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Questions tagged [amortized-analysis]

A method in analysis of algorithms that considers the overall cost of a sequence of operations.

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Amortized Analysis of extract-min-operation of Fibonacci Heap

I am studying the operations of the Fibonacci heap. While going through min-extraction operation every step and its complexities are fairly clear to me. In short, it is: The potential before ...
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Analyzing Hybrid Merge and Insertion Sort

We know that merge sort takes O(n log n) and insertion sort takes (n^2) for worst case. The combination of these two algorithm is to speed up and reduce key comparisons, as for a subarray with small ...
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Amortized cost depending on the number of operations

Considering a dynamic array that grows by a constant factor $k$ (the new array has $k$ more cells than the last one) each time the array is full which initially has $n$ elements in it. Calculating the ...
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A formal definition for amortized time

Several (tutoring) students have asked me for a formal definition of amortized time and I've never been able to find one online. All the literature I've found usually outlines the three most common ...
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42 views

General Proof on Potential Method and Amortized Analysis

Let $T$ be an arbitrary data structure for a dynamic set. For every state T of $T$, let $d_t \in \mathbb{N}$. Observe two Operations $O_1, O_2$ on $T$ whose runtimes are proportional to $d_t$ and $...
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Changing binary counter structure such that increament and decreament methods will work in O(1) amortized

Just trying to solve the second part of a question with two parts. First part was to prove that you can't add decrement method to a standart binary counter without hurting the amortized complexity and ...
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61 views

Accounting method - dynamic array

I want to compute the amortize time of a type of dynamic array (inserting such that if i have no place to insert i am multipling the array by (1+a) (a is between 0 to 1). I need to compute the time ...
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26 views

Amortized analysis - adding operations to a data structure

One of the finer points of amortized analysis about which I have been able to find relatively little information is the broad question of what happens to the amortized cost of a structure's existing ...
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234 views

Does this data structure already exist?

I was working on a problem for some time now, and I made a data structure to solve it. To my surprise, I could not find any instance of this data structure on the internet (though I am certain someone ...
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31 views

Can you delete-min from fibonacci heap in O(1) amortized?

I just had a data-structures exam. One of the questions asked us to create a data structure which allowed insert operations in O(logn) amortized and delete-max (or min, doesn't matter) in O(1) ...
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44 views

Fibonacci Heap that consolidates after every step

The lecturer of my graduate algorithms course suggested that, even if a Fibonacci Heap would consolidate its tree list after every operation (not just when doing deleteMin()), most operations would ...
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49 views

Dynamic array with 4x growth factor: Potential Method

I am curious on the use of the potential method for amortized analysis for a dynamic array which quadruples in size after it becomes full. I understand how the potential function is defined and used ...
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109 views

design a strange data structure, is it possible?

I need a FIFO QUEUE that can do Insert and Remove from Queue in amortized $O(1)$ but extract min in $O(log n)$. is it possible? When just find min is important (not removing) there is lots of $O(1)$ ...
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55 views

When do you use amortized time complexity and when to use unamortized?

This is my guess: -Use amortized because we want to know the "averaged" complexity over n operations assuming the ...
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46 views

Amortized Analysis on Binary Heap, Potential Method

The potential function of a Binary Heap is given as the sum of levels of every node in the Binary Heap. For example,potential of a Binary Heap with 6 nodes is 0 + 2 ∗ 1 + 3 ∗ 2 = 8. It is given that ...
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30 views

sum selected nodes of a set in $\log n$ time

Given a sum operation of a dynamic set $S$ of length $n$ which includes integer pairs $(x, y)$. The sum operation is defined as taking two inputs $a$ and $b$ such that $a \leq b$. The sum operation ...
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Create a potential function for an abstract queue data structure to show constant amortized-time complexity

Consider a variation of a Queue called MaxQueue, Q, that has the following operations: dequeue(Q): removes and returns the first element of Q enqueue(Q, s): Appends the integer s to the end of Q ...
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Closest point in embedded simplicial complex

Suppose I have a simplicial $k$-complex $\mathcal S$ whose vertices are embedded in Euclidean space $\mathbb R^n$, for roughly $k< n\leq 6$. Examples include triangle mesh surfaces ($k=2$) embedded ...
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Difficulty in few steps in proof of "Amortized cost of $\text{Find-Set}$ operation is $\Theta(\alpha(n))$"assuming union by rank, path compression

I was reading the section of data structures for disjoint sets from the text Introduction to Algorithms by Cormen et. al .I faced difficulty in understanding few steps in the proof of the lemma as ...
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118 views

Intuition behind the entire (amortized) concept of Fibonacci Heap operations

The following excerpts are from the section Fibonacci Heap from the text Introduction to Algorithms by Cormen et. al The potential function for the Fibonacci Heaps $H$ is defined as follows: $$\Phi(H)...
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Splay tree amortized analysis cost using Access Lemma

Currently studying for an algorithms exam and I came across this question and solution, but I can't understand the solution where it references nodes of depth less than $4\log n$ and not restructuring....
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81 views

$\Phi_1=1$ or $\Phi_1=2$ for the dynamic $\text{Table-Insert}$ , where $\Phi_i$ is the potential function after $i$ th operation, as per CLRS

The following comes from section Dynamic Tables, Introduction to Algorithms by Cormen. et. al. In the following pseudocode, we assume that $T$ is an object representing the table. The field $table[T]$...
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Is there a known way to make an efficient, compact, and fully persistent stack or queue?

In the world of mutable/ephemeral data structures and imperative programming languages, one of the classic ways to implement a stack or queue is to use array doubling: use mutation to fill up or empty ...
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Amortized time for dynamic array

I'm struggling to understand one part from the book "Cracking the coding interview". The author states inserting an element in a dynamic array is $O(1)$ most of the time, except when the array is full ...
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Amortized analysis for disjoint sets' find-set(x) function (from CLRS)

I start off by apologizing for the fact that I don't really know how to use latex/markdown. My question, however, is directly from the Introduction To Algorithms book by Cormen et al. The topic ...
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Does amortized algorithm analysis make any assumptions about the sequence of function calls?

I have read that average case analysis makes some assumptions about the inputs to the data structure, and amortized analysis makes no such assumptions. Does amortized analysis make any assumptions ...
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Is using Fibonacci Heaps in Huffman Code, better than a regular Min-heap?

When using Huffman Code, to generate prefix-code trees for a sequence of letters, CLRS choose to use a normal Min-heap data structure. Using Fibonacci-heaps instead, are we not able to achieve a ...
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118 views

Amortized value of dequeue with potential function

A deque is implemented with 3 stacks. one for the head, one for the tail and one is always empty. Pushing is therefor O(1), light popping (in case the head/tail respectively aren't empty) is also O(1)....
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343 views

Can someone let me know if my understanding of amortized run time in a dynamic array list is correct?

Am I right in my understanding for amortized time for insertion in a dynamic array list? (dynamic means create a copy double its size and copy existing elements to new one WHEN we reach the current ...
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68 views

converting for loop into mathematical notation

I am a bit confused with a problem that I am having.I am trying to do an amoritized analysis and I am able to represent the function that I want to use using code. I coded the function as follows: <...
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When was the dynamic array first introduced as an example for amortized analysis?

I'm writing a report on amortized analysis, and I'm using the example of a dynamic array to explain each method. I think it would be nice to add a reference to when this example was first used, as it ...
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60 views

What time complexity is more significant? [closed]

A certain algorithm executes $n$ operations of three types: insert, delete, and find. We know that $n/10$ of the operations are inserts, and the rest are deletes and finds. You are given two ...
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1k views

Amortized analysis - increment in ternary counter [closed]

What is the amortized analysis of increment action in a ternary counter that is initialized to 0?
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2k views

Can an algorithm with $\Theta(n^2)$ run time be faster than an algorithm with $\Theta(n\log n)$ run time?

This is a question posted for extra practice (i.e., not for credit): Can an algorithm with $\Theta(n^2)$ run time be faster than an algorithm with $\Theta(n\log n)$ run time? Explain. I'm not sure ...
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57 views

Best known (state of science) time complexity of an array access problem

Consider an Array $A$ with $n$ values and the following operations: get(i): Returns the value of $A[i]$ insert (x): Insert the element x into the any free place in A (not necessarily in $A[x]$ or the ...
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Algorithm to optimize redistribution of balls amongst urns [closed]

Here is the question: Say we have k urns with 1 ball in each urn. At each iteration of the game, I pick one urn and redistribute its contents amongst other urns and each urn can receive at most one ...
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251 views

Constant factor of an array

In Elements of Programming Interviews in Python by Aziz, Lee and Prakash, they state on page 41: Insertion into a full array can be handled by resizing, i.e., allocating a new array with ...
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Complexity of many constant time steps with occasional logarithmic steps

I have a data structure that can perform a task $T$ in constant time, $O(1)$. However, every $k$th invocation requires $O(\log{n})$, where $k$ is constant. Is it possible for this task to ever take ...
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359 views

general question amortized cost and worst case

lets say a data structure has operations called insert and delete both of which take O(log(n)) worst case. Suppose the amortized cost of insert is O(log(n)) and the amortized cost of delete is O(1). ...
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253 views

How to do if a potential function Does not work? Amortized analysis

Here is an example taken from CLRS. q)Consider an ordinary binary min-heap data structure with n elements supporting the instructions INSERT and EXTRACT-MIN in O(lg n) worst-case time. Give a ...
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147 views

if binary heap potential function is c*size(binary heap)) then insert will not take O(logn)and extract min will not take O(1) amortized time

So i want to prove that if i choose a potential function for binary heap as any constant*size of the binary heap (n is the number of nodes) then my insert will not have O(logn) amortized cost and ...
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436 views

What does $O(\alpha(n))$ amortized time mean?

DELETE(S, i): Delete integer $i$ from the set $S$. if $i \notin S$, there is no effect. from a set of consectutive integers like $S = \{1,2,3,5,6\}$ Provide a data structure and an algorithm for ...
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452 views

Amortized analysis of max-heap

Consider an ordinary binary max-heap data structure with $n$ elements that supports insert and extract-max in $O(\log n)$ worst-case time. Question: If extract max is $O(1)$ amortized does that ...
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353 views

amortized analysis of max heap

Q) Consider an ordinary binary max-heap data structure with n elements that supports insert and extract-max in $O(log(n))$ worst-case time. Give a potential function $\Phi$ such that the amortized ...
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317 views

Amortized time complexity for double stack emulated queue

Assume that we have a data type $stack$ which has two operation $push$ and $pop$, both operations' time complexity is $O(1)$ in worst case. The $stack$ also has a property $size$ indicate how many ...
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133 views

The validity of the potential function for splay tree

The paper "Self-Adjusting Binary Search Trees" defines (Page 658) the potential function for analyzing the amortized cost of a sequence of $m$ splay operations as the sum of the ranks of all nodes in ...
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940 views

Incremental strongly connected components

For a changing directed graph, I would like to maintain information about strongly connected components. The graph operations are incremental: only vertex addition and edge addition. What data ...
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143 views

Sequence of operations of Union-Find of length $m$ ($n$ being the number of Make-Set operations) with time complexity in $\Omega(m\log n)$

In Union-Find with link-by-rank but no path compression find a sequence of operations Make-Set, Find, Union of length $m$, containing $n$ Make-Set operations, and with time complexity in $\Omega(m\log ...
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631 views

Accounting method vs Potential method for analysing an augmented stack and differences with standard complexity analysis

With reference to chapter 17 of CLRS, (Amortized analysis). I'm trying to understand the differences between the accounting method and the potential method. Let's start with standard analysis of the ...
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Formal justification of the accounting method and its meaning

I'm reading through CLRS again and I was wondering if there's a formal justification or construction of the accounting method, explaining why it works. For some reason it seems to me that CLRS ...