\fi
\chapter{Fine Details of Computation}
+\id{ramchap}
\section{Models and machines}
-Traditionally, computer scientists use a~variety of computational models
+Traditionally, computer scientists have been using a~variety of computational models
as a~formalism in which their algorithms are stated. If we were studying
-NP-completeness, we could safely assume that all the models are equivalent,
+NP-complete\-ness, we could safely assume that all these models are equivalent,
possibly up to polynomial slowdown which is negligible. In our case, the
differences between good and not-so-good algorithms are on a~much smaller
-scale. In this chapter, we will replace the usual ``yardsticks'' by a~micrometer,
+scale. In this chapter, we will replace the usual ``tape measure'' by a~micrometer,
state our computation models carefully and develop a repertoire of basic
-data structures taking advantage of the fine details of the models.
+data structures tailor-made for the fine details of the models.
We would like to keep the formalism close enough to the reality of the contemporary
computers. This rules out Turing machines and similar sequentially addressed
time to access a~single element of an~$n$-element array. It is hard to say which
way is superior --- while most ``real'' computers have instructions for constant-time
indexing, it seems to be physically impossible to fulfil this promise regardless of
-the size of memory. Indeed, at the level of logical gates inside the computer,
+the size of addressable memory. Indeed, at the level of logical gates inside the computer,
the depth of the actual indexing circuits is logarithmic.
In recent decades, most researchers in the area of combinatorial algorithms
\para
The \df{Random Access Machine (RAM)} is not a~single coherent model, but rather a~family
-of closely related machines, sharing the following properties.
+of closely related machines which share the following properties.
(See Cook and Reckhow \cite{cook:ram} for one of the usual formal definitions
and Hagerup \cite{hagerup:wordram} for a~thorough description of the differences
between the RAM variants.)
The \df{memory} of the machine is represented by an~array of \df{memory cells}
-addressed by non-negative integers, each of them containing a~single non-negative integer.
+addressed by non-negative integers. Each cell contains a~single non-negative integer.
The \df{program} is a~finite sequence of \df{instructions} of two basic kinds: calculation
instructions and control instructions.
counting not only the values, but also the addresses of the respective memory
cells.
\:Place a~limit on the size of the numbers ---define the \df{word size~$W$,}
- the number of bits available in the memory cells--- and keep the cost of
+ the number of bits available in each memory cell--- and keep the cost of
instructions and memory cells constant. The word size must not be constant,
since we can address only~$2^W$ cells of memory. If the input of the algorithm
is stored in~$N$ cells, we need~$W\ge\log N$ just to be able to read the input.
On the other hand, we are interested in polynomial-time algorithms only, so $\Theta(\log N)$-bit
numbers should be sufficient. In practice, we pick~$W$ to be the larger of
$\Theta(\log N)$ and the size of integers used in the algorithm's input and output.
- We will call an integer which fits in a~single memory cell a~\df{machine word.}
+ We will call an integer that fits in a~single memory cell a~\df{machine word.}
\endlist
Both restrictions easily avoid the problems of unbounded parallelism. The first
standard complexity classes, but the calculations of time and space complexity tend
to be somewhat tedious. What more, when compared with the RAM with restricted
word size, the complexities are usually exactly $\Theta(W)$ times higher.
-This does not hold in general (consider a~program which uses many small numbers
+This does not hold in general (consider a~program that uses many small numbers
and $\O(1)$ large ones), but it is true for the algorithms we are interested in.
Therefore we will always assume that the operations have unit cost and we make
sure that all numbers are limited by the available word size.
fan-in and fan-out. This includes all operations of the Word-RAM except for multiplication,
division and remainders, and also many other operations like computing the Hamming
weight (number of bits set in a~given number).
-\:Both restrictions at once.
+\:Both restrictions combined.
\endlist
-Thorup discusses the usual techniques employed by RAM algorithms in~\cite{thorup:aczero}
+Thorup \cite{thorup:aczero} discusses the usual techniques employed by RAM algorithms
and he shows that they work on both Word-RAM and ${\rm AC}^0$-RAM, but the combination
of the two restrictions is too weak. On the other hand, the intersection of~${\rm AC}^0$
with the instruction set of modern processors is already strong enough (e.g., when we
When speaking of the \df{RAM,} we implicitly mean the version with numbers limited
by a~specified word size of $W$~bits, unit cost of operations and memory cells and the instruction
set of the Word-RAM. This corresponds to the usage in recent algorithmic literature,
-although the authors rarely mention the details. In some cases, a~non-uniform variant
-of the Word-RAM is considered as well (e.g., in~\cite{hagerup:dd}):
+although the authors rarely mention the details.
+
+In some cases, a~non-uniform variant
+of the Word-RAM is considered as well (e.g., by Hagerup \cite{hagerup:dd}):
\defn\id{nonuniform}%
A~Word-RAM is called \df{weakly non-uniform,} if it is equipped with $\O(1)$-time
access to a~constant number of word-sized constants, which depend only on the word
size. These are called \df{native constants} and they are available in fixed memory
cells when the program starts. (By analogy with the high-level programming languages,
-these constants can be thought of as computed at ``compile time.'')
+these constants can be thought of as computed at ``compile time''.)
\para
-The \df{Pointer Machine (PM)} also does not have any well established definition. The
-various kinds of pointer machines are mapped by Ben-Amram in~\cite{benamram:pm},
+The \df{Pointer Machine (PM)} also does not seem to have any well established definition. The
+various kinds of pointer machines are examined by Ben-Amram in~\cite{benamram:pm},
but unlike the RAM's they turn out to be equivalent up to constant slowdown.
Our definition will be closely related to the \em{linking automaton} proposed
by Knuth in~\cite{knuth:fundalg}, we will only adapt it to use RAM-like
The PM works with two different types of data: \df{symbols} from a~finite alphabet
and \df{pointers}. The memory of the machine consists of a~fixed amount of \df{registers}
(some of them capable of storing a~single symbol, each of the others holds a~single pointer)
-and an~arbitrary amount of \df{cells}. The structure of all cells is the same: each of them
+and an~arbitrary amount of \df{cells}. The structure of all cells is the same: each cell
again contains a~fixed number of fields for symbols and pointers. Registers can be addressed
directly, the cells only via pointers --- by using a~pointer stored either in a~register,
-or in a~cell pointed to by a~register (longer chains of pointers cannot be followed in
-constant time).
+or in a~cell pointed to by a~register. Longer chains of pointers cannot be followed in
+constant time.
We can therefore view the whole memory as a~directed graph, whose vertices
correspond to the cells (the registers are stored in a~single special cell).
The outgoing edges of each vertex correspond to pointer fields of the cells and they are
-labelled with distinct labels drawn from a~finite set. In addition to that,
-each vertex contains a~fixed amount of symbols. The program can directly access
+labeled with distinct labels drawn from a~finite set. In addition to that,
+each vertex contains a~fixed amount of symbols. The machine can directly access
vertices within distance~2 from the register vertex.
The program is a~finite sequence of instructions of the following kinds:
\itemize\ibull
\:\df{symbol instructions,} which read a~pair of symbols, apply an~arbitrary
- function on them and write the result to a~symbol register or field;
+ function to them and write the result to a~symbol register or field;
\:\df{pointer instructions} for assignment of pointers to pointer registers/fields
- and for creation of new memory cells (a~pointer to the new cell is assigned
+ and for creation of new memory cells (a~pointer to the new cell is stored into a~register
immediately);
\:\df{control instructions} --- similarly to the RAM; conditional jumps can decide
- on~arbitrary unary relations on symbols and compare pointers for equality.
+ arbitrary unary relations on symbols and compare pointers for equality.
\endlist
Time and space complexity are defined in the straightforward way: all instructions
and arithmetic instructions. We can emulate both with poly-logarithmic slowdown,
but it will turn out that they are rarely needed in graph algorithms. We are
also going to prove that the RAM is strictly stronger, so we will prefer to
-formulate our algorithms in the PM model and use RAM only when necessary.
+formulate our algorithms for the PM and use the RAM only when necessary.
\thm
Every program for the Word-RAM with word size~$W$ can be translated to a~PM program
algorithms, but the quadratic bound is good enough for our purposes.}
\qed
-\FIXME{Add references, especially to the unbounded parallelism remark.}
-
\thm
Every program for the PM running in polynomial time can be translated to a~program
computing the same on the Word-RAM with only $\O(1)$ slowdown.
\para
There are also \df{randomized} versions of both machines. These are equipped
with an~additional instruction for generating a~single random bit. The standard
-techniques of design and analysis of randomized algorithms apply (see for
+methods of design and analysis of randomized algorithms can be used (see for
example Motwani and Raghavan~\cite{motwani:randalg}).
-\FIXME{Consult sources. Does it make more sense to generate random words at once on the RAM?}
-
\rem
-There is one more interesting machine: the \df{Immutable Pointer Machine} (see
+There is one more interesting machine: the \df{Immutable Pointer Machine} (mentioned for example in
the description of LISP machines in \cite{benamram:pm}). It differs from the
ordinary PM by the inability to modify existing memory cells. Only the contents
of the registers are allowed to change. All cell modifications thus have to
persistent --- it is possible to return to a~previously visited state by recalling
the $\O(1)$ values of the registers (everything else could not have changed
since that time) and ``fork'' the computations. This corresponds to the semantics
-of pure functional languages, e.g., Haskell~\cite{jones:haskell}.
+of pure functional languages, e.g., of Haskell~\cite{jones:haskell}.
Unless we are willing to accept a~logarithmic penalty in execution time and space
(in fact, our emulation of the Word-RAM on the PM can be easily made immutable),
the design of efficient algorithms for the immutable PM requires very different
-techniques. Therefore, we will concentrate on the imperative models instead
+techniques. Therefore, we will be interested in the imperative models only
and refer the interested reader to the thorough treatment of purely functional
data structures in the Okasaki's monograph~\cite{okasaki:funcds}.
%--------------------------------------------------------------------------------
-\section{Bucket sorting and contractions}\id{bucketsort}%
+\section{Bucket sorting and unification}\id{bucketsort}%
-The Contractive Bor\o{u}vka's algorithm (\ref{contbor}) needed to contract a~given
-set of edges in the current graph and flatten it afterwards, all this in time $\O(m)$.
-We have spared the technical details for this section and they will be useful
-in further algorithms, too.
+The Contractive Bor\o{u}vka's algorithm (\ref{contbor}) needs to contract a~given
+set of edges in the current graph and then flatten the graph, all this in time $\O(m)$.
+We have spared the technical details for this section, in which we are going to
+explain several rather general techniques based on bucket sorting.
-As already suggested, the contractions can be performed by building an~auxiliary
-graph and finding its connected components. Thus we will take care of the flattening
-only.
+As we have already suggested in the proof of Lemma \ref{contiter}, contractions
+can be performed in linear time by building an~auxiliary graph and finding its
+connected components. We will thus take care only of the subsequent flattening.
-\para
-On the RAM, it is sufficient to sort the edges lexicographically (each edge viewed
-as an~ordered pair of vertex identifiers with the smaller of the identifiers placed
-first). We can do that by a two-pass bucket sort with~$n$ buckets corresponding
-to the vertex identifiers.
+\paran{Flattening on RAM}%
+On the RAM, we can view the edges as ordered pairs of vertex identifiers with the
+smaller of the identifiers placed first. We sort these pairs lexicographically. This brings
+parallel edges together, so that a~simple linear scan suffices to find each bunch
+of parallel edges and to remove all but the lightest one.
+Lexicographic sorting of pairs can be accomplished in linear time by a~two-pass
+bucket sort with $n$~buckets corresponding to the vertex identifiers.
-However, there is a~catch in this. Suppose that we use the standard representation
+However, there is a~catch. Suppose that we use the standard representation
of graphs by adjacency lists whose heads are stored in an array indexed by vertex
identifiers. When we contract and flatten the graph, the number of vertices decreases,
but if we inherit the original vertex identifiers, the arrays will still have the
-same size. Hence we spend a~super-linear amount of time on scanning the increasingly
+same size. We could then waste a~super-linear amount of time by scanning the increasingly
sparse arrays, most of the time skipping unused entries.
-To avoid this, we have to renumber the vertices after each contraction to component
-identifiers from the auxiliary graph and we create a~new vertex array. This way,
-the representation of the graph will be kept linear with respect to the size of the
-current graph.
+To avoid this problem, we have to renumber the vertices after each contraction to component
+identifiers from the auxiliary graph and create a~new vertex array. This helps
+keep the size of the representation of the graph linear with respect to its current
+size.
-\para
-The pointer representation of graphs does not suffer from sparsity as the vertices
+\paran{Flattening on PM}%
+The pointer representation of graphs does not suffer from sparsity since the vertices
are always identified by pointers to per-vertex structures. Each such structure
then contains all attributes associated with the vertex, including the head of its
adjacency list. However, we have to find a~way how to perform bucket sorting
-without arrays.
+without indexing of arrays.
+
+We will keep a~list of the per-vertex structures and we will use it to establish the order of~vertices.
+Each such structure will be endowed with a~pointer to the head of the list of items in
+the corresponding bucket. Inserting an~edge to a~bucket can be then done in constant time
+and scanning the contents of all~$n$ buckets takes $\O(n+m)$ time.
+
+At last, we must not forget that while it was easy to \df{normalize} the pairs on the RAM
+by putting the smaller identifier first, this fails on the PM because we can directly
+compare the identifiers only for equality. We can work around this again by bucket-sorting:
+we sort the multiset $\{ (x,i) \mid \hbox{$x$~occurs in the $i$-th pair} \}$ on~$x$.
+Then we reset all pairs and re-insert the values back in their increasing order.
+This also takes $\O(n+m)$.
+
+\paran{Tree isomorphism}%
+Another nice example of pointer-based radix sorting is a~Pointer Machine algorithm for
+deciding whether two rooted trees are isomorphic. Let us assume for a~moment that
+the outdegree of each vertex is at most a~fixed constant~$k$. We begin by sorting the subtrees
+of both trees by their depth. This can be accomplished by running depth-first search to calculate
+the depths and bucket-sorting them with $n$~buckets afterwards.
+
+Then we proceed from depth~0 to the maximum depth and for each depth we identify
+the isomorphism equivalence classes of the particular subtrees. We will assign
+unique \df{codes} (identifiers) to all such classes; at most~$n+1$ of them are needed as there are
+$n+1$~subtrees in the tree (including the empty subtree). As the PM does not
+have numbers as an~elementary type, we create a~``\df{yardstick}'' ---a~list
+of $n+1$~distinct items--- and we use pointers to these ``ticks'' as identifiers.
+When we are done, isomorphism of the whole trees can be decided by comparing the
+codes assigned to their roots.
+
+Suppose that classes of depths $0,\ldots,d-1$ are already computed and we want
+to identify those of depth~$d$. We will denote their count of~$n_d$. We take
+a~root of every such tree and label it with an~ordered $k$-tuple of codes
+of its subtrees; when it has less than $k$ sons, we pad the tuple with empty
+subtrees. Tuples corresponding to isomorphic subtrees are identical up to
+reordering of elements. We therefore sort the codes inside each tuple and then
+sort the tuples, which brings the equivalent tuples together.
+
+The first sort (inside the tuples) would be easy on the RAM, but on the PM we
+have to use the normalization trick mentioned above. The second sort is
+a~straightforward $k$-pass bucket sort.
+
+If we are not careful, a~single sorting pass takes $\O(n_d + n)$ time, because
+while we have only $n_d$~items to sort, we have to scan all $n$~buckets. This can
+be easily avoided if we realize that the order of the buckets does not need to be
+fixed --- in every pass, we can use a~completely different order and it still
+does bring the equivalent tuples together. Thus we can keep a~list of buckets
+that are used in the current pass and look only inside these buckets. This way,
+we reduce the time spent in a~single pass to $\O(n_d)$ and the whole algorithm
+takes just $\O(\sum_d n_d) = \O(n)$.
+
+Our algorithm can be easily modified for trees with unrestricted degrees.
+We replace the fixed $d$-tuples by general sequences of codes. The first
+sort does not need any changes. In the second sort, we proceed from the first
+position to the last one and after each bucket-sorting pass we put aside the sequences
+that have just ended. They are obviously not equivalent to any other sequences.
+The time complexity of the second sort is linear in the sum of the lengths of the sequences, which is
+$n_{d+1}$ for depth~$d$. We can therefore decide isomorphism of the whole trees
+in time $\O(\sum_d (n_d + n_{d+1})) = \O(n)$.
+
+The unification of sequences by bucket sorting will be useful in many
+other situations, so we will state it as a~separate lemma:
+
+\lemman{Sequence unification}\id{suniflemma}%
+Partitioning of a~collection of sequences $S_1,\ldots,S_n$, whose elements are
+arbitrary pointers and symbols from a~finite alphabet, to equality classes can
+be performed on the Pointer Machine in time $\O(n + \sum_i \vert S_i \vert)$.
+
+\rem
+The first linear-time algorithm that partitions all subtrees to isomorphism equivalence
+classes is probably due to Zemlayachenko \cite{zemlay:treeiso}, but it lacks many
+details. Dinitz et al.~\cite{dinitz:treeiso} have recast this algorithm in modern
+terminology and filled the gaps. Our algorithm is easier to formulate than those,
+because it replaces the need for auxiliary data structures by more elaborate bucket
+sorting.
+
+\paran{Topological graph computations}%
+Many graph algorithms are based on the idea of so called \df{micro/macro decomposition:}
+We decompose a~graph to subgraphs on roughly~$k$ vertices and solve the problem
+separately inside these ``micrographs'' and in the ``macrograph'' obtained by
+contraction of the micrographs. If $k$~is small enough, many of the micrographs
+are isomorphic, so we can compute the result only once for each isomorphism class
+and recycle it for all micrographs of that class. On the other hand, the macrograph
+is roughly $k$~times smaller than the original graph, so we can use a~less efficient
+algorithm and it will still run in linear time with respect to the size of the original
+graph.
+
+This kind of decomposition is traditionally used for trees, especially in the
+algorithms for the Lowest Common Ancestor problem (cf.~Section \ref{verifysect}
+and the survey paper \cite{alstrup:nca}) and for online maintenance of marked ancestors
+(cf.~Alstrup et al.~\cite{alstrup:marked}). Let us take a~glimpse at what happens when
+we decompose a~tree with $k$ set to~$1/4\cdot\log n$. There are at most $2^{2k} = \sqrt n$ non-isomorphic subtrees of size~$k$,
+because each isomorphism class is uniquely determined by the sequence of $2k$~up/down steps
+performed by depth-first search of the tree. Suppose that we are able to decompose the input and identify
+the equivalence classes of microtrees in linear time, then solve the problem in time $\O(\poly(k))$ for
+each microtree and finally in $\O(n'\log n')$ for the macrotree of size $n'=n/k$. When we put these pieces
+together, we get an~algorithm for the whole problem which achieves time complexity $\O(n
++ \sqrt{n}\cdot\poly(\log n) + n/\log n\cdot\log(n/\log n)) = \O(n)$.
+
+Decompositions are usually implemented on the RAM, because subgraphs can be easily
+encoded in numbers, and these can be then used to index arrays containing the precomputed
+results. As the previous algorithm for subtree isomorphism shows, indexing is not strictly
+required for identifying equivalent microtrees and it can be replaced by bucket
+sorting on the Pointer Machine. Buchsbaum et al.~\cite{buchsbaum:verify} have extended
+this technique to general graphs in form of so called topological graph computations.
+Let us define them.
+
+\defn
+A~\df{graph computation} is a~function that takes a~\df{labeled undirected graph} as its input. The labels of
+vertices and edges can be arbitrary symbols drawn from a~finite alphabet. The output
+of the computation is another labeling of the same graph. This time, the vertices and
+edges can be labeled with not only symbols of the alphabet, but also with pointers to the vertices
+and edges of the input graph, and possibly also with pointers to outside objects.
+A~graph computation is called \df{topological} if it produces isomorphic
+outputs for isomorphic inputs. The isomorphism of course has to preserve not only
+the structure of the graph, but also the labels in the obvious way.
+
+\obs
+The topological graph computations cover a~great variety of graph problems, ranging
+from searching for matchings or Eulerian tours to finding Hamilton circuits.
+The MST problem itself however does not belong to this class, because we do not have any means
+of representing the edge weights as labels, unless there is only a~fixed amount
+of possible values.
+
+As in the case of tree decompositions, we would like to identify the equivalent subgraphs
+and process only a~single instance from each equivalence class. We need to be careful
+with the definition of the equivalence classes, because
+graph isomorphism is known to be computationally hard (it is one of the few
+problems that are neither known to lie in~$\rm P$ nor to be $\rm NP$-complete;
+see Arvind and Kurur \cite{arvind:isomorph} for recent results on its complexity).
+We will therefore manage with a~weaker form of equivalence, based on some sort
+of graph encodings:
+
+\defn
+A~\df{canonical encoding} of a~given labeled graph represented by adjacency lists
+is obtained by running the depth-first search on the graph and recording its traces.
+We start with an~empty encoding. When we enter
+a~vertex, we assign an~identifier to it (again using a~yardstick to represent numbers)
+and we append the label of this vertex to the encoding. Then we scan all back edges
+going from this vertex and append the identifiers of their destinations, accompanied
+by the edges' labels. Finally we append a~special terminator to mark the boundary
+between the code of this vertex and its successor.
+
+\obs
+The canonical encoding is well defined in the sense that non-iso\-morphic graphs always
+receive different encodings. Obviously, encodings of isomorphic graphs can differ,
+depending on the order of vertices and also of the adjacency lists. A~graph
+on~$n$ vertices with $m$~edges is assigned an~encoding of length at most $2n+2m$ ---
+for each vertex, we record its label and a~single terminator; edges contribute
+by identifiers and labels. These encodings can be constructed in linear time and
+in the same time we can also create a~graph corresponding to a~given encoding.
+We will use the encodings for our unification of graphs:
+
+\defn
+For a~collection~$\C$ of graphs, we define $\vert\C\vert$ as the number of graphs in
+the collection and $\Vert\C\Vert$ as their total size, i.e., $\Vert\C\Vert = \sum_{G\in\C} n(G) + m(G)$.
+
+\lemman{Graph unification}\id{guniflemma}%
+A~collection~$\C$ of labeled graphs can be partitioned into classes which share the same
+canonical encoding in time $\O(\Vert\C\Vert)$ on the Pointer Machine.
+
+\proof
+Construct canonical encodings of all the graphs and then apply the Sequence unification lemma
+(\ref{suniflemma}) on them.
+\qed
+
+\para
+When we want to perform a~topological computation on a~collection~$\C$ of graphs
+with $k$~vertices, we first precompute its result for a~collection~$\cal G$ of \df{generic graphs}
+corresponding to all possible canonical encodings on $k$~vertices. Then we use unification to match
+the \df{actual graphs} in~$\C$ to the generic graphs in~$\cal G$. This gives us the following
+theorem:
+
+\thmn{Topological computations, Buchsbaum et al.~\cite{buchsbaum:verify}}\id{topothm}%
+Suppose that we have a~topological graph computation~$\cal T$ that can be performed in time
+$T(k)$ for graphs on $k$~vertices. Then we can run~$\cal T$ on a~collection~$\C$
+of labeled graphs on~$k$ vertices in time $\O(\Vert\C\Vert + (k+s)^{k(k+2)}\cdot (T(k)+k^2))$,
+where~$s$ is a~constant depending only on the number of symbols used as vertex/edge labels.
-We will keep a~list of the per-vertex structures which defines the order of~vertices.
-Each such structure will contain a~pointer to the head of the corresponding bucket,
-again stored as a~list. Putting an~edge to a~bucket can be done in constant time then,
-scanning all~$n$ buckets takes $\O(n+m)$ time.
+\proof
+A~graph on~$k$ vertices has less than~$k^2/2$ edges, so the canonical encodings of
+all such graphs are shorter than $2k + 2k^2/2 = k(k+2)$. Each element of the encoding
+is either a~vertex identifier, or a~symbol, or a~separator, so it can attain at most $k+s$
+possible values for some fixed~$s$.
+We can therefore enumerate all possible encodings and convert them to a~collection $\cal G$
+of all generic graphs such that $\vert{\cal G}\vert \le (k+s)^{k(k+2)}$ and $\Vert{\cal G}\Vert
+\le \vert{\cal G}\vert \cdot k^2$.
+
+We run the computation on all generic graphs in time $\O(\vert{\cal G}\vert \cdot T(k))$
+and then we use the Unification lemma (\ref{guniflemma}) on the union of the collections
+$\C$ and~$\cal G$ to match the generic graphs with the equivalent actual graphs in~$\C$
+in time $\O(\Vert\C\Vert + \Vert{\cal G}\Vert)$.
+Finally we create a~copy of the generic result for each of the actual graphs.
+If the computation uses pointers to the input vertices in its output, we have to
+redirect them to the actual input vertices, which we can do by associating
+the output vertices that refer to an~input vertex with the corresponding places
+in the encoding of the input graph. This way, the whole output can be generated in time
+$\O(\Vert\C\Vert + \Vert{\cal G}\Vert)$.
+\looseness=1 %%HACK%%
+\qed
-\FIXME{Add an example of locally determined orders, e.g., tree isomorphism?}
+\rem
+The topological computations and the Graph unification lemma will play important
+roles in Sections \ref{verifysect} and \ref{optalgsect}.
%--------------------------------------------------------------------------------
\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.
+the data structure is 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 is the tree of van Emde Boas~\cite{boas:vebt}
+which represents a~subset of the integers $\{0,\ldots,U-1\}$. It allows 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.
-
-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 $\O(\log_W n)$
-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
-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.}
+where $w_{max}$ is the maximum weight.
+
+A~real breakthrough has however been made by Fredman and Willard who introduced
+the Fusion trees~\cite{fw:fusion}. These trees also offer membership and predecessor
+operations on a~set of $n$~word-sized integers, but they reach time complexity $\O(\log_W n)$
+per operation on a~Word-RAM with $W$-bit words. As $W$ must be at least~$\log n$,
+the operations take $\O(\log n/\log\log n)$ time each and thus we are able to sort
+$n$~integers in time~$o(n\log n)$. (Of course, when $W=\Theta(\log n)$, we can even
+do that in linear time using radix-sort in base~$n$; it is the cases with large~$W$
+that is important.)
+Since then, a~long sequence of faster and faster sorting algorithms has
+emerged, culminating with the work of 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.
+
+The Fusion trees themselves have very limited use in graph algorithms, but the
+principles behind them are ubiquitous 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 RAM 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.
%--------------------------------------------------------------------------------
-\section{Bits and vectors}
+\section{Bits and vectors}\id{bitsect}
In this rather technical section, we will show how the RAM can be used as a~vector
computer to operate in parallel on multiple elements, as long as these elements
Another approach to division is using the improved elementary school algorithm as described
by Knuth in~\cite{knuth:seminalg}. It uses $\O(k^2)$ steps, but the steps involve
calculation of the most significant bit set in a~word. We will show below that it
-can be done in constant time, but we have to be careful to avoid division instructions.
+can be done in constant time, but we have to be careful to avoid division instructions in it.
\qed
\notan{Bit strings}\id{bitnota}%
We will work with binary representations of natural numbers by strings over the
alphabet $\{\0,\1\}$: we will use $\(x)$ for the number~$x$ written in binary,
-$\(x)_b$ for the same padded to exactly $b$ bits by adding zeroes at the beginning
-and $x[k]$ for the value of the $k$-th bit of~$x$.
+$\(x)_b$ for the same padded to exactly $b$ bits by adding leading zeroes,
+and $x[k]$ for the value of the $k$-th bit of~$x$ (with a~numbering of bits such that $2^k[k]=1$).
The usual conventions for operations on strings will be utilized: When $s$
and~$t$ are strings, we write $st$ for their concatenation and
$s^k$ for the string~$s$ repeated $k$~times.
\defn
The \df{bitwise encoding} of a~vector ${\bf x}=(x_0,\ldots,x_{d-1})$ of~$b$-bit numbers
-is an~integer~$x$ such that $\(x)=\0\(x_{d-1})_b\0\(x_{d-2})_b\0\ldots\0\(x_0)_b = \sum_i 2^{(b+1)i}\cdot x_i$.
-(We have interspersed the elements with \df{separator bits.})
+is an~integer~$x$ such that $\(x)=\(x_{d-1})_b\0\(x_{d-2})_b\0\ldots\0\(x_0)_b$. In other
+words, $x = \sum_i 2^{(b+1)i}\cdot x_i$. (We have interspersed the elements with \df{separator bits.})
\notan{Vectors}\id{vecnota}%
We will use boldface letters for vectors and the same letters in normal type
-for their encodings. The elements of a~vector~${\bf x}$ will be denoted as
+for the encodings of these vectors. The elements of a~vector~${\bf x}$ will be written as
$x_0,\ldots,x_{d-1}$.
\para
-If we wish for the whole vector to fit in a~single word, we need $(b+1)d\le W$.
+If we want to fit the whole vector in a~single word, the parameters $b$ and~$d$ must satisfy
+the condition $(b+1)d\le W$.
By using multiple-precision arithmetics, we can encode all vectors satisfying $bd=\O(W)$.
-We will now describe how to translate simple vector manipulations to $\O(1)$ RAM operations
-on their codes. As we are interested in asymptotic complexity only, we prefer clarity
-of the algorithms over saving instructions. Among other things, we freely use calculations
+We will now describe how to translate simple vector manipulations to sequences of $\O(1)$ RAM operations
+on their codes. As we are interested in asymptotic complexity only, we will prefer clarity
+of the algorithms over saving instructions. Among other things, we will freely use calculations
on words of size $\O(bd)$, assuming that the Multiple-precision lemma comes to save us
when necessary.
\para
First of all, let us observe that we can use $\band$ and $\bor$ with suitable constants
-to write zeroes or ones to an~arbitrary set of bit positions in constant time. These operations
-are usually called \df{bit masking}. Also, an~element of a~vector can be extracted or replaced
-using masking and shifts.
+to write zeroes or ones to an~arbitrary set of bit positions at once. These operations
+are usually called \df{bit masking}. Also, any element of a~vector can be extracted or
+replaced by a~different value in $\O(1)$ time by masking and shifts.
\newdimen\slotwd
\def\setslot#1{\setbox0=#1\slotwd=\wd0}
\medskip
}
-\algn{Operations on vectors with $d$~elements of $b$~bits each}
+\algn{Operations on vectors with $d$~elements of $b$~bits each}\id{vecops}
\itemize\ibull
-\:$\<Replicate>(\alpha)$ --- creates a~vector $(\alpha,\ldots,\alpha)$:
+\:$\<Replicate>(\alpha)$ --- Create a~vector $(\alpha,\ldots,\alpha)$:
\alik{\<Replicate>(\alpha)=\alpha\cdot(\0^b\1)^d. \cr}
-\:$\<Sum>(x)$ --- calculates the sum of the elements of~${\bf x}$, assuming that
-it fits in~$b$ bits:
+\:$\<Sum>(x)$ --- Calculate the sum of the elements of~${\bf x}$, assuming that
+the result fits in $b$~bits:
\alik{\<Sum>(x) = x \bmod \1^{b+1}. \cr}
-This works because when we work modulo~$\1^{b+1}$, the number $2^{b+1}=\1\0^{b+1}$
+This is correct because when we calculate modulo~$\1^{b+1}$, the number $2^{b+1}=\1\0^{b+1}$
is congruent to~1 and thus $x = \sum_i 2^{(b+1)i}\cdot x_i \equiv \sum_i 1^i\cdot x_i \equiv \sum_i x_i$.
-As the result should fit in $b$~bits, the modulo cannot change it.
+As the result should fit in $b$~bits, the modulo makes no difference.
If we want to avoid division, we can use double-precision multiplication instead:
\[r_{d-1}] \dd \[r_2] \[r_1] \[s_d] \dd \[s_3] \[s_2] \[s_1] \cr
}
-This way, we even get the vector of all partial sums:
+This way, we also get all partial sums:
$s_k=\sum_{i=0}^{k-1}x_i$, $r_k=\sum_{i=k}^{d-1}x_i$.
-\:$\<Cmp>(x,y)$ --- element-wise comparison of~vectors ${\bf x}$ and~${\bf y}$,
-i.e., a~vector ${\bf z}$ such that $z_i=1$ iff $x_i<y_i$.
+\:$\<Cmp>(x,y)$ --- Compare vectors ${\bf x}$ and~${\bf y}$ element-wise,
+i.e., make a~vector~${\bf z}$ such that $z_i=1$ if $x_i<y_i$ and $z_i=0$ otherwise.
We replace the separator zeroes in~$x$ by ones and subtract~$y$. These ones
change back to zeroes exactly at the positions where $x_i<y_i$ and they stop
}
It only remains to shift the separator bits to the right positions, negate them
-and zero out all other bits.
+and mask out all other bits.
-\:$\<Rank>(x,\alpha)$ --- return the number of elements of~${\bf x}$ which are less than~$\alpha$,
+\:$\<Rank>(x,\alpha)$ --- Return the number of elements of~${\bf x}$ which are less than~$\alpha$,
assuming that the result fits in~$b$ bits:
\alik{
\<Rank>(x,\alpha) = \<Sum>(\<Cmp>(x,\<Replicate>(\alpha))). \cr
}
-\:$\<Insert>(x,\alpha)$ --- insert~$\alpha$ into a~sorted vector $\bf x$:
+\:$\<Insert>(x,\alpha)$ --- Insert~$\alpha$ into a~sorted vector $\bf x$:
-Calculate $k = \<Rank>(x,\alpha)$ first, then insert~$\alpha$ as the $k$-th
-field of~$\bf x$ using masking operations.
+We calculate the rank of~$\alpha$ in~$x$ first, then we insert~$\alpha$ into the particular
+field of~$\bf x$ using masking operations and shifts.
-\:$\<Unpack>(\alpha)$ --- create a~vector whose elements are the bits of~$\(\alpha)_d$.
-In other words, we insert blocks~$\0^b$ between the bits of~$\alpha$. Assuming that $b\ge d$,
+\algo
+\:$k\=\<Rank>(x,\alpha)$.
+\:$\ell\=x \band \1^{(b+1)(n-k)}\0^{(b+1)k}$. \cmt{``left'' part of the vector}
+\:$r=x \band \1^{(b+1)k}$. \cmt{``right'' part}
+\:Return $(\ell\shl (b+1)) \bor (\alpha\shl ((b+1)k)) \bor r$.
+\endalgo
+
+\:$\<Unpack>(\alpha)$ --- Create a~vector whose elements are the bits of~$\(\alpha)_d$.
+In other words, insert blocks~$\0^b$ between the bits of~$\alpha$. Assuming that $b\ge d$,
we can do it as follows:
\algo
\:$x\=\<Replicate>(\alpha)$.
\:$y\=(2^{b-1},2^{b-2},\ldots,2^0)$. \cmt{bitwise encoding of this vector}
\:$z\=x \band y$.
-\:Return $\<Cmp>(z,y)$.
+\:Return $\<Cmp>(z,y) \bxor (\0^b\1)^d$.
\endalgo
Let us observe that $z_i$ is either zero or equal to~$y_i$ depending on the value
of the $i$-th bit of the number~$\alpha$. Comparing it with~$y_i$ normalizes it
-to either zero or one.
+to either zero or one, but in the opposite way than we need, so we flip the bits
+by an~additional $\bxor$.
-\:$\<Unpack>_\varphi(\alpha)$ --- like \<Unpack>, but changes the order of the
-bits according to a~fixed permutation~$\varphi$: The $i$-th element of the
+\:$\<Unpack>_\pi(\alpha)$ --- Like \<Unpack>, but change the order of the
+bits according to a~fixed permutation~$\pi$: The $i$-th element of the
resulting vector is equal to~$\alpha[\pi(i)]$.
-Implemented as above, but with mask~$y=(2^{\pi(b-1)},\ldots,2^{\pi(0)})$.
+Implemented as above, but with a~mask $y=(2^{\pi(b-1)},\ldots,2^{\pi(0)})$.
-\:$\<Pack>(x)$ --- the inverse of \<Unpack>: given a~vector of zeroes and ones,
-it produces a~number whose bits are the elements of the vector (in other words,
+\:$\<Pack>(x)$ --- The inverse of \<Unpack>: given a~vector of zeroes and ones,
+produce a~number whose bits are the elements of the vector (in other words,
it crosses out the $\0^b$ blocks).
We interpret the~$x$ as an~encoding of a~vector with elements one bit shorter
-and sum these elements. For example, when $n=4$ and~$b=4$:
+and we sum these elements. For example, when $n=4$ and~$b=4$:
\setslot{\hbox{$x_3$}}
\def\z{\[\0]}
However, this ``reformatting'' does not produce a~correct encoding of a~vector,
because the separator zeroes are missing. For this reason, the implementation
-of~\<Sum> by modulo does not work correctly (it produces $\1^b$ instead of $\0^b$).
-We use the technique based on multiplication instead, which does not need
+of~\<Sum> using modulo does not work correctly (it produces $\0^b$ instead of $\1^b$).
+We therefore use the technique based on multiplication instead, which does not need
the separators. (Alternatively, we can observe that $\1^b$ is the only case
affected, so we can handle it separately.)
\endlist
-\para
-We can use the above tricks to perform interesting operations on individual
+\paran{Scalar operations}%
+We can use the aforementioned tricks to perform interesting operations on individual
numbers in constant time, too. Let us assume for a~while that we are
operating on $b$-bit numbers and the word size is at least~$b^2$.
This enables us to make use of intermediate vectors with $b$~elements
\itemize\ibull
-\:$\<Weight>(\alpha)$ --- compute the Hamming weight of~$\alpha$, i.e., the number of ones in~$\(\alpha)$.
+\:$\<Weight>(\alpha)$ --- Compute the Hamming weight of~$\alpha$, i.e., the number of ones in~$\(\alpha)$.
-Perform \<Unpack> and then \<Sum>.
+We perform \<Unpack> and then \<Sum>.
-\:$\<Permute>_\pi(\alpha)$ --- shuffle the bits of~$\alpha$ according
+\:$\<Permute>_\pi(\alpha)$ --- Shuffle the bits of~$\alpha$ according
to a~fixed permutation~$\pi$.
-
Perform $\<Unpack>_\pi$ and \<Pack> back.
+
We perform $\<Unpack>_\pi$ and \<Pack> back.
-\:$\<LSB>(\alpha)$ --- find the least significant bit of~$\alpha$,
+\:$\<LSB>(\alpha)$ --- Find the least significant bit of~$\alpha$,
i.e., the smallest~$i$ such that $\alpha[i]=1$.
By a~combination of subtraction with $\bxor$, we create a~number
-which contain ones exactly at positions below $\<LSB>(\alpha)$:
+that contains ones exactly at the position of $\<LSB>(\alpha)$ and below:
\alik{
\alpha&= \9\9\9\9\9\1\0\0\0\0\cr
\alpha\bxor(\alpha-1)&= \0\9\9\9\0\1\1\1\1\1\cr
}
-Then calculate the \<Weight> of the result.
+Then we calculate the \<Weight> of the result and subtract~1.
-\:$\<MSB>(\alpha)$ --- find the most significant bit (the position
-of the highest bit set in~$\alpha$).
+\:$\<MSB>(\alpha)$ --- Find the most significant bit of~$\alpha$ (the position
+of the highest bit set).
-Reverse the bits of the number
first by~\<Permute>, then apply \<LSB>
+Reverse the bits of the number
~$\alpha$ first by calling \<Permute>, then apply \<LSB>
and subtract the result from~$b-1$.
\endlist
-\rem
+\paran{LSB and MSB}%
As noted by Brodnik~\cite{brodnik:lsb} and others, the space requirements of
-the \<LSB> operation can be reduced to linear. We split the input to $\sqrt{b}$
-blocks of $\sqrt{b}$ bits each. Then we determine which blocks are non-zero and
+the \<LSB> operation can be lowered to linear. We split the $w$-bit input to $\sqrt{w}$
+blocks of $\sqrt{w}$ bits each. Then we determine which blocks are non-zero and
identify the lowest such block (this is a~\<LSB> of a~number whose bits
correspond to the blocks). Finally we calculate the \<LSB> of this block. In
-both calls to \<LSB,> we have a $\sqrt{b}$-bit number in a~$b$-bit word, so we
-can use the previous algorithm. The same trick of course works for finding the
-\<MSB> as well.
+both calls to \<LSB,> we have a $\sqrt{w}$-bit number in a~$w$-bit word, so we
+can use the previous algorithm. The same trick of course applies to for finding the
+\<MSB>, too.
-The following algorithm shows the details.
+The following algorithm shows the details:
\algn{LSB in linear workspace}
\algo\id{lsb}
\algin A~$w$-bit number~$\alpha$.
-\:$b\=\lceil\sqrt{w}\rceil$. \cmt{size of a~block}
+\:$b\=\lceil\sqrt{w}\,\rceil$. \cmt{the size of a~block}
\:$\ell\=b$. \cmt{the number of blocks is the same}
-\:$x\=(\alpha \band (\0\1^b)^\ell) \bor (\alpha \band (\1\0^b)^\ell)$.
+\:$x\=(\alpha \band (\0\1^b)^\ell) \bor ((\alpha \band (\1\0^b)^\ell) \shr 1)$.
\hfill\break
-\cmt{$x_i\ne 0$ iff the $i$-th block is non-zero}%
+\cmt{encoding of a~vector~${\bf x}$ such that $x_i\ne 0$ iff the $i$-th block is non-zero}%
\foot{Why is this so complicated? It is tempting to take $\alpha$ itself as a~code of this vector,
-but we unfortunately need the separator bits between elements, so we create them and
+but we must not forget the separator bits between elements, so we create them and
relocate the bits we have overwritten.}
\:$y\=\<Cmp>(0,x)$. \cmt{$y_i=1$ if the $i$-th block is non-zero, otherwise $y_0=0$}
\:$\beta\=\<Pack>(y)$. \cmt{each block compressed to a~single bit}
\algout $\<LSB>(\alpha) = bp+q$.
\endalgo
-\rem
-We have used a~plenty of constants which depend on the format of the vectors.
+\paran{Constants}%
+We have used a~plenty of constants that depend on the format of the vectors.
Either we can write non-uniform programs (see \ref{nonuniform}) and use native constants,
or we can observe that all such constants can be easily manufactured. For example,
-$(\0^b\1)^d = \1^{bd} / \1^{b+1} = (2^{bd}-1)/(2^{b+1}-1)$. The only exceptions
+$(\0^b\1)^d = \1^{(b+1)d} / \1^{b+1} = (2^{(b+1)d}-1)/(2^{b+1}-1) = ((1 \shl (b+1)d)-1) / ((2\shl b) - 1)$. The only exceptions
are the~$w$ and~$b$ in the LSB algorithm \ref{lsb}, which we are unable to produce
-in constant time.
+in constant time. In practice we use the ``bit tricks'' as frequently called subroutines
+in an~encompassing algorithm, so we usually can afford spending a~lot of time on the precalculation
+of constants performed once during algorithm startup.
+
+\paran{History}%
+The history of combining arithmetic and logical operations to obtain fast programs for various
+interesting functions is blurred. Many of the bit tricks, which we have described, have been
+discovered independently by numerous people in the early ages of digital computers.
+Since then, they have become a~part of the computer science folklore. Probably the
+earliest documented occurrence is in the 1972's memo of the MIT Artificial Intelligence
+Lab \cite{hakmem}. However, until the work of Fredman and Willard nobody seemed to
+realize the full consequences.
+
+%--------------------------------------------------------------------------------
+
+\section{Q-Heaps}\id{qheaps}%
+
+We have shown how to perform non-trivial operations on a~set of values
+in constant time, but so far only under the assumption that the number of these
+values is small enough and that the values themselves are also small enough
+(so that the whole set fits in $\O(1)$ machine words). Now we will show how to
+lift the restriction on the magnitude of the values and still keep constant time
+complexity. We will describe a~slightly simplified version of the Q-heaps developed by
+Fredman and Willard in~\cite{fw:transdich}.
+
+The Q-heap represents a~set of at most~$k$ word-sized integers, where $k\le W^{1/4}$
+and $W$ is the word size of the machine. It will support insertion, deletion, finding
+of minimum and maximum, and other operations described below, in constant time, provided that
+we are willing to spend~$\O(2^{k^4})$ time on preprocessing.
+
+The exponential-time preprocessing may sound alarming, but a~typical application uses
+Q-heaps of size $k=\log^{1/4} N$, where $N$ is the size of the algorithm's input.
+This guarantees that $k\le W^{1/4}$ and $\O(2^{k^4}) = \O(N)$. Let us however
+remark that the whole construction is primarily of theoretical importance ---
+the huge multiplicative constants hidden in the~$\O$ make these heaps useless
+in practical algorithms. Despite this, many of the tricks we develop have proven
+themselves useful even in real-life data structures.
+
+Spending so much time on preprocessing makes it possible to precompute tables of
+almost arbitrary functions and then assume that the functions can be evaluated in
+constant time:
+
+\lemma\id{qhprecomp}%
+When~$f$ is a~function computable in polynomial time, $\O(2^{k^4})$ time is enough
+to precompute a~table of the values of~$f$ for all arguments whose size is $\O(k^3)$ bits.
+
+\proof
+There are $2^{\O(k^3)}$ possible combinations of arguments of the given size and for each of
+them we spend $\poly(k)$ time on calculating the function. It remains
+to observe that $2^{\O(k^3)}\cdot \poly(k) = \O(2^{k^4})$.
+\qed
+
+\paran{Tries and ranks}%
+We will first develop an~auxiliary construction based on tries and then derive
+the real definition of the Q-heap from it.
+
+\nota
+\itemize\ibull
+\:$W$ --- the word size of the RAM,
+\:$k = \O(W^{1/4})$ --- the limit on the size of the heap,
+\:$n\le k$ --- the number of elements stored in the heap,
+\:$X=\{x_1, \ldots, x_n\}$ --- the elements themselves: distinct $W$-bit numbers
+indexed in a~way that $x_1 < \ldots < x_n$,
+\:$g_i = \<MSB>(x_i \bxor x_{i+1})$ --- the position of the most significant bit in which $x_i$ and~$x_{i+1}$ differ,
+\:$R_X(x)$ --- the rank of~$x$ in~$X$, that is the number of elements of~$X$ which are less than~$x$
+(where $x$~itself need not be an~element of~$X$).\foot{We will dedicate the whole Chapter~\ref{rankchap} to the
+study of various ranks.}
+\endlist
+
+\defn
+A~\df{trie} for a~set of strings~$S$ over a~finite alphabet~$\Sigma$ is
+a~rooted tree whose vertices are the prefixes of the strings in~$S$ and there
+is an~edge going from a~prefix~$\alpha$ to a~prefix~$\beta$ iff $\beta$ can be
+obtained from~$\alpha$ by appending a~single symbol of the alphabet. The edge
+will be labeled with that particular symbol. We will also define the~\df{letter depth}
+of a~vertex as the length of the corresponding prefix. We mark the vertices
+which match a~string of~$S$.
+
+A~\df{compressed trie} is obtained by removing the vertices of outdegree~1
+except for the root and the marked vertices.
+Wherever there is a~directed path whose internal vertices have outdegree~1 and they carry
+no mark, we replace this path by a~single edge labeled with the concatenation
+of the original edges' labels.
+
+In both kinds of tries, we order the outgoing edges of every vertex by their labels
+lexicographically.
+
+\obs
+In both tries, the root of the tree is the empty word. Generally, the prefix
+in a~vertex is equal to the concatenation of edge labels on the path
+leading from the root to that vertex. The letter depth of the vertex is equal to
+the total size of these labels. All leaves correspond to strings in~$S$, but so can
+some internal vertices if there are two strings in~$S$ such that one is a~prefix
+of the other.
+
+Furthermore, the labels of all edges leaving a~common vertex are always
+distinct and when we compress the trie, no two such labels share their initial
+symbols. This allows us to search in the trie efficiently: when looking for
+a~string~$x$, we follow the path from the root and whenever we visit
+an~internal vertex of letter depth~$d$, we test the $d$-th character of~$x$,
+we follow the edge whose label starts with this character, and we check that the
+rest of the label matches.
+
+The compressed trie is also efficient in terms of space consumption --- it has
+$\O(\vert S\vert)$ vertices (this can be easily shown by induction on~$\vert S\vert$)
+and all edge labels can be represented in space linear in the sum of the
+lengths of the strings in~$S$.
+
+\defn
+For our set~$X$, we define~$T$ as a~compressed trie for the set of binary
+encodings of the numbers~$x_i$, padded to exactly $W$~bits, i.e., for $S = \{ \(x)_W \mid x\in X \}$.
+
+\float{\valign{\vfil#\vfil\cr
+\hbox{\epsfbox{pic/qheap.eps}}\cr
+\noalign{\qquad\quad}
+ \halign{#\hfil&\quad#\hfil\cr
+ $x_1 = \0\0\0\0\1\1$ & $g_1=3$ \cr
+ $x_2 = \0\0\1\0\1\0$ & $g_2=4$ \cr
+ $x_3 = \0\1\0\0\0\1$ & $g_3=2$ \cr
+ $x_4 = \0\1\0\1\0\1$ & $g_4=5$ \cr
+ $x_5 = \1\0\0\0\0\0$ & $g_5=0$ \cr
+ $x_6 = \1\0\0\0\0\1$ \cr
+ }\cr
+}}{Six numbers stored in a~compressed trie}
+
+\obs
+The trie~$T$ has several interesting properties. Since all words in~$S$ have the same
+length, the leaves of the trie correspond to these exact words, that is to the numbers~$x_i$.
+The depth-first traversal of the trie enumerates the words of~$S$ in lexicographic order
+and therefore also the~$x_i$'s in the order of their values. Between each
+pair of leaves $x_i$ and~$x_{i+1}$ it visits an~internal vertex whose letter depth
+is exactly~$W-1-g_i$.
+
+\para
+Let us now modify the algorithm for searching in the trie and make it compare
+only the first symbols of the edges. In other words, we will test only the bits~$g_i$
+which will be called \df{guides} (as they guide us through the tree). For $x\in
+X$, the modified algorithm will still return the correct leaf. For all~$x$ outside~$X$
+it will no longer fail and instead it will land on some leaf~$x_i$. At the
+first sight the number~$x_i$ may seem unrelated, but we will show that it can be
+used to determine the rank of~$x$ in~$X$, which will later form a~basis for all
+Q-heap operations:
+
+\lemma\id{qhdeterm}%
+The rank $R_X(x)$ is uniquely determined by a~combination of:
+\itemize\ibull
+\:the trie~$T$,
+\:the index~$i$ of the leaf found when searching for~$x$ in~$T$,
+\:the relation ($<$, $=$, $>$) between $x$ and $x_i$,
+\:the bit position $b=\<MSB>(x\bxor x_i)$ of the first disagreement between~$x$ and~$x_i$.
+\endlist
+\proof
+If $x\in X$, we detect that from $x_i=x$ and the rank is obviously~$i-1$.
+Let us assume that $x\not\in X$ and imagine that we follow the same path as when
+searching for~$x$,
+but this time we check the full edge labels. The position~$b$ is the first position
+where~$\(x)$ disagrees with a~label. Before this point, all edges not taken by
+the search were leading either to subtrees containing elements all smaller than~$x$
+or all larger than~$x$ and the only values not known yet are those in the subtree
+below the edge that we currently consider. Now if $x[b]=0$ (and therefore $x<x_i$),
+all values in that subtree have $x_j[b]=1$ and thus they are larger than~$x$. In the other
+case, $x[b]=1$ and $x_j[b]=0$, so they are smaller.
+\qed
+
+\paran{A~better representation}%
+The preceding lemma shows that the rank can be computed in polynomial time, but
+unfortunately the variables on which it depends are too large for a~table to
+be efficiently precomputed. We will carefully choose an~equivalent representation
+of the trie which is compact enough.
+
+\lemma\id{citree}%
+The compressed trie is uniquely determined by the order of the guides~$g_1,\ldots,g_{n-1}$.
+
+\proof
+We already know that the letter depths of the trie vertices are exactly
+the numbers~$W-1-g_i$. The root of the trie must have the smallest of these
+letter depths, i.e., it must correspond to the highest numbered bit. Let
+us call this bit~$g_i$. This implies that the values $x_1,\ldots,x_i$
+must lie in the left subtree of the root and $x_{i+1},\ldots,x_n$ in its
+right subtree. Both subtrees can be then constructed recursively.\foot{This
+construction is also known as the \df{cartesian tree} for the sequence
+$g_1,\ldots,g_{n-1}$ and it is useful in many other algorithms as it can be
+built in $\O(n)$ time. A~nice application on the Lowest Common Ancestor
+and Range Minimum problems has been described by Bender et al.~in \cite{bender:lca}.}
+\qed
+
+\para
+Unfortunately, the vector of the $g_i$'s is also too long (is has $k\log W$ bits
+and we have no upper bound on~$W$ in terms of~$k$), so we will compress it even
+further:
+
+\nota\id{qhnota}%
+\itemize\ibull
+\:$B = \{g_1,\ldots,g_n\}$ --- the set of bit positions of all the guides, stored as a~sorted array,
+\:$G : \{1,\ldots,n\} \rightarrow \{1,\ldots,n\}$ --- a~function mapping
+the guides to their bit positions in~$B$: $g_i = B[G(i)]$,
+\:$x[B]$ --- a~bit string containing the bits of~$x$ originally located
+at the positions given by~$B$, i.e., the concatenation of bits $x[B[1]],
+x[B[2]],\ldots, x[B[n]]$.
+\endlist
+
+\obs\id{qhsetb}%
+The set~$B$ has $\O(k\log W)=\O(W)$ bits, so it can be stored in a~constant number
+of machine words in the form of a~sorted vector. The function~$G$ can be also stored as a~vector
+of $\O(k\log k)$ bits. We can change a~single~$g_i$ in constant time using
+vector operations: First we delete the original value of~$g_i$ from~$B$ if it
+is not used anywhere else. Then we add the new value to~$B$ if it was not
+there yet and we write its position in~$B$ to~$G(i)$. Whenever we insert
+or delete a~value in~$B$, the values at the higher positions shift one position
+up or down and we have to update the pointers in~$G$. This can be fortunately
+accomplished by adding or subtracting a~result of vector comparison.
+
+In this representation, we can reformulate our lemma on ranks as follows:
+
+\lemma\id{qhrank}%
+The rank $R_X(x)$ can be computed in constant time from:
+\itemize\ibull
+\:the function~$G$,
+\:the values $x_1,\ldots,x_n$,
+\:the bit string~$x[B]$,
+\:$x$ itself.
+\endlist
+
+\proof
+Let us prove that all ingredients of Lemma~\ref{qhdeterm} are either small
+enough or computable in constant time.
+
+We know that the shape of the trie~$T$ is uniquely determined by the order of the $g_i$'s
+and therefore by the function~$G$ since the array~$B$ is sorted. The shape of
+the trie together with the bits in $x[B]$ determine the leaf~$x_i$ found when searching
+for~$x$ using only the guides. This can be computed in polynomial time and it
+depends on $\O(k\log k)$ bits of input, so according to Lemma~\ref{qhprecomp}
+we can look it up in a~precomputed table.
+
+The relation between $x$ and~$x_i$ can be obtained directly as we know the~$x_i$.
+The bit position of the first disagreement can be calculated in constant time
+using the Brodnik's LSB/MSB algorithm (\ref{lsb}).
+
+All these ingredients can be stored in $\O(k\log k)$ bits, so we may assume
+that the rank can be looked up in constant time as well.
+\qed
+
+\para
+We would like to store the set~$X$ as a~sorted array together
+with the corresponding trie, which will allow us to determine the position
+for a~newly inserted element in constant time. However, the set is too large
+to fit in a~vector and we cannot perform insertion on an~ordinary array in
+constant time. This can be worked around by keeping the set in an~unsorted
+array and storing a~separate vector containing the permutation that sorts the array.
+We can then insert a~new element at an~arbitrary place in the array and just
+update the permutation to reflect the correct order.
+
+\paran{The Q-heap}%
+We are now ready for the real definition of the Q-heap and for the description
+of the basic operations on it.
+
+\defn
+A~\df{Q-heap} consists of:
+\itemize\ibull
+\:$k$, $n$ --- the capacity of the heap and the current number of elements (word-sized integers),
+\:$X$ --- the set of word-sized elements stored in the heap (an~array of words in an~arbitrary order),
+\:$\varrho$ --- a~permutation on~$\{1,\ldots,n\}$ such that $X[\varrho(1)] < \ldots < X[\varrho(n)]$
+(a~vector of $\O(n\log k)$ bits; we will write $x_i$ for $X[\varrho(i)]$),
+\:$B$ --- a~set of ``interesting'' bit positions
+(a~sorted vector of~$\O(n\log W)$ bits),
+\:$G$ --- the function that maps the guides to the bit positions in~$B$
+(a~vector of~$\O(n\log k)$ bits),
+\:precomputed tables of various functions.
+\endlist
+
+\algn{Search in the Q-heap}\id{qhfirst}%
+\algo
+\algin A~Q-heap and an~integer~$x$ to search for.
+\:$i\=R_X(x)+1$, using Lemma~\ref{qhrank} to calculate the rank.
+\:If $i\le n$ return $x_i$, otherwise return {\sc undefined.}
+\algout The smallest element of the heap which is greater or equal to~$x$.
+\endalgo
+
+\algn{Insertion to the Q-heap}
+\algo
+\algin A~Q-heap and an~integer~$x$ to insert.
+\:$i\=R_X(x)+1$, using Lemma~\ref{qhrank} to calculate the rank.
+\:If $x=x_i$, return immediately (the value is already present).
+\:Insert the new value to~$X$:
+\::$n\=n+1$.
+\::$X[n]\=x$.
+\::Insert~$n$ at the $i$-th position in the permutation~$\varrho$.
+\:Update the $g_j$'s:
+\::Move all~$g_j$ for $j\ge i$ one position up. \hfil\break
+ This translates to insertion in the vector representing~$G$.
+\::Recalculate $g_{i-1}$ and~$g_i$ according to the definition.
+ \hfil\break Update~$B$ and~$G$ as described in~\ref{qhsetb}.
+\algout The updated Q-heap.
+\endalgo
+
+\algn{Deletion from the Q-heap}
+\algo
+\algin A~Q-heap and an~integer~$x$ to be deleted from it.
+\:$i\=R_X(x)+1$, using Lemma~\ref{qhrank} to calculate the rank.
+\:If $i>n$ or $x_i\ne x$, return immediately (the value is not in the heap).
+\:Delete the value from~$X$:
+\::$X[\varrho(i)]\=X[n]$.
+\::Find $j$ such that~$\varrho(j)=n$ and set $\varrho(j)\=\varrho(i)$.
+\::$n\=n-1$.
+\:Update the $g_j$'s like in the previous algorithm.
+\algout The updated Q-heap.
+\endalgo
+
+\algn{Finding the $i$-th smallest element in the Q-heap}\id{qhlast}%
+\algo
+\algin A~Q-heap and an~index~$i$.
+\:If $i<1$ or $i>n$, return {\sc undefined.}
+\:Return~$x_i$.
+\algout The $i$-th smallest element in the heap.
+\endalgo
+
+\paran{Extraction}%
+The heap algorithms we have just described have been built from primitives
+operating in constant time, with one notable exception: the extraction
+$x[B]$ of all bits of~$x$ at positions specified by the set~$B$. This cannot be done
+in~$\O(1)$ time on the Word-RAM, but we can implement it with ${\rm AC}^0$
+instructions as suggested by Andersson in \cite{andersson:fusion} or even
+with those ${\rm AC}^0$ instructions present on real processors (see Thorup
+\cite{thorup:aczero}). On the Word-RAM, we need to make use of the fact
+that the set~$B$ is not changing too much --- there are $\O(1)$ changes
+per Q-heap operation. As Fredman and Willard have shown \cite{fw:transdich},
+it is possible to maintain a~``decoder'', whose state is stored in $\O(1)$
+machine words and which helps us to extract $x[B]$ in a~constant number of
+operations:
+
+\lemman{Extraction of bits, Fredman and Willard \cite{fw:transdich}}\id{qhxtract}%
+Under the assumptions on~$k$, $W$ and the preprocessing time as in the Q-heaps,\foot{%
+Actually, this is the only place where we need~$k$ to be as low as $W^{1/4}$.
+In the ${\rm AC}^0$ implementation, it is enough to ensure $k\log k\le W$.
+On the other hand, we need not care about the exponent because it can
+be increased arbitrarily using the Q-heap trees described below.}
+it is possible to maintain a~data structure for a~set~$B$ of bit positions,
+which allows~$x[B]$ to be extracted in $\O(1)$ time for an~arbitrary~$x$.
+When a~single element is inserted to~$B$ or deleted from~$B$, the structure
+can be updated in constant time, as long as $\vert B\vert \le k$.
+\qed
+
+\para
+This was the last missing bit of the mechanics of the Q-heaps. We are
+therefore ready to conclude this section by the following theorem
+and its consequences:
+
+\thmn{Q-heaps, Fredman and Willard \cite{fw:transdich}}\id{qh}%
+Let $W$ and~$k$ be positive integers such that $k=\O(W^{1/4})$. Let~$Q$
+be a~Q-heap of at most $k$-elements of $W$~bits each. Then the Q-heap
+operations \ref{qhfirst} to \ref{qhlast} on~$Q$ (insertion, deletion,
+search for a~given value and search for the $i$-th smallest element)
+run in constant time on a~Word-RAM with word size~$W$, after spending
+time $\O(2^{k^4})$ on the same RAM on precomputing of tables.
+
+\proof
+Every operation on the Q-heap can be performed in a~constant number of
+vector operations and calculations of ranks. The ranks are computed
+in $\O(1)$ steps involving again $\O(1)$ vector operations, binary
+logarithms and bit extraction. All these can be calculated in constant
+time using the results of Section \ref{bitsect} and Lemma \ref{qhxtract}.
+\qed
+
+\paran{Combining Q-heaps}%
+We can also use the Q-heaps as building blocks of more complex structures
+like Atomic heaps and AF-heaps (see once again \cite{fw:transdich}). We will
+show a~simpler, but often sufficient construction, sometimes called the \df{\hbox{Q-heap} tree.}
+Suppose we have a~Q-heap of capacity~$k$ and a~parameter $d\in{\bb N}^+$. We
+can build a~balanced $k$-ary tree of depth~$d$ such that its leaves contain
+a~given set and every internal vertex keeps the minimum value in the subtree
+rooted in it, together with a~Q-heap containing the values in all its sons.
+This allows minimum to be extracted in constant time (it is placed in the root)
+and when any element is changed, it is sufficient to recalculate the values
+from the path from this element to the root, which takes $\O(d)$ Q-heap
+operations.
+
+\corn{Q-heap trees}\id{qhtree}%
+For every positive integer~$r$ and $\delta>0$ there exists a~data structure
+capable of maintaining the minimum of a~set of at most~$r$ word-sized numbers
+under insertions and deletions. Each operation takes $\O(1)$ time on a~Word-RAM
+with word size $W=\Omega(r^{\delta})$, after spending time
+$\O(2^{r^\delta})$ on precomputing of tables.
+
+\proof
+Choose $\delta' \le \delta$ such that $r^{\delta'} = \O(W^{1/4})$. Build
+a~Q-heap tree of depth $d=\lceil \delta/\delta'\rceil$ containing Q-heaps of
+size $k=r^{\delta'}$. \qed
+
+\rem\id{qhtreerem}%
+When we have an~algorithm with input of size~$N$, the word size is at least~$\log N$
+and we can spend time $\O(N)$ on preprocessing, so we can choose $r=\log N$ and
+$\delta=1$ in the above corollary and get a~heap of size $\log N$ working in
+constant time per operation.
\endpart
projects / saga.git / blobdiff