# Questions tagged [amortized-analysis]

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

149 questions
Filter by
Sorted by
Tagged with
1 vote
60 views

### Can there exist a deque like data structure that supports amortized $O(1)$ random access?

A lot of modern languages usually have a "list" or "vector" structure which allows for amortized $O(1)$ append and removal from back as well as amortized $O(1)$ random access. I'm ...
93 views

### Prove with potential method that dynamic table with $q > 1$ expansion runs in amortized constant time

Suppose I have a dynamic table supporting $Insert$ procedure, which sets an input value after the tail of the dynamic table. If the underlying table is already full, we multiply its size by $q > 1$....
1 vote
47 views

92 views

### Amortized cost for Stack Operations

In this problem we consider two stacks $A$ and $B$ manipulated using the following operations ($n$ denotes the size of $A$ and $m$ the size of $B$):   PushA($x$): Push element $x$ on stack $A$.   ...
1 vote
61 views

### Amortized analysis of dynamic array insertion

I learned the amortized analysis of Prof Demaine's 6.006 videos. The Erik's thesis was, If we reallocate memory by doubling capacity => $$1 +2+4 + 8 + 16 +..+n = \theta(2^{lgn}) = \theta(n)$$ From ...
1 vote
93 views

### Amortised cost - transferring tokens

I'm trying to solve a problem from one of the older exams. Question: There's an infinite, one-dimensional board, with fields numbered consecutively $\ldots, -2, -1, 0, 1, 2, \ldots$ A move in the ...
191 views

### Algorithm with amortized time complexity

While I understand the process of considering/observing an algorithm and finding an average time, necessary to perform an operation that happens in this algorithm, I still cannot quite gasp the idea, ...
39 views

### Amortized analysis on skew heap arbitrary deletion

A practice problem in my textbook asks to proof the amortized complexity for a sequence of insert, delete min, and decrease-key operations on an initially empty skew heap. Insert and delete min both ...
1 vote
65 views

### Understanding David Pisinger's balanced algorithm for the subset-sum problem with bounded weights

I'm trying to understand David Pisinger's balanced algorithm for the subset-sum problem with bounded weights, which can be found on page 5 of his paper Linear Time Algorithms for Knapsack Problems ...
44 views

### Splay Trees - Sequential Access Theorem & lower bound for comparison-based sorting

The following theorem was proven by R.E. Tarjan in 1984: Theorem (Sequential Access Theorem). If we access each of the nodes of an arbitrary initial tree once, in symmetric order, the total time ...
123 views

### Difficulty in last sentence in proof of "Amortized cost of $\text{Find-Set}$ operation is $O(\alpha(n))$" from CLRS

I was reading the section of Data Structures for Disjoint Sets from the text Introduction to Algorithms by Cormen et. al. I made it through the proof, but I'm not sure I understand the very last ...
511 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 ...
119 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 $table[T]$...
1 vote
189 views

### Aggregate method for dynamic table (amortized analysis)

For amortized analysis (aggregate method), dynamic table insertion cost can be divided into: if no expansion, then cost = 1 if we expand the table, then cost = i (if i-1 is an exact power of 2) then ...
42 views

### What are the actual costs $c_i$ in the potential method?

In "Introduction to Algorithms" by Cormen et al. the Potential Method is explained. For example, we have the following representation for the amortized costs of the i-th operation with ...
1 vote
22 views

### Algorithm for an incremental update to cut vertex set

There is a classic linear algorithm to find every cut vertex (AKA articulation point) in a graph. I have a usecase that does this computation after every time a non-articulation-point is inserted or ...
302 views

### Designing a Data Structure that allows both insertion and extracting a number lower than median in amoritzed O(1) cost?

Consider a data structure that has only two functions. extract_lowerthan_median() and insert(). How can we design it in a way that the amortized cost for both the operation is O(1)? Using a 2 tree, a ...
517 views

### Potential Method For Decimal Counter

There is a counter that counts the number of items in the store. For every increase in item or item that has been inserted, the cost is a + kb where k is the number of digits that has been changed in ...
263 views

### Prove: Self-organizing list that uses Move-to-Front is 2-Competitive

Preparing for my finals in my "advances algorithms" course. Usually there is a question to prove one of the theorems that was given over the course. I'm currently trying to write a full ...
1 vote
85 views

1 vote
549 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 ...
836 views

### Why isn't the time complexity of constructing a Fenwick tree tighter than $O(n\lg n)$?

Intuition: Suppose I have an array of nonzero integer values $A[n]$ and a partially constructed Fenwick tree of this array: $F[k], n>k$. I can see why inserting the next element would be worst ...
204 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 ...
138 views

### In Amortized Analysis, can we chose how big $n$ is?

Suppose I want to show by contradiction that the amortized cost of a data structure with some operations cannot be less then $\Theta(k)$. I assume for the sake of contradiction that it is possible. ...
1 vote
237 views

### 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 ...
112 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) ...
1 vote
100 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 ...
1 vote
244 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 ...
31k views

### Why is push_back in C++ vectors constant amortized?

I am learning C++ and noticed that the running time for the push_back function for vectors is constant "amortized." The documentation further notes that "If a reallocation happens, the reallocation is ...
156 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 $O(1)$ ...
173 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 ...
650 views

### how to verify permutation generated in constant amortized time?

Here is an algorithm that generates the next permutation in lexicographic order, changing the given permutation in-place: Find the largest index k such that a[k] < a[k+1]. If no such index exists, ...
103 views

### 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 ...
128 views

### 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 ...
2k views

### Red-black tree amortized cost of the rebalancing

I've read in different sources that the amortized cost of a red-black tree rebalancing is constant (at least during the tree creation using only insertions). How can it be proven?
1 vote
218 views

### 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 ...