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### The amortized cost of finding the minimum [chapter]

Haim Kaplan, Or Zamir, Uri Zwick
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
More specifically, we show that for any fixed > 0, where n is the number of items in the priority queue and A(insert), A(delete) and A(find-min) are the amortized costs of the insert, delete and find-min  ...  We obtain an essentially optimal tradeoff between the amortized cost of the three basic priority queue operations insert, delete and find-min in the comparison model.  ...  Acknowledgement We would like to thank Bob Tarjan for helpful discussions and Mikkel Thorup for bringing the Brodal et al. [5] paper to our attention.  ...

### A Polylogarithmic Approximation of the Minimum Bisection

Uriel Feige, Robert Krauthgamer
2002 SIAM journal on computing (Print)
Finding the bisection of minimum cost is NP-hard. We present an algorithm that finds a bisection whose cost is within ratio of Ç´ÐÓ ¾ Òµ from the optimal.  ...  A bisection of a graph with Ò vertices is a partition of its vertices into two sets, each of size Ò ¾. The bisection cost is the number of edges connecting the two sets.  ...  The first author is the Incumbent of the Joseph and Celia Reskin Career Development Chair. Work of the second author was supported in part by the Open University of Israel.  ...

### A Polylogarithmic Approximation of the Minimum Bisection

Robert Krauthgamer, Uriel Feige
2006 SIAM Review
It is known that finding a bisection of minimum cost is NP-hard. We present an algorithm that finds a bisection whose cost is within ratio of O(log 2 n) from the minimum.  ...  A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n/2. The bisection cost is the number of edges connecting the two sets.  ...  Acknowledgements We thank Kobbi Nissim for his part in bootstrapping this research, and the anonymous referees for comments that improved the presentation.  ...

### Page 682 of American Society of Civil Engineers. Collected Journals Vol. 116, Issue HY6 [page]

1990 American Society of Civil Engineers. Collected Journals
Z* = the sum of (1) Amortized cost of installed pipe in stage tf; and (2) incremental amortized cost, for stages 1—t, of installing the stronger pressure class pipe associated with S, over that of the  ...  *, giving the minimum cost.  ...

### Melding priority queues

Ran Mendelson, Robert E. Tarjan, Mikkel Thorup, Uri Zwick
2006 ACM Transactions on Algorithms
into a priority queue data structure that also supports fast meld operations with essentially no increase in the amortized cost of the other operations.  ...  , or O( √ log log n) expected amortized cost per operation, respectively.  ...  (The analysis of [37] only showed that the extra cost is tiny.)  ...

### Melding Priority Queues [chapter]

Ran Mendelson, Robert E. Tarjan, Mikkel Thorup, Uri Zwick
2004 Lecture Notes in Computer Science
into a priority queue data structure that also supports fast meld operations with essentially no increase in the amortized cost of the other operations.  ...  , or O( √ log log n) expected amortized cost per operation, respectively.  ...  (The analysis of [37] only showed that the extra cost is tiny.)  ...

### Selection of a Farm Machinery Replacement Criterion Using Simulation

Darrel Kletke
1969 Southern Journal of Agricultural Economics
The economic life of a machine is here defined as the interval of time during which that machine reaches its minimum average yearly cost.  ...  This is the same as saying that when economic life is reached, the actual yearly cost (marginal cost) is equal to the average yearly cost of the machine.  ...  This average is checked against the minimum amortized average cost of the proposed replacement.  ...

### Layered Heaps [chapter]

Amr Elmasry
2004 Lecture Notes in Computer Science
This latter priority queue achieves, in the amortized sense, constant cost per insertion, minimum finding and decrease-key operations, and logarithmic cost with at most 1.44 log n + O(log log n) comparisons  ...  In particular, we give a priority queue with constant cost per insertion and minimum finding, and logarithmic cost with at most log n + O(log log n) 1 comparisons per deletion and minimum deletion, improving  ...  This motivated the improvements in Section 4 as it appears in this version of the paper.  ...

### STRONGER QUICKHEAPS

GONZALO NAVARRO, RODRIGO PAREDES, PATRICIO V. POBLETE, PETER SANDERS
2011 International Journal of Foundations of Computer Science
We prove that these offer logarithmic expected amortized complexity for all operations regardless of the input distribution.  ...  It has also been shown to offer logarithmic expected amortized complexity for all of its operations.  ...  properties assumed in the analysis of the original QHs do not hold.  ...

### Maintaining Minimum Spanning Forests in Dynamic Graphs

Monika R. Henzinger, Valerie King
2001 SIAM journal on computing (Print)
We present the first fully dynamic algorithm for maintaining a minimum spanning forest in time o( √ n) per operation.  ...  To be precise, the algorithm uses O(n 1/3 log n) amortized time per update operation. The algorithm is fairly simple and deterministic.  ...  To summarize the cost of initialization when amortized over Ω(m in ) operations is O(m 1/3 in ) and the cost per deletion of an edge and finding replacement edges, when amortized over Ω(m in ) operations  ...

### Page 134 of POWER Vol. 90, Issue [page]

1946 POWER
a de- sirable minimum rate of return.  ...  Looking up intersection of 0.200 and 6° ordinates on chart find 6.2 years by inter- polating between dashed curves. This can be checked by finding the amortization rate — 0.200 — 0.060 0.140.  ...

### A potential-based amortized analysis of the union-find data structure

Gregory C. Harfst, Edward M. Reingold
2000 ACM SIGACT News
Traditionally the amortized analysis of this structure is done using the accounting method . We present a simple, new potential-based analysis covering several of the most interestin g results.  ...  We present a simple, new potential-based amortized analysis for the standar d union-find data structure . The standard union-find data structure is simple to describe, yet has a complex analysis .  ...  Hence the amortized cost of a single find is once again O(a(m, n) ) .  ...

### Maintaining minimum spanning trees in dynamic graphs [chapter]

Monika R. Henzinger, Valerie King
1997 Lecture Notes in Computer Science
We present the first fully dynamic algorithm for maintaining a minimum spanning tree in time o( √ n) per operation.  ...  To be precise, the algorithm uses O(n 1/3 log n) amortized time per update operation. The algorithm is fairly simple and deterministic.  ...  To summarize the cost of initialization when amortized over (m in ) operations is O(m 1/3 in ) and the cost per deletion of an edge and finding replacement edges, when amortized over (m in )) operations  ...

### Fully persistent arrays [chapter]

Paul F. Dietz
1989 Lecture Notes in Computer Science
To insert into the data structure, find the bucket whose minimum element is the greatest minimum element less than the new element, and put the new element into the bucket.  ...  If the sum of the costs of the elements of L are always O(IILII), and we assume it takes time cost(x) to relabel x, we now there is an algorithm for WMLP that takes O(I~llog IILII) amortized time for ChangeCost  ...

### Pairing Heaps with Costless Meld [article]

Amr Elmasry
2009 arXiv   pre-print
Improving the structure and analysis in elm0, we give a variation of the pairing heaps that has amortized zero cost per meld (compared to an O(n) in elm0) and the same amortized bounds for all other operations  ...  More precisely, the new pairing heap requires: no cost per meld, O(1) per find-min and insert, O(n) per delete-min, and O(n) per decrease-key.  ...  find-min No potential or credit changes are required. The actual work of find-min is O(1). It follows that the worst-case cost of find-min is O(1).  ...
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