Shadow heap explained
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
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
O(|A|+min\{log|B|loglog|B|,log|A|log|B|\})
.
Algorithm
We wish to merge the two binary min-heaps
and
. The algorithm is as follows:
- Concatenate the array
at the end of the array
to obtain an array
.
- Identify the shadow of
in
; that is, the ancestors of the last
nodes in
which destroy the heap property.
- Identify the following two parts of the shadow from
:
-
: the set of nodes in the shadow for which there are at most 2 at any depth of
;
-
: the remainder of the shadow.
- Extract and sort the smallest
nodes from the shadow into an array
.
- Transform
as follows:
-
, then starting from the smallest element in the sorted array, sequentially insert each element of
into
, replacing them with
's smallest elements.
-
, then extract and sort the
smallest elements from
, and merge this sorted list with
.
- Replace the elements of
into their original positions in
.
- Make a heap out of
.
Running time
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
O(|A|+min\{log|A|log|B|,log|B|loglog|B|\})
.
Structure
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, 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
We are given a shadow heap
, with threshold function
log|H|\leqf(H)\leqlog|H|loglog|H|
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:
\Psi(H)=|A|(1+min\{log|B|loglog|B|,log|B|log|A|\}/f(H))
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
O(1+min\{log|B|loglog|B|,log|B|log|A|\}/f(H))
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.
Related algorithms & data structures
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.
[2] 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, shadow merge remains efficient.
Notes and References
- Kuszmaul . Christopher L. . Efficient Merge and Insert Operations for Binary Heaps and Trees . NASA Advanced Supercomputing Division . 2000 . 00-003 .
- .