The Median Maintenance Problem: Difference between revisions
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The problem can be resolved by repeatedly solving the [[Selection_Problem#Overview|selection problem]] on the set of numbers seen so far, but the selection problem has a running time of O(n), so repeating the selection algorithm for each number will have a running time of O(n<sup>2</sup>). The median maintenance problem is a canonical use case for a heap. | The problem can be resolved by repeatedly solving the [[Selection_Problem#Overview|selection problem]] on the set of numbers seen so far, but the selection problem has a running time of O(n), so repeating the selection algorithm for each number will have a running time of O(n<sup>2</sup>). The median maintenance problem is a canonical use case for a heap. | ||
=Solution= | =Solution= | ||
Use two heaps: a [[Heap#Min_Heap_and_Max_Heap|max heap]] H<sub>low</sub> that supports the <tt>[[Heap#REMOVE-MAX|REMOVE-MAX]]</tt> operation and a [[Heap#Min_Heap_and_Max_Heap|min heap]] H<sub>high</sub> that supports the <tt>[[Heap#REMOVE-MIN|REMOVE-MIN]]</tt> operation. Maintain the first smallest half elements in H<sub>low</sub> | Use two heaps: a [[Heap#Min_Heap_and_Max_Heap|max heap]] H<sub>low</sub> that supports the <tt>[[Heap#REMOVE-MAX|REMOVE-MAX]]</tt> operation and a [[Heap#Min_Heap_and_Max_Heap|min heap]] H<sub>high</sub> that supports the <tt>[[Heap#REMOVE-MIN|REMOVE-MIN]]</tt> operation. | ||
Maintain the first smallest half elements in H<sub>low</sub> and the largest half of the elements in H<sub>high</sub>. When a new element comes, if its smaller than the current median insert it in H<sub>low</sub>, if it is higher than the current median, insert it in H<sub>high</sub>. Keep track of heap sizes and rebalance when necessary, so the total number of elements is evenly split among the heaps. | |||
More details: {{External|https://www.coursera.org/learn/algorithms-graphs-data-structures/lecture/iIzo8/heaps-operations-and-applications}} | |||
Revision as of 22:40, 9 October 2021
Internal
Problem
Give a sequence of numbers x1, x2, .... xn, report at each step the median number of numbers seen so far, at each step i.
Discussion
The problem can be resolved by repeatedly solving the selection problem on the set of numbers seen so far, but the selection problem has a running time of O(n), so repeating the selection algorithm for each number will have a running time of O(n2). The median maintenance problem is a canonical use case for a heap.
Solution
Use two heaps: a max heap Hlow that supports the REMOVE-MAX operation and a min heap Hhigh that supports the REMOVE-MIN operation.
Maintain the first smallest half elements in Hlow and the largest half of the elements in Hhigh. When a new element comes, if its smaller than the current median insert it in Hlow, if it is higher than the current median, insert it in Hhigh. Keep track of heap sizes and rebalance when necessary, so the total number of elements is evenly split among the heaps.
More details: