In computer science, a shadow heap is a mergeable heap data structure which supports efficient heap merging in the amortized sense. More specifically, shadow heaps make use of the shadow merge algorithm to achieve insertion in O(f(n)) amortized time and deletion in O((log n log log n)/f(n)) amortized time, for any choice of 1 ≤ f(n) ≤ log log n.[1]

Throughout this article, it is assumed that A and B are binary heaps with |A| ≤ |B|.

Shadow merge

edit

Shadow merge is an algorithm for merging two binary heaps efficiently if these heaps are implemented as arrays. Specifically, the running time of shadow merge on two heaps   and   is  .

Algorithm

edit

We wish to merge the two binary min-heaps   and  . The algorithm is as follows:

  1. Concatenate the array   at the end of the array   to obtain an array  .
  2. Identify the shadow of   in  ; that is, the ancestors of the last   nodes in   which destroy the heap property.
  3. Identify the following two parts of the shadow from  :
    • The path  : the set of nodes in the shadow for which there are at most 2 at any depth of  ;
    • The subtree  : the remainder of the shadow.
  4. Extract and sort the smallest   nodes from the shadow into an array  .
  5. Transform   as follows:
    • If  , then starting from the smallest element in the sorted array, sequentially insert each element of   into  , replacing them with  's smallest elements.
    • If  , then extract and sort the   smallest elements from  , and merge this sorted list with  .
  6. Replace the elements of   into their original positions in  .
  7. Make a heap out of  .

Running time

edit

Again, let   denote the path, and   denote the subtree of the concatenated heap  . The number of nodes in   is at most twice the depth of  , which is  . Moreover, the number of nodes in   at depth   is at most 3/4 the number of nodes at depth  , so the subtree has size  . Since there are at most 2 nodes at each level on  , then reading the smallest   elements of the shadow into the sorted array   takes   time.

If  , then combining   and   as in step 5 above takes time  . Otherwise, the time taken in this step is  . Finally, making a heap of the subtree   takes   time. This amounts to a total running time for shadow merging of  .

Structure

edit

A shadow heap   consists of threshold function  , and an array for which the usual array-implemented binary heap property is upheld in its first entries, and for which the heap property is not necessarily upheld in the other entries. Thus, the shadow heap is essentially a binary heap   adjacent to an array  . To add an element to the shadow heap, place it in the array  . If the array becomes too large according to the specified threshold, we first build a heap out of   using Floyd's algorithm for heap construction,[2] and then merge this heap with   using shadow merge. Finally, the merging of shadow heaps is simply done through sequential insertion of one heap into the other using the above insertion procedure.

Analysis

edit

We are given a shadow heap  , with threshold function   as above. Suppose that the threshold function is such that any change in   induces no larger a change than in  . We derive the desired running time bounds for the mergeable heap operations using the potential method for amortized analysis. The potential   of the heap is chosen to be:

 

Using this potential, we can obtain the desired amortized running times:

create(H): initializes a new empty shadow heap  

Here, the potential   is unchanged, so the amortized cost of creation is  , the actual cost.

insert(x, H): inserts   into the shadow heap  

There are two cases:
  • If the merge is employed, then the drop in the potential function is exactly the actual cost of merging   and  , so the amortized cost is  .
  • If the merge is not done, then the amortized cost is  
By choice of the threshold function, we thus obtain that the amortized cost of insertion is:
 

delete_min(H): deletes the minimum priority element from  

Finding and deleting the minimum takes actual time  . Moreover, the potential function can only increase after this deletion if the value of   decreases. By choice of  , we have that the amortized cost of this operation is the same as the actual cost.
edit

A naive binary heap merging algorithm will merge the two heaps   and   in time   by simply concatenating both heaps and making a heap out of the resulting array using Floyd's algorithm for heap construction. Alternatively, the heaps can simply be merged by sequentially inserting each element of   into  , taking time  .

Sack and Strothotte proposed an algorithm for merging the binary heaps in   time.[3] Their algorithm is known to be more efficient than the second naive solution described above roughly when  . Shadow merge performs asymptotically better than their algorithm, even in the worst case.

There are several other heaps which support faster merge times. For instance, Fibonacci heaps can be merged in   time. Since binary heaps require   time to merge,[4] shadow merge remains efficient.

References

edit
  1. ^ Kuszmaul, Christopher L. (2000). Efficient Merge and Insert Operations for Binary Heaps and Trees (PDF) (Technical report). NASA Advanced Supercomputing Division. 00-003.
  2. ^ Suchenek, Marek A. (2012), "Elementary Yet Precise Worst-Case Analysis of Floyd's Heap-Construction Program", Fundamenta Informaticae, 120 (1), IOS Press: 75–92, doi:10.3233/FI-2012-751
  3. ^ Sack, Jörg-R.; Strothotte, Thomas (1985), "An Algorithm for Merging Heaps", Acta Informatica, 22 (2), Springer-Verlag: 171–186, doi:10.1007/BF00264229, S2CID 6427624.
  4. ^ Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. (1990). Introduction to Algorithms (1st ed.). MIT Press and McGraw-Hill. ISBN 0-262-03141-8.