Questions tagged [amortized-analysis]
A method in analysis of algorithms that considers the overall cost of a sequence of operations.
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questions with no upvoted or accepted answers
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Choosing potential function in amortized analysis
How should I think to choose the potential function in the amortized analysis?
More specifically are there techniques or tips for choosing optimal or good potential functions?
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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 ...
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Analysis of Weighted Quick Union with Path Compression
I have searched the internet for an analysis of why WQUPC is amortized $O( m \alpha (n) ) $ for m operations on n nodes ( $\alpha ( n) $ is the inverse Ackerman function).
I understand why it is $O ( ...
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Amortized time complexity of append on a dynamic array that resizes according to geometric base 1.25?
I'm trying to prove that the amortized time complexity of appending to a dynamic array that resizes in accordance with capacity = $N$ to $N+\lceil{\frac{N}{4}}\rceil$ is $O(1)$. I'm assuming that ...
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Array counter with mimimum find
I need to implement data strucure such as array, but with the following interface:
GetMin() - Returns the minimum from the array
IncRight(index) - Increases all values from specified index to the end ...
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How to compute amortized complexity of n runs of Dijkstra's algorithm?
I'm trying to figure out how to compute an amortized complexity/ or complexity of this algorithm. We have a Graph which is oriented. And we are going to run Dijkstra's algorithm for finding a shortest ...
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What is hiding behind amortized constant delay enumeration?
The following may contain errors. It is precisely because I am not
sure I understand the topic that I am asking questions. I do not have
books about it and could not find an adequate reference on the ...
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0answers
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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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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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0answers
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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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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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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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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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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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573 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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amortized analysis
I am trying to find a general solution for the given A,B,C in dynamic arrays. Those veriables presents factors in the following operations :
given :
c_i the size of the array after operation O_i (...
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Why is maximum size of root is 2n + 1 in Splay trees?
In the amortized analysis of Splaying in Dynamic trees, let us consider a splay tree $T$ with $n$ keys and $v$ be a node of $T$. We define $size(v)$ as the number of nodes in the subtree rooted at $...
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Calculating the cost for each operation in amortized analysis
According to what I've read in the CLRS book , we calculate the amortized cost for a complete set , and not for a single operation.But in an exam question , it was asked about an operation amortized ...
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Is it true that the potential method is usually the inverse of the accounting method in the context of amortized analysis?
I remember reading this somewhere (will post the source as soon as I find it), but was wondering, if anyone knows why this is true or has maybe an example to portray why this is the case.
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Potential method analysis for Insert and Extract-max on a Max heap data structure
Suppose that you do some sequence of operations on a max heap, in this case only Insert and Extract-max. Whenever the heap ...
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0answers
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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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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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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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Essence of the cost benifit obtained by using “markings” in Fibonacci Heaps (by using a mathematical approach)
The following excerpts are from the section Fibonacci Heap from the text Introduction to Algorithms by Cormen et. al
The authors deal with a notion of marking the nodes of Fibonacci Heaps with the ...
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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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Choosing potential function in amortized analysis of the dynamic array
Suppose we have a dynamic array with some initial capacity $c_{initial}$ (i.e., the dynamic array with zero elements will have this capacity). The $add$ operation is modified the following way:
New ...
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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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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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240 views
Analysis of Union-Find(Disjoint Sets)
I have been trying to learn more about amortized analysis. Recently I came across the Disjoint Sets or Union-Find structures. I am using union by rank and path comparison. The potential of such data ...
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Amortized cost of decimal counter
Can somebody tell me what the lowest amortized cost for the increment operation of a decimal counter is?
I can show the costs are O(1) and with max amortized costs of 2 (similar to a binary counter), ...
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Amortised complexity of dynamic array using potential function
I'm trying to find out how potential function works. I'm trying to compute an amortised complexity of $n$ operations on dynamic array. To make it simple, assume, that we can't delete items and we can ...
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1answer
416 views
What is the amortized time complexity of inserting an element to this heap?
Assume you implement a heap using an array and each time the array is full, you copy it to an array double its size. What is the amortized time complexity (for the worst case) of inserting elements ...
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1answer
301 views
Custom binary counter supports only increment in $2^i$ values amortized analysis
I'm a having trouble analyzing this algorithm.
This is a binary counter that supports only increments in $2^i$ values
it's implemented in this way:
starting from the $i$-th location change all the ...
