->cnt - tr[u << 1 | 1]->cnt;
    } else {
      tr[u]->x = tr[u << 1 | 1]->x;
      tr[u]->cnt = tr[u << 1 | 1]->cnt - tr[u << 1]->cnt;
    }
  }
};

class MajorityChecker {
public:
  MajorityChecker(vector<int>& arr) {
    tree = new SegmentTree(arr);
    for (int i = 0; i < arr.size(); ++i) {
      d[arr[i]].push_back(i);
    }
  }

  int query(int left, int right, int threshold) {
    int x = tree->query(1, left + 1, right + 1).first;
    auto l = lower_bound(d[x].begin(), d[x].end(), left);
    auto r = lower_bound(d[x].begin(), d[x].end(), right + 1);
    return r - l >= threshold ? x : -1;
  }

private:
  unordered_map<int, vector<int>> d;
  SegmentTree* tree;
};

/**
 * Your MajorityChecker object will be instantiated and called as such:
 * MajorityChecker* obj = new MajorityChecker(arr);
 * int param_1 = obj->query(left,right,threshold);
 */

type node struct {
  l, r, x, cnt int
}

type segmentTree struct {
  nums []int
  tr   []*node
}

type pair struct{ x, cnt int }

func newSegmentTree(nums []int) *segmentTree {
  n := len(nums)
  tr := make([]*node, n<<2)
  for i := range tr {
    tr[i] = &node{}
  }
  t := &segmentTree{nums, tr}
  t.build(1, 1, n)
  return t
}

func (t *segmentTree) build(u, l, r int) {
  t.tr[u].l, t.tr[u].r = l, r
  if l == r {
    t.tr[u].x = t.nums[l-1]
    t.tr[u].cnt = 1
    return
  }
  mid := (l + r) >> 1
  t.build(u<<1, l, mid)
  t.build(u<<1|1, mid+1, r)
  t.pushup(u)
}

func (t *segmentTree) query(u, l, r int) pair {
  if t.tr[u].l >= l && t.tr[u].r <= r {
    return pair{t.tr[u].x, t.tr[u].cnt}
  }
  mid := (t.tr[u].l + t.tr[u].r) >> 1
  if r <= mid {
    return t.query(u<<1, l, r)
  }
  if l > mid {
    return t.query(u<<1|1, l, r)
  }
  left, right := t.query(u<<1, l, r), t.query(u<<1|1, l, r)
  if left.x == right.x {
    left.cnt += right.cnt
  } else if left.cnt >= right.cnt {
    left.cnt -= right.cnt
  } else {
    right.cnt -= left.cnt
    left = right
  }
  return left
}

func (t *segmentTree) pushup(u int) {
  if t.tr[u<<1].x == t.tr[u<<1|1].x {
    t.tr[u].x = t.tr[u<<1].x
    t.tr[u].cnt = t.tr[u<<1].cnt + t.tr[u<<1|1].cnt
  } else if t.tr[u<<1].cnt >= t.tr[u<<1|1].cnt {
    t.tr[u].x = t.tr[u<<1].x
    t.tr[u].cnt = t.tr[u<<1].cnt - t.tr[u<<1|1].cnt
  } else {
    t.tr[u].x = t.tr[u<<1|1].x
    t.tr[u].cnt = t.tr[u<<1|1].cnt - t.tr[u<<1].cnt
  }
}

type MajorityChecker struct {
  tree *segmentTree
  d  map[int][]int
}

func Constructor(arr []int) MajorityChecker {
  tree := newSegmentTree(arr)
  d := map[int][]int{}
  for i, x := range arr {
    d[x] = append(d[x], i)
  }
  return MajorityChecker{tree, d}
}

func (this *MajorityChecker) Query(left int, right int, threshold int) int {
  x := this.tree.query(1, left+1, right+1).x
  l := sort.SearchInts(this.d[x], left)
  r := sort.SearchInts(this.d[x], right+1)
  if r-l >= threshold {
    return x
  }
  return -1
}

/**
 * Your MajorityChecker object will be instantiated and called as such:
 * obj := Constructor(arr);
 * param_1 := obj.Query(left,right,threshold);
 */