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The amortized cost of finding the minimum
[chapter]

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

doi:10.1137/1.9781611973730.51
dblp:conf/soda/KaplanZZ15
fatcat:7chciwpzdrcatkjceybqxsmpb4
*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. ...##
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A Polylogarithmic Approximation of the Minimum Bisection

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

2006
*
SIAM Review
*

It is known that

doi:10.1137/050640904
fatcat:vebchn4xanglve5ub2hdpso6gy
*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

2006
*
ACM Transactions on Algorithms
*

into a priority queue data structure that also supports fast meld operations with essentially no increase in

doi:10.1145/1198513.1198517
fatcat:ox4auyefabcivepe4cnnoa33ma
*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]

2004
*
Lecture Notes in Computer Science
*

into a priority queue data structure that also supports fast meld operations with essentially no increase in

doi:10.1007/978-3-540-27810-8_20
fatcat:qv3yicinlrfxlmiombp5gmhkdm
*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

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]

2004
*
Lecture Notes in Computer Science
*

This latter priority queue achieves, in

doi:10.1007/978-3-540-27810-8_19
fatcat:6ujxkkngpzbdhegbly5k3q2ut4
*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

2011
*
International Journal of Foundations of Computer Science
*

We prove that these offer logarithmic expected

doi:10.1142/s0129054111008507
fatcat:y45545cg4jbtjb63vyndlu4lmy
*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

2001
*
SIAM journal on computing (Print)
*

We present

doi:10.1137/s0097539797327209
fatcat:sqfj7ldpargyldr6k5o2uswoom
*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

2000
*
ACM SIGACT News
*

Traditionally

doi:10.1145/356458.356463
fatcat:zajraxw7wzffpfemq6jkzhlhgu
*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]

1997
*
Lecture Notes in Computer Science
*

We present

doi:10.1007/3-540-63165-8_214
fatcat:b6r3hyfduvbmraphz6xmlppddm
*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]

1989
*
Lecture Notes in Computer Science
*

To insert into

doi:10.1007/3-540-51542-9_8
fatcat:6uig7nqlejaxjndlwdx4tsnucm
*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]

2009
*
arXiv
*
pre-print

Improving

arXiv:0903.4130v2
fatcat:o26mps2gyvenfkysnk625hqwiu
*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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