91
votes
Accepted
What's the difference between a binary search tree and a binary heap?
Heap just guarantees that elements on higher levels are greater (for max-heap) or smaller (for min-heap) than elements on lower levels, whereas BST guarantees order (from "left" to "right"). If you ...
- 1,092
49
votes
37
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What's the difference between a binary search tree and a binary heap?
Both binary search trees and binary heaps are tree-based data structures.
Heaps require the nodes to have a priority over their children. In a max heap, each node's children must be less than itself. ...
- 1,711
27
votes
Accepted
What is the advantage of heaps over sorted arrays?
$\small \texttt{find-min}$ (resp. $\small \texttt{find-max}$), $\small \texttt{delete-min}$ (resp. $\small \texttt{delete-max}$) and $\small \texttt{insert}$ are the three most important operations of ...
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14
votes
Accepted
Why use heap over red-black tree?
Its about pragmatic efficiency.
The big-O notation tends to simplify many aspects of the machine that the algorithm is executing on. It leaves out the constant multipliers, and constant additions. It ...
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13
votes
What's the difference between a binary search tree and a binary heap?
With data structure one has to distinguish levels of concern.
The abstract data structures (objects stored, their operations) in this question are different. One implements a priority queue, the ...
- 27.6k
8
votes
Searching through a heap complexity
You are correct: it's $\Theta(n)$ in the worst case. Suppose you're looking for something that's no bigger than the smallest value in a max-heap. The max-heap property (that the value of every node is ...
- 80.2k
8
votes
Accepted
Why can't we sort an Array in O(n) using Fibonacci Heap?
Increase-key is not a $O(1)$ operation on Fibonacci heaps. You're thinking of decrease-key.
Exercise: Why can't increase-key be a $O(1)$ operation on this data structure?
- 18.9k
7
votes
Accepted
Heap-like data structure allowing peek at largest & smallest
One simple approach is to use a max-heap. Separately keep track of the minimum element stored in the heap. Then all operations can be done relatively efficiently:
Load from array can be done in $O(...
D.W.♦
- 141k
7
votes
Extract Max for a max-heap in $\log n + \log\log n$ comparisons
Explanation for log n + log log n comparisons:
1) First, remove the max element from the heap, leaving a hole at the root.
2) Next, remove the last element of the heap (at the bottom right), and ...
- 406
6
votes
Accepted
Why clear the child's and not the parent's mark in Fibonacci heaps?
To understand Fibonacci heaps, it may help to understand binomial heaps first.
A binomial heap is a forest of heap-ordered binomial trees. A binomial tree of degree k is a node whose children are ...
- 18.9k
6
votes
What is the advantage of heaps over sorted arrays?
To answer your questions, you have to define which different actions you will perform and how often, and you have to evaluate the time complexity of each action.
Which method is performing better ...
- 25.2k
6
votes
What is the time complexity for getting the size of a heap?
If you're talking about an ADT, you can't really say. It depends on the implementation. You can certainly do it in O(1) (for example by keeping a counter).
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Is there a name for this priority queue data structure?
This is essentially a Segment tree which is a data structure that augments an array with a binary tree as you describe such that:
You have fast set and get at any index
You have fast "aggregate" ...
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6
votes
Accepted
What are the disadvantages of Fibonacci Heaps?
$O(1)$ merely means that no matter how large your heap grows, the operation will always take roughly the same time to execute. It doesn't mean "the fastest".
Wikipedia article you linked has ...
6
votes
If both could be implemented with the other, what are the differences between priority queues and binary heaps?
Based on standard usage of the terms, a heap is a specific data structure, with a specific representation in memory. A priority queue is an abstract data type: it identifies some operations that must ...
D.W.♦
- 141k
5
votes
Accepted
Merging Sorted lists using Heap Data Structure
Hint: Try to merge the lists in the correct way, using the fact that merging lists of size $m_1,m_2$ costs $O(m_1+m_2)$. If you do this correctly, you will obtain an $O(n\log\log n)$ algorithm.
We ...
- 270k
5
votes
Best and worse case inputs for heap sort and quick sort?
Since nobody's really addressed heapSort yet:
Assuming you're using a max heap represented as an array and inserting your max elements backwards into your output array/into the back of your array if ...
- 51
5
votes
Proving that an $n$-element heap has at most $\lceil \frac{n}{2^{h+1}-1} \rceil$ nodes
Probably, you mean this:
A heap of size $n$ has at most $\lceil \frac{n}{2^{h+1}} \rceil$ nodes with height $h$.
Proof can be found for example here:
http://www.cs.sfu.ca/CourseCentral/307/petra/2009/...
5
votes
Accepted
Why do you need to fill the first element of array when implementing heap?
It is not some magic element but $-\infty$, which is because this is min-heap, in max-heap it would be $\infty$.
The sole purpose is consistent representation of all elements inserted - it guarantees ...
- 9,325
5
votes
To find median of $k$ sorted arrays of $n$ elements each in less than $O(nk\log k)$
Let us denote the arrays by $A_1,\ldots,A_k$, their sizes by $|A_1|,\ldots,|A_k|$, their medians by $m_1,\ldots,m_k$, and their union by $\mathbf{A}$. We will try to solve the following more general ...
- 270k
5
votes
Why is Binary Heap never unbalanced?
You must refer to the definition of a Binary Heap:
A Binary heap is by definition a complete binary tree ,that is, all levels ...
- 1,135
5
votes
Accepted
Min Fibonacci Heap - increase key
First note that $\text{increase-key}$ must be $O(\log n)$ if we wish for $\text{insert}$ and $\text{find-min}$ to stay $O(1)$ as they are in a Fibonacci heap.
If it weren't you'd be able to sort in $...
- 12.3k
4
votes
Accepted
Show that the running time of the build_heap function is $O(n)$
The main point is to do the analysis exactly without being sloppy. Let me define the depth of a node in this way. Depth of any leaf is 0 and depth of any non-leaf node is maximum depth among its ...
4
votes
Accepted
Can we create binomial heaps in linear time?
Wikipedia claims that insertion takes $O(1)$ amortized time, and so converting an array of numbers into a binomial heap should indeed take time $O(n)$. This is also supported by these lecture notes, ...
- 270k
4
votes
Build-Max-Heap vs. HeapSort
HeapSort: A procedure which sorts an array in place.
Can be in-place or not in-place, the point is it's using a heap data structure to help sort.
Building heap is linear with a careful analysis
See ...
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4
votes
Finding the $k$-smallest elements in a min-heap
If you mean the usual binary heap represented in an array, this is answered in "An Optimal Algorithm for Selection in a Min-Heap", by Frederickson. It is pretty complex.
Other priority queues have ...
- 3,200
4
votes
Accepted
Randomized Meldable Heap - Expected Height
In the paper, $h_Q$ isn't the height. It's the length of a random walk away from the root in a full binary tree (they insist every leaf is "nil"), so the expression they have is the right thing.
...
- 2,886
4
votes
Is it possible to build a heap from the root to the leaves?
Binary heaps are one possible implementations for priority queues. Although their operations are best understood as binary trees, their implementation as arrays is essential. They are stored ...
- 27.6k
4
votes
Ideal value of d in a d-ary heap for Dijkstra's algorithm
Please note that $ m \leq n(n-1)/2 $, thus $m/n < n/2$. So the example $m=3, n=1$ does not happen.
First, the article clearly claims that this improvement is achieved whenever $ m $ is larger than ...
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