Question

480. Sliding Window Median

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Description

The median is the middle value in an ordered integer list. If the size of the list is even, there is no middle value. So the median is the mean of the two middle values.

• For examples, if arr = [2,3,4], the median is 3.
• For examples, if arr = [1,2,3,4], the median is (2 + 3) / 2 = 2.5.

You are given an integer array nums and an integer k. There is a sliding window of size k which is moving from the very left of the array to the very right. You can only see the k numbers in the window. Each time the sliding window moves right by one position.

Return _the median array for each window in the original array_. Answers within 10-5 of the actual value will be accepted.

 

Example 1:

Input: nums = [1,3,-1,-3,5,3,6,7], k = 3
Output: [1.00000,-1.00000,-1.00000,3.00000,5.00000,6.00000]
Explanation: 
Window position        Median
---------------        -----
[1  3  -1] -3  5  3  6  7    1
 1 [3  -1  -3] 5  3  6  7     -1
 1  3 [-1  -3  5] 3  6  7     -1
 1  3  -1 [-3  5  3] 6  7    3
 1  3  -1  -3 [5  3  6] 7    5
 1  3  -1  -3  5 [3  6  7]     6

Example 2:

Input: nums = [1,2,3,4,2,3,1,4,2], k = 3
Output: [2.00000,3.00000,3.00000,3.00000,2.00000,3.00000,2.00000]

 

Constraints:

• 1 <= k <= nums.length <= 105
• -231 <= nums[i] <= 231 - 1

Solutions

Solution 1

class MedianFinder:
  def __init__(self, k: int):
    self.k = k
    self.small = []
    self.large = []
    self.delayed = defaultdict(int)
    self.small_size = 0
    self.large_size = 0

  def add_num(self, num: int):
    if not self.small or num <= -self.small[0]:
      heappush(self.small, -num)
      self.small_size += 1
    else:
      heappush(self.large, num)
      self.large_size += 1
    self.rebalance()

  def find_median(self) -> float:
    return -self.small[0] if self.k & 1 else (-self.small[0] + self.large[0]) / 2

  def remove_num(self, num: int):
    self.delayed[num] += 1
    if num <= -self.small[0]:
      self.small_size -= 1
      if num == -self.small[0]:
        self.prune(self.small)
    else:
      self.large_size -= 1
      if num == self.large[0]:
        self.prune(self.large)
    self.rebalance()

  def prune(self, pq: List[int]):
    sign = -1 if pq is self.small else 1
    while pq and sign * pq[0] in self.delayed:
      self.delayed[sign * pq[0]] -= 1
      if self.delayed[sign * pq[0]] == 0:
        self.delayed.pop(sign * pq[0])
      heappop(pq)

  def rebalance(self):
    if self.small_size > self.large_size + 1:
      heappush(self.large, -heappop(self.small))
      self.small_size -= 1
      self.large_size += 1
      self.prune(self.small)
    elif self.small_size < self.large_size:
      heappush(self.small, -heappop(self.large))
      self.large_size -= 1
      self.small_size += 1
      self.prune(self.large)


class Solution:
  def medianSlidingWindow(self, nums: List[int], k: int) -> List[float]:
    finder = MedianFinder(k)
    for x in nums[:k]:
      finder.add_num(x)
    ans = [finder.find_median()]
    for i in range(k, len(nums)):
      finder.add_num(nums[i])
      finder.remove_num(nums[i - k])
      ans.append(finder.find_median())
    return ans

class MedianFinder {
  private PriorityQueue<Integer> small = new PriorityQueue<>(Comparator.reverseOrder());
  private PriorityQueue<Integer> large = new PriorityQueue<>();
  private Map<Integer, Integer> delayed = new HashMap<>();
  private int smallSize;
  private int largeSize;
  private int k;

  public MedianFinder(int k) {
    this.k = k;
  }

  public void addNum(int num) {
    if (small.isEmpty() || num <= small.peek()) {
      small.offer(num);
      ++smallSize;
    } else {
      large.offer(num);
      ++largeSize;
    }
    rebalance();
  }

  public double findMedian() {
    return (k & 1) == 1 ? small.peek() : ((double) small.peek() + large.peek()) / 2;
  }

  public void removeNum(int num) {
    delayed.merge(num, 1, Integer::sum);
    if (num <= small.peek()) {
      --smallSize;
      if (num == small.peek()) {
        prune(small);
      }
    } else {
      --largeSize;
      if (num == large.peek()) {
        prune(large);
      }
    }
    rebalance();
  }

  private void prune(PriorityQueue<Integer> pq) {
    while (!pq.isEmpty() && delayed.containsKey(pq.peek())) {
      if (delayed.merge(pq.peek(), -1, Integer::sum) == 0) {
        delayed.remove(pq.peek());
      }
      pq.poll();
    }
  }

  private void rebalance() {
    if (smallSize > largeSize + 1) {
      large.offer(small.poll());
      --smallSize;
      ++largeSize;
      prune(small);
    } else if (smallSize < largeSize) {
      small.offer(large.poll());
      --largeSize;
      ++smallSize;
      prune(large);
    }
  }
}

class Solution {
  public double[] medianSlidingWindow(int[] nums, int k) {
    MedianFinder finder = new MedianFinder(k);
    for (int i = 0; i < k; ++i) {
      finder.addNum(nums[i]);
    }
    int n = nums.length;
    double[] ans = new double[n - k + 1];
    ans[0] = finder.findMedian();
    for (int i = k; i < n; ++i) {
      finder.addNum(nums[i]);
      finder.removeNum(nums[i - k]);
      ans[i - k + 1] = finder.findMedian();
    }
    return ans;
  }
}

class MedianFinder {
public:
  MedianFinder(int k) {
    this->k = k;
  }

  void addNum(int num) {
    if (small.empty() || num <= small.top()) {
      small.push(num);
      ++smallSize;
    } else {
      large.push(num);
      ++largeSize;
    }
    reblance();
  }

  void removeNum(int num) {
    ++delayed[num];
    if (num <= small.top()) {
      --smallSize;
      if (num == small.top()) {
        prune(small);
      }
    } else {
      --largeSize;
      if (num == large.top()) {
        prune(large);
      }
    }
    reblance();
  }

  double findMedian() {
    return k & 1 ? small.top() : ((double) small.top() + large.top()) / 2.0;
  }

private:
  priority_queue<int> small;
  priority_queue<int, vector<int>, greater<int>> large;
  unordered_map<int, int> delayed;
  int smallSize = 0;
  int largeSize = 0;
  int k;

  template <typename T>
  void prune(T& pq) {
    while (!pq.empty() && delayed[pq.top()]) {
      if (--delayed[pq.top()] == 0) {
        delayed.erase(pq.top());
      }
      pq.pop();
    }
  }

  void reblance() {
    if (smallSize > largeSize + 1) {
      large.push(small.top());
      small.pop();
      --smallSize;
      ++largeSize;
      prune(small);
    } else if (smallSize < largeSize) {
      small.push(large.top());
      large.pop();
      ++smallSize;
      --largeSize;
      prune(large);
    }
  }
};

class Solution {
public:
  vector<double> medianSlidingWindow(vector<int>& nums, int k) {
    MedianFinder finder(k);
    for (int i = 0; i < k; ++i) {
      finder.addNum(nums[i]);
    }
    vector<double> ans = {finder.findMedian()};
    for (int i = k; i < nums.size(); ++i) {
      finder.addNum(nums[i]);
      finder.removeNum(nums[i - k]);
      ans.push_back(finder.findMedian());
    }
    return ans;
  }
};

type MedianFinder struct {
  small        hp
  large        hp
  delayed        map[int]int
  smallSize, largeSize int
  k          int
}

func Constructor(k int) MedianFinder {
  return MedianFinder{hp{}, hp{}, map[int]int{}, 0, 0, k}
}

func (this *MedianFinder) AddNum(num int) {
  if this.small.Len() == 0 || num <= -this.small.IntSlice[0] {
    heap.Push(&this.small, -num)
    this.smallSize++
  } else {
    heap.Push(&this.large, num)
    this.largeSize++
  }
  this.rebalance()
}

func (this *MedianFinder) FindMedian() float64 {
  if this.k&1 == 1 {
    return float64(-this.small.IntSlice[0])
  }
  return float64(-this.small.IntSlice[0]+this.large.IntSlice[0]) / 2
}

func (this *MedianFinder) removeNum(num int) {
  this.delayed[num]++
  if num <= -this.small.IntSlice[0] {
    this.smallSize--
    if num == -this.small.IntSlice[0] {
      this.prune(&this.small)
    }
  } else {
    this.largeSize--
    if num == this.large.IntSlice[0] {
      this.prune(&this.large)
    }
  }
  this.rebalance()
}

func (this *MedianFinder) prune(pq *hp) {
  sign := 1
  if pq == &this.small {
    sign = -1
  }
  for pq.Len() > 0 && this.delayed[sign*pq.IntSlice[0]] > 0 {
    this.delayed[sign*pq.IntSlice[0]]--
    if this.delayed[sign*pq.IntSlice[0]] == 0 {
      delete(this.delayed, sign*pq.IntSlice[0])
    }
    heap.Pop(pq)
  }
}

func (this *MedianFinder) rebalance() {
  if this.smallSize > this.largeSize+1 {
    heap.Push(&this.large, -heap.Pop(&this.small).(int))
    this.smallSize--
    this.largeSize++
    this.prune(&this.small)
  } else if this.smallSize < this.largeSize {
    heap.Push(&this.small, -heap.Pop(&this.large).(int))
    this.smallSize++
    this.largeSize--
    this.prune(&this.large)
  }
}

func medianSlidingWindow(nums []int, k int) []float64 {
  finder := Constructor(k)
  for _, num := range nums[:k] {
    finder.AddNum(num)
  }
  ans := []float64{finder.FindMedian()}
  for i := k; i < len(nums); i++ {
    finder.AddNum(nums[i])
    finder.removeNum(nums[i-k])
    ans = append(ans, finder.FindMedian())
  }
  return ans
}

type hp struct{ sort.IntSlice }

func (h hp) Less(i, j int) bool { return h.IntSlice[i] < h.IntSlice[j] }
func (h *hp) Push(v any)    { h.IntSlice = append(h.IntSlice, v.(int)) }
func (h *hp) Pop() any {
  a := h.IntSlice
  v := a[len(a)-1]
  h.IntSlice = a[:len(a)-1]
  return v
}