Find running median from a stream of integers

Question

Possible Duplicate    Rolling median algorithm in C          Given that integers are read from a data stream  Find median of elements read so far in efficient way     Solution I have read  We can use a max heap on left side to represent elements that are less than the effective median  and a min heap on right side to represent elements that are greater than the effective median   After processing an incoming element  the number of elements in heaps differ at most by 1 element  When both heaps contain the same number of elements  we find the average of heap s root data as effective median  When the heaps are not balanced  we select the effective median from the root of heap containing more elements   But how would we construct a max heap and min heap i e  how would we know the effective median here  I think that we would insert 1 element in max-heap and then the next 1 element in min-heap  and so on for all the elements  Correct me If I am wrong here

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Efficient is a word that depends on context. The solution to this problem depends on the amount of queries performed relative to the amount of insertions. Suppose you are inserting N numbers and K times towards the end you were interested in the median. The heap based algorithm's complexity would be O(N log N + K).

Consider the following alternative. Plunk the numbers in an array, and for each query, run the linear selection algorithm (using the quicksort pivot, say). Now you have an algorithm with running time O(K N).

Now if K is sufficiently small (infrequent queries), the latter algorithm is actually more efficient and vice versa.

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If you can t hold all the items in memory at once  this problem becomes much harder  The heap solution requires you to hold all the elements in memory at once  This is not possible in most real world applications of this problem    Instead  as you see numbers  keep track of the count of the number of times you see each integer  Assuming 4 byte integers  that s 2 32 buckets  or at most 2 33 integers  key and count for each int   which is 2 35 bytes or 32GB  It will likely be much less than this because you don t need to store the key or count for those entries that are 0  ie  like a defaultdict in python   This takes constant time to insert each new integer   Then at any point  to find the median  just use the counts to determine which integer is the middle element  This takes constant time  albeit a large constant  but constant nonetheless

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An intuitive way to think about this is that if you had a full balanced binary search tree  then the root would be the median element  since there there would be the same number of smaller and greater elements   Now  if the tree isn t full this won t be quite the case since there will be elements missing from the last level    So what we can do instead is have the median  and two balanced binary trees  one for elements less than the median  and one for elements greater than the median  The two trees must be kept at the same size   When we get a new integer from the data stream  we compare it to the median  If it s greater than the median  we add it to the right tree  If the two tree sizes differ more than 1  we remove the min element of the right tree  make it the new median  and put the old median in the left tree  Similarly for smaller

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Here is my simple but efficient algorithm  in C    for calculating running median from a stream of integers   include lt algorithm gt   include lt fstream gt   include lt vector gt   include lt list gt   using namespace std   void runningMedian std  ifstream amp  ifs  std  ofstream amp  ofs  const unsigned bufSize        if  bufSize  lt  1          throw exception  quot Wrong buffer size  quot        bool evenSize   bufSize   2    0   true   false      list lt int gt  q      vector lt int gt  nums      int n      unsigned count   0      while  ifs good              ifs  gt  gt  n          q push back n           auto ub   std  upper bound nums begin    nums end    n           nums insert ub  n           count            if  nums size    gt   bufSize                auto it   std  find nums begin    nums end    q front                 nums erase it               q pop front                if  evenSize                  ofs  lt  lt  count  lt  lt   quot    quot   lt  lt   static cast lt double gt  nums nums size     2 - 1                    static cast lt double gt  nums nums size     2       2 0  lt  lt    n               else                 ofs  lt  lt  count  lt  lt   quot    quot   lt  lt  static cast lt double gt  nums nums size     2                       The bufferSize specifies the size of the numbers sequence  on which the running median must be calculated  When reading numbers from the input stream ifs the vector of the size bufferSize is maintained in sorted order  The median is calculated by taking the middle of the sorted vector  if bufferSize is odd  or the sum of the two middle elements divided by 2  when bufferSize is even  Additinally  I maintain a list of last bufferSize elements read from input  When a new element is added  I put it in the right place in sorted vector and remove from the vector the element added bufferSize steps before  the value of the element retained in the front of the list   In the same time I remove the old element from the list  every new element is placed on the back of the list  every old element is removed from the front  After reaching the bufferSize  both the list and the vector stop to grow  and every insertion of a new element is compensated be deletion of an old element  placed in the list bufferSize steps before  Note  I do not care  whether I remove from the vector exactly the element  placed bufferSize steps before  or just an element that has the same value  For the value of median it does not matter  All calculated median values are output in the output stream

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The most efficient way to calculate a percentile of a stream that I have found is the P   algorithm  Raj Jain  Imrich Chlamtac  The P   Algorithm for Dynamic Calculation of Quantiiles and Histograms Without Storing Observations  Commun  ACM 28 10   1076-1085  1985   The algorithm is straight forward to implement and works extremely well   It is an estimate  however  so keep that in mind   From the abstract      A heuristic algorithm is proposed for dynamic calculation qf the median and other quantiles  The estimates are produced dynamically as the observations are generated  The observations are not stored  therefore  the algorithm has a very small and fixed storage requirement regardless of the number of observations  This makes it ideal for implementing in a quantile chip that can be used in industrial controllers and recorders  The algorithm is further extended to histogram plotting  The accuracy of the algorithm is analyzed

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If the variance of the input is statistically distributed  e g  normal  log-normal  etc   then reservoir sampling is a reasonable way of estimating percentiles medians from an arbitrarily long stream of numbers   int n   0      Running count of elements observed so far    define SIZE 10000 int reservoir SIZE      while streamHasData        int x   readNumberFromStream       if  n  lt  SIZE             reservoir n      x                 else            int p   random   n      Choose a random number 0  gt   p  lt  n       if  p  lt  SIZE                     reservoir p    x                   reservoir  is then a running  uniform  fair   sample of all input - regardless of size  Finding the median  or any percentile  is then a straight-forward matter of sorting the reservoir and polling the interesting point    Since the reservoir is fixed size  the sort can be considered to be effectively O 1  - and this method runs with both constant time and memory consumption

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Can t you do this with just one heap  Update  no  See the comment   Invariant  After reading 2 n inputs  the min-heap holds the n largest of them   Loop  Read 2 inputs  Add them both to the heap  and remove the heap s min  This reestablishes the invariant   So when 2n inputs have been read  the heap s min is the nth largest  There ll need to be a little extra complication to average the two elements around the median position and to handle queries after an odd number of inputs

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If we want to find the median of the n most recently seen elements  this problem has an exact solution that only needs the n most recently seen elements to be kept in memory   It is fast and scales well   An indexable skiplist supports O ln n  insertion  removal  and indexed search of arbitrary elements while maintaining sorted order    When coupled with a FIFO queue that tracks the n-th oldest entry  the solution is simple   class RunningMedian       Fast running median with O lg n  updates where n is the window size       def   init   self  n  iterable           self it   iter iterable          self queue   deque islice self it  n           self skiplist   IndexableSkiplist n          for elem in self queue              self skiplist insert elem       def   iter   self           queue   self queue         skiplist   self skiplist         midpoint   len queue     2         yield skiplist midpoint          for newelem in self it              oldelem   queue popleft               skiplist remove oldelem              queue append newelem              skiplist insert newelem              yield skiplist midpoint    Here are links to complete working code  an easy-to-understand class version and an optimized generator version with the indexable skiplist code inlined     http   code activestate com recipes 576930-efficient-running-median-using-an-indexable-skipli  http   code activestate com recipes 577073

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There are a number of different solutions for finding running median from streamed data  I will briefly talk about them at the very end of the answer    The question is about the details of the a specific solution  max heap min heap solution   and how heap based solution works is explained below   For the first two elements add smaller one to the maxHeap on the left  and bigger one to the minHeap on the right  Then process stream data one by one    Step 1  Add next item to one of the heaps     if next item is smaller than maxHeap root add it to maxHeap     else add it to minHeap  Step 2  Balance the heaps  after this step heaps will be either balanced or    one of them will contain 1 more item      if number of elements in one of the heaps is greater than the other by    more than 1  remove the root element from the one containing more elements and    add to the other one   Then at any given time you can calculate median like this      If the heaps contain equal amount of elements       median    root of maxHeap   root of minHeap  2    Else      median   root of the heap with more elements   Now I will talk about the problem in general as promised in the beginning of the answer  Finding running median from a stream of data is a tough problem  and finding an exact solution with memory constraints efficiently is probably impossible for the general case  On the other hand  if the data has some characteristics we can exploit  we can develop efficient specialized solutions  For example  if we know that the data is an integral type  then we can use counting sort  which can give you a constant memory constant time algorithm  Heap based solution is a more general solution because it can be used for other data types  doubles  as well  And finally  if the exact median is not required and an approximation is enough  you can just try to estimate a probability density function for the data and estimate median using that

[algorithm] Find running median from a stream of integers

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