Questions tagged [average-case]
79 questions
16
votes
1 answer
On "The Average Height of Planted Plane Trees" by Knuth, de Bruijn and Rice (1972)
I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" want to prove that the average height $h_n$ of a…
Christian Rinderknecht
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14
votes
1 answer
Expected number of swaps in bubble sort
Given an array $A$ of $N$ integers, each element in the array can be increased by a fixed number $b$ with some probability $p[i]$, $0 \leq i < n$. I have to find the expected number of swaps that will take place to sort the array using bubble…
TheRock
13
votes
1 answer
Proof that a randomly built binary search tree has logarithmic height
How do you prove that the expected height of a randomly built binary search tree with $n$ nodes is $O(\log n)$? There is a proof in CLRS Introduction to Algorithms (chapter 12.4), but I don't understand it.
user1675999
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12
votes
4 answers
Evaluating the average time complexity of a given bubblesort algorithm.
Considering this pseudo-code of a bubblesort:
FOR i := 0 TO arraylength(list) STEP 1
switched := false
FOR j := 0 TO arraylength(list)-(i+1) STEP 1
IF list[j] > list[j + 1] THEN
switch(list,j,j+1)
switched…
Sim
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10
votes
1 answer
What is the average-case complexity of trial division?
The trial division algorithm for checking if a number $N$ is prime works by trying to divide $N$ by all integers in the range 2, 3, ..., $\lfloor \sqrt{n} \rfloor$. If any of them cleanly divide $N$, then we say that $N$ is composite; otherwise we…
templatetypedef
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8
votes
1 answer
Proving that the average case complexity of binary search is O(log n)
I know that the both the average and worst case complexity of binary search is O(log n) and I know how to prove the worst case complexity is O(log n) using recurrence relations.
But how would I go about proving that the average case complexity of…
cj1996
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7
votes
2 answers
Complexity of keeping track of $K$ smallest integers in a stream
I need to analyze the time complexity of an online algorithm to keep track of minimum $K$ numbers from a stream of $R$ numbers. The algorithm is
Suppose the $i$th number in the stream is $S_i$.
Keep a max heap of size $K$.
If the heap contains…
Piyush
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6
votes
3 answers
What is the time complexity of this atrocious algorithm?
This grew out of a discussion of deliberately bad algorithms; credit to benneh on the xkcd forums for the pseudocode algorithm, which I've translated to Python so you can actually run it:
def sort(list):
if len(list) < 2:
return list
…
Wildcard
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6
votes
3 answers
Average depth of a Binary Search Tree and AVL Tree
My professor recently mentioned that the average depth of the nodes in a binary search tree will be $O(log(n))$ where $n$ is the amount of nodes in the tree. I ended up drawing out a bunch of binary search trees and I don't think I am understanding…
Nickknack
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6
votes
1 answer
Completeness of formal definition of 'hardness on the average'
While reading a cryptography textbook, i find the definition of a function that is hard on the average.(More precisely, it is 'hard on the average but easy with auxiliary input', but i omit latter for simplicity.)
Definition : Hard on the average…
euna
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6
votes
1 answer
Can expected "depth" of an element and expected "height" differ significantly?
When analysing treaps (or, equivalently, BSTs or Quicksort), it is not too hard to show that
$\qquad\displaystyle \mathbb{E}[d(k)] \in O(\log n)$
where $d(k)$ is the depth of the element with rank $k$ in the set of $n$ keys.
Intuitively, this seems…
Raphael
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6
votes
4 answers
Are there NP COMPLETE problems that are "easy" in practice?
NP COMPLETE problems are hard in the worst case (assuming $P \neq NP$). What that means is that for every polynomial $p$, sufficiently large integer $n$, and algorithm $A$, there is an instance $x$ of size $n$ for which the algorithm takes more than…
wlad
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5
votes
0 answers
Average redundancy in Huffman or Hu-Tucker codes on random symbol probabilities
Huffman and Hu-Tucker codes are well-known compression schemes, which both come close to the entropy lower bound. It is known that if $L_1$ and $L_2$ are the lengths of a Huffman resp. Hu-Tucker code, then $H\le L_1 \le H+1$ and $H\le L_2<…
Sebastian
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5
votes
3 answers
Why does this mergesort variant not do Θ(n) comparisons on average?
A comparison sort cannot require fewer than $\Theta (n\log n)$ comparisons on average. However, consider this sorting algorithm:
sort(array):
if length(array) < 2:
return array
unsorted ← empty_array
i ← 0
while i <…
Mets
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5
votes
1 answer
What is the expected number of nodes at depth d of a tree after i random insertions
Suppose one wanted to build a tree at random. Let the first insertion at step $i = 1$ be the root node. From here, nodes are inserted into the tree at random one at a time. How would one go about calculating the expected number of nodes $E(d)$ at…
Bryce Thomas
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