o MST verification
o Linear-time verification
o A randomized algorithm
- . ?? Chazelle ??
- . ?? Pettie ??
o Special cases and related problems
-* Ranking combinatorial objects
+* Approaching Optimality
+
+ . Soft heaps
+ . Robust contractions
+ . An optimal algorithm
+ . Decision trees
+
+* Ranking Combinatorial Objects
o Ranking and unranking
o Ranking of permutations
- practical considerations: katriel:cycle, moret:practice (mention pairing heaps)
- parallel algorithms: p243-cole (see also remarks in Karger and pettie:minirand), pettie:parallel
- bounded expansion classes?
-- restricted cases and arborescences
+- degree-restricted cases and arborescences
- Pettie's paper on random bits (pettie:minirand)
+- random sampling (propp:randommst)
Models:
- consequences of Q-Heaps: Thorup's undirected SSSP etc.
- add more context from thorup:aczero, also mention FP operations
- expand the section on radix-sorting, mention Buchsbaum
+- move Q-Heaps to the chapter on the MST's?
Ranking:
year = {2002},
pages = {1879--1895},
}
+
+@article{ chazelle:softheap,
+ author = {Bernard Chazelle},
+ title = {The soft heap: an approximate priority queue with optimal error rate},
+ journal = {J. ACM},
+ volume = {47},
+ number = {6},
+ year = {2000},
+ issn = {0004-5411},
+ pages = {1012--1027},
+ doi = {http://doi.acm.org/10.1145/355541.355554},
+ publisher = {ACM},
+ address = {New York, NY, USA},
+}
+
+@article{ propp:randommst,
+ author = {James Gary Propp and David Bruce Wilson},
+ title = {How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph},
+ journal = {J. Algorithms},
+ volume = {27},
+ number = {2},
+ year = {1998},
+ issn = {0196-6774},
+ pages = {170--217},
+ doi = {http://dx.doi.org/10.1006/jagm.1997.0917},
+ publisher = {Academic Press, Inc.},
+ address = {Duluth, MN, USA},
+}
+
+@article{ vuillemin:binheap,
+ title={{A Data Structure for Manipulating Priority Queues}},
+ author={Vuillemin, Jean},
+ journal={Communications of the ACM},
+ volume = 21,
+ number = 4,
+ pages={309--315},
+ year={1978}
+}
--- /dev/null
+\ifx\endpart\undefined
+\input macros.tex
+\fi
+
+\chapter{Approaching Optimality}
+
+\section{Soft heaps}
+
+Recently, Chazelle \cite{chazelle:ackermann} and Pettie \cite{pettie:ackermann}
+have presented algorithms for the MST with worst-case time complexity
+$\O(m\timesalpha(m,n))$. We will devote this chapter to their results
+and especially to another algorithm by Pettie and Ramachandran \cite{pettie:optimal},
+which is provably optimal.
+
+At the very heart of all these algorithms lies the \df{soft heap} discovered by
+Chazelle \cite{chazelle:softheap}. It is a~meldable priority queue, roughly
+similar to the Vuillemin's binomial heaps \cite{vuillemin:binheap} or Fredman's
+and Tarjan's Fibonacci heaps \cite{ft:fibonacci}. The soft heaps run faster at
+the expense of \df{corrupting} a~fraction of the inserted elements by raising
+their values (the values are however never decreased). This allows for
+an~trade-off between accuracy and speed controlled by a~parameter~$\varepsilon$.
+The heap operations take $\O(\log(1/\varepsilon))$ amortized time and at every
+moment at most~$\varepsilon n$ elements of the~$n$ elements inserted can be
+corrupted.
+
+\defnn{Soft heap operations}%
+The soft heap contains distinct elements from a~totally ordered universe and it
+supports the following operations:
+\itemize\ibull
+\:$\<Create>(\varepsilon)$ --- create an~empty soft heap with the given accuracy parameter,
+\:$\<Insert>(H,x)$ --- insert a~new element~$x$ to the heap~$H$,
+\:$\<Meld>(H_1,H_2)$ --- form a~new soft heap containing the elements stored in two disjoint
+ heaps $H_1$ and~$H_2$
+ (both heaps must have the same~$\varepsilon$ and they are destroyed in the process),
+\:$\<DeleteMin>(H)$ --- delete the minimum element of the heap~$H$ and return its value
+ (optionally signalling that the value has been corrupted).
+\endlist
+
+\defnn{Soft queues}
+The soft heap is built from \df{soft queues} (we will omit the adjective soft
+in the rest of this section). Each queue has a~shape of a~binary tree.\foot{%
+Actually, Chazelle defines the queues as binomial trees, but he transforms them in ways that are
+somewhat counter-intuitive, albeit well-defined. We prefer describing the queues as binary
+trees with a~special distribution of values. In fact, we get this exact type of binary
+trees from the natural correspondence between general rooted trees and binary trees
+(as described for example in Knuth's Fundamental Algorithms \cite{knuth:fundalg}).
+Also, the original C~code in the Chazelle's paper uses this representation internally.}
+Each vertex~$v$ of the tree remembers a~linked list $\<list>(v)$ of values. The first value in the list
+is called the \df{controlling key} of the vertex, denoted by $\<ckey>(v)$ and defined to be~$+\infty$ if the
+list is empty. The item list in every left son will be used only temporarily and it
+will be kept empty between operations. Only right sons and the root have their lists
+permanently occupied. (See the picture.)
+
+Each vertex is also assigned its \df{rank,} which is a~non-negative integer $\<rank>(v)$.
+The rank of leaves is always zero, the rank of every internal vertex must be strictly
+greater than the ranks of its sons.
+
+\obs
+The rank of the root.....
+Complete binary trees.....
+The master tree.....
+
+
+
+
+\endpart
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