year={1998},
publisher={Springer}
}
+
+@article{ thorup:usssp,
+ title={{Undirected Single-Source Shortest Paths with Positive Integer Weights in Linear Time}},
+ author={Thorup, Mikkel},
+ journal={Journal of the ACM},
+ volume={46},
+ number={3},
+ pages={362--394},
+ year={1999}
+}
+
+@article{ thorup:sssp,
+ author = {Mikkel Thorup},
+ title = {Integer priority queues with decrease key in constant time and the single source shortest paths problem},
+ journal = {Journal of Computer and System Sciences},
+ volume = {69},
+ number = {3},
+ year = {2004},
+ issn = {0022-0000},
+ pages = {330--353},
+ doi = {http://dx.doi.org/10.1016/j.jcss.2004.04.003},
+ publisher = {Academic Press, Inc.},
+ address = {Orlando, FL, USA},
+}
+
+@inproceedings{ hagerup:sssp,
+ author = {Torben Hagerup},
+ title = {Improved Shortest Paths on the Word RAM},
+ booktitle = {ICALP '00: Proceedings of the 27th International Colloquium on Automata, Languages and Programming},
+ year = {2000},
+ isbn = {3-540-67715-1},
+ pages = {61--72},
+ publisher = {Springer-Verlag},
+ address = {London, UK},
+}
\section{Data structures on the RAM}
\id{ramdssect}
-There is a~lot of data structures designed specifically for the RAM, taking
-advantage of both indexing and arithmetics. In many cases, they surpass the known
-lower bounds for the same problem on the~PM and they often achieve constant time
+There is a~lot of data structures designed specifically for the RAM. These structures
+take advantage of both indexing and arithmetics and they often surpass the known
+lower bounds for the same problem on the~PM. In many cases, they achieve constant time
per operation, at least when either the magnitude of the values or the size of
the data structure are suitably bounded.
-A~classical result of this type are the trees of van Emde Boas~\cite{boas:vebt},
-which represent a~subset of the integers $\{0,\ldots,U-1\}$, allowing insertion,
+A~classical result of this type are the trees of van Emde Boas~\cite{boas:vebt}
+which represent a~subset of the integers $\{0,\ldots,U-1\}$. They allow insertion,
deletion and order operations (minimum, maximum, successor etc.) in time $\O(\log\log U)$,
regardless of the size of the subset. If we replace the heap used in the Jarn\'\i{}k's
algorithm (\ref{jarnik}) by this structure, we immediately get an~algorithm
for finding the MST in integer-weighted graphs in time $\O(m\log\log w_{max})$,
-where $w_{max}$ is the maximum weight. We will show later that it is even
-possible to achieve linear time complexity for arbitrary integer weights.
+where $w_{max}$ is the maximum weight.
-A~real breakthrough has been made by Fredman and Willard, who introduced
-the Fusion trees~\cite{fw:fusion} which again perform membership and predecessor
-operation on a~set of $n$~integers, but this time with complexity $\O(\log_W n)$
+A~real breakthrough has been however made by Fredman and Willard who introduced
+the Fusion trees~\cite{fw:fusion}. They again perform membership and predecessor
+operation on a~set of $n$~integers, but with time complexity $\O(\log_W n)$
per operation on a~Word-RAM with $W$-bit words. This of course assumes that
each element of the set fits in a~single word. As $W$ must at least~$\log n$,
-the operations take $\O(\log n/\log\log n)$ and we are able to sort $n$~integers
+the operations take $\O(\log n/\log\log n)$ time and thus we are able to sort $n$~integers
in time~$o(n\log n)$. This was a~beginning of a~long sequence of faster and
faster sorting algorithms, culminating with the work by Thorup and Han.
They have improved the time complexity of integer sorting to $\O(n\log\log n)$ deterministically~\cite{han:detsort}
and expected $\O(n\sqrt{\log\log n})$ for randomized algorithms~\cite{hanthor:randsort},
both in linear space.
-Despite the recent progress, the corner-stone of most RAM data structures
-is still the representation of data structures by integers introduced by Fredman
-and Willard. It will also form a~basis for the rest of this chapter.
-
-\FIXME{Add more history.}
+The Fusion trees themselves have very limited use in graph algorithms, but the
+principles behind them are ubiquitious in many other data structures and these
+will serve us well and often. We are going to build the theory of Q-heaps in
+Section \ref{qheaps}, which will later lead to a~linear-time MST algorithm
+for arbitrary integer weights in Section \ref{iteralg}. Other such structures
+will help us in building linear-time RAM algorithms for computing the ranks
+of various combinatorial structures in Chapter~\ref{rankchap}.
+
+Outside our area, important consequences of these data structures include the
+Thorup's $\O(m)$ algorithm for single-source shortest paths in undirected
+graphs with positive integer weights \cite{thorup:usssp} and his $\O(m\log\log
+n)$ algorithm for the same problem in directed graphs \cite{thorup:sssp}. Both
+algorithms have been then significantly simplified by Hagerup
+\cite{hagerup:sssp}.
+
+Despite the progress in the recent years, the corner-stone of all RAM structures
+is still the representation of combinatorial objects by integers introduced by
+Fredman and Willard. It will also form a~basis for the rest of this chapter.
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