Cleaned up topological computations and unification.

[saga.git] / ram.tex

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\chapter{Fine Details of Computation}
\id{ramchap}

\section{Models and machines}

Traditionally, computer scientists use 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,
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 ``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.

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
models, but even the remaining models are subtly different from each other. For example, some of them
allow indexing of arrays in constant time, while on the others,
arrays have to be emulated with pointer structures, requiring $\Omega(\log n)$
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 depth of the actual indexing circuits is logarithmic.

In recent decades, most researchers in the area of combinatorial algorithms
have been considering two computational models: the Random Access Machine and the Pointer
Machine. The former is closer to the programmer's view of a~real computer,
the latter is slightly more restricted and ``asymptotically safe.''
We will follow this practice and study our algorithms in both models.

\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.
(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.
The \df{program} is a~finite sequence of \df{instructions} of two basic kinds: calculation
instructions and control instructions.

\df{Calculation instructions} have two source arguments and one destination
argument, each \df{argument} being either an~immediate constant (not available
as destination), a~directly addressed memory cell (specified by its number)
or an~indirectly addressed memory cell (its address is stored in a~directly
addressed memory cell).

\df{Control instructions} include branches (to a~specific instruction in
the program), conditional branches (e.g., jump if two arguments specified as
in the calculation instructions are equal) and an~instruction to halt the program.

At the beginning of the computation, the memory contains the input data
in specified cells and arbitrary values in all other cells.
Then the program is executed one instruction at a~time. When it halts,
specified memory cells are interpreted as the program's output.

\para\id{wordsize}%
In the description of the RAM family, we have omitted several details
on~purpose, because different members of the family define them differently.
These are: the size of the available integers, the time complexity of a~single
instruction, the space complexity assigned to a~single memory cell and the set
of operations available in calculation instructions.

If we impose no limits on the magnitude of the numbers and we assume that
arithmetic and logical operations work on them in constant time, we get
a~very powerful parallel computer --- we can emulate an~exponential number
of parallel processors using arithmetics and suddenly almost everything can be
computed in constant time, modulo encoding and decoding of input and output.
Such models are unrealistic and there are two basic possibilities how to
avoid this behavior:

\numlist\ndotted
\:Keep unbounded numbers, but increase costs of instructions: each instruction
  consumes time proportional to the number of bits of the numbers it processes,
  including memory addresses. Similarly, space usage is measured in bits,
  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
  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 that fits in a~single memory cell a~\df{machine word.}
\endlist

Both restrictions easily avoid the problems of unbounded parallelism. The first
choice is theoretically cleaner and Cook et al.~show nice correspondences to the
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 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.

\para
As for the choice of RAM operations, the following three instruction sets are often used:

\itemize\ibull
\:\df{Word-RAM} --- allows the ``C-language operators'', i.e., addition,
  subtraction, multiplication, division, remainder, bitwise $\band$, $\bor$, exclusive
  $\bor$ ($\bxor$) and negation ($\bnot$), and bitwise shifts ($\shl$ and~$\shr$).
\:\df{${\rm AC}^0$-RAM} --- allows all operations from the class ${\rm AC}^0$, i.e.,
  those computable by constant-depth polynomial-size boolean circuits with unlimited
  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.
\endlist

Thorup discusses the usual techniques employed by RAM algorithms in~\cite{thorup:aczero}
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
add some floating-point operations and multimedia instructions available on the Intel's
Pentium~4~\cite{intel:pentium}).

We will therefore use the Word-RAM instruction set, mentioning differences from the
${\rm AC}^0$-RAM where necessary.

\nota
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}):

\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.'')

\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},
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
instructions instead of an~opaque control unit.

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
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).

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
labeled 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
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;
\:\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
  immediately);
\:\df{control instructions} --- similarly to the RAM; conditional jumps can decide
  on~arbitrary unary relations on symbols and compare pointers for equality.
\endlist

Time and space complexity are defined in the straightforward way: all instructions
have unit cost and so do all memory cells.

Both input and output of the machine are passed in the form of a~linked structure
pointed to by a~designated register. For example, we can pass graphs back and forth
without having to encode them as strings of numbers or symbols. This is important,
because with the finite alphabet of the~PM, symbolic representations of graphs
generally require super-linear space and therefore also time.\foot{%
The usual representation of edges as pairs of vertex labels uses $\Theta(m\log n)$ bits
and as a~simple counting argument shows, this is asymptotically optimal for general
sparse graphs. On the other hand, specific families of sparse graphs can be stored
more efficiently, e.g., by a~remarkable result of Tur\'an~\cite{turan:succinct},
planar graphs can be encoded in~$\O(n)$ bits. Encoding of dense graphs is of
course trivial as the adjacency matrix has only~$\Theta(n^2)$ bits.}

\para
Compared to the RAM, the PM lacks two important capabilities: indexing of arrays
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.

\thm
Every program for the Word-RAM with word size~$W$ can be translated to a~PM program
computing the same with $\O(W^2)$ slowdown (given a~suitable encoding of inputs and
outputs, of course). If the RAM program does not use multiplication, division
and remainder operations, $\O(W)$~slowdown is sufficient.

\proofsketch
Represent the memory of the RAM by a~balanced binary search tree or by a~radix
trie of depth~$\O(W)$. Values are encoded as~linked lists of symbols pointed
to by the nodes of the tree. Both direct and indirect accesses to the memory
can therefore be done in~$\O(W)$ time. Use standard algorithms for arithmetic
on big numbers: $\O(W)$ per operation except for multiplication, division and
remainders which take $\O(W^2)$.\foot{We could use more efficient arithmetic
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.

\proofsketch
Encode each cell of the PM's memory to $\O(1)$ integers. Store the encoded cells to
the memory of the RAM sequentially and use memory addresses as pointers. As the symbols
are finite and there is only a~polynomial number of cells allocated during execution
of the program, $\O(\log N)$-bit integers suffice ($N$~is the size of the program's input).
\qed

\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
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
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
be performed by creating a~copy of the particular cell with some fields changed.
This in turn requires the pointers to the cell to be updated, possibly triggering
a~cascade of further cell copies. For example, when a~node of a~binary search tree is
updated, all nodes on the path from that node to the root have to be copied.

One of the advantages of this model is that the states of the machine are
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}.

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
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 unification}\id{bucketsort}%

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 we have already suggested in the proof of Lemma \ref{contbor}, 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.

\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 and sort them lexicographically. This brings
parallel edges together, so that a~simple linear scan suffices to find each bunch
of parallel edges and 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
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. 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 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.

\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 indexing of arrays.

We will keep a~list of the per-vertex structures that defines 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 is also $\O(n+m)$.

\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 of them we identify
the isomorphism equivalence classes of subtrees of that particular depth. We will assign
unique identifiers 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 a~first-class 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
identifiers 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 identifiers
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 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
which 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 $\O(\sum_d n_d) = \O(n)$.

Our algorithm can be easily modified for trees with unrestricted degrees.
We replace the fixed $d$-tuples by general sequences of identifiers. 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 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 in 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. 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, which 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 its
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 of course 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. The obstacle is that
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 adjancency 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 any 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{Batched 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.

\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
$\O(\Vert\C\Vert + \Vert{\cal G}\Vert)$.
\qed

\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
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,
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.}

%--------------------------------------------------------------------------------

\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
fit in a~single machine word. At the first sight this might seem useless, because we
cannot require large word sizes, but surprisingly often the elements are small
enough relative to the size of the algorithm's input and thus also relative to
the minimum possible word size. Also, as the following lemma shows, we can
easily emulate slightly longer words:

\lemman{Multiple-precision calculations}
Given a~RAM with $W$-bit words, we can emulate all calculation and control
instructions of a~RAM with word size $kW$ in time depending only on the~$k$.
(This is usually called \df{multiple-precision arithmetics.})

\proof
We split each word of the ``big'' machine to $W'$-bit blocks, where $W'=W/2$, and store these
blocks in $2k$ consecutive memory cells. Addition, subtraction, comparison and
bitwise logical operations can be performed block-by-block. Shifts by a~multiple
of~$W'$ are trivial, otherwise we can combine each block of the result from
shifted versions of two original blocks.
To multiply two numbers, we can use the elementary school algorithm using the $W'$-bit
blocks as digits in base $2^{W'}$ --- the product of any two blocks fits
in a~single word.

Division is harder, but Newton-Raphson iteration (see~\cite{ito:newrap})
converges to the quotient in a~constant number of iterations, each of them
involving $\O(1)$ multiple-precision additions and multiplications. A~good
starting approximation can be obtained by dividing the two most-significant
(non-zero) blocks of both numbers.

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.
\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 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.
When the meaning is clear from the context,
we will use $x$ and $\(x)$ interchangeably to avoid outbreak of symbols.

\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)=\(x_{d-1})_b\0\(x_{d-2})_b\0\ldots\0\(x_0)_b$, i.e.,
$x = \sum_i 2^{(b+1)i}\cdot x_i$. (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 written as
$x_0,\ldots,x_{d-1}$.

\para
If we want to fit the whole vector in a~single word, the parameters $b$ and~$d$ must satisty
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 sequences of $\O(1)$ 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
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 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}
\def\slot#1{\hbox to \slotwd{\hfil #1\hfil}}
\def\[#1]{\slot{$#1$}}
\def\9{\rack{\0}{\hss$\cdot$\hss}}

\def\alik#1{%
\medskip
\halign{\hskip 0.15\hsize\hfil $ ##$&\hbox to 0.6\hsize{${}##$ \hss}\cr
#1}
\medskip
}

\algn{Operations on vectors with $d$~elements of $b$~bits each}\id{vecops}

\itemize\ibull

\:$\<Replicate>(\alpha)$ --- creates 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
the result fits in $b$~bits:

\alik{\<Sum>(x) = x \bmod \1^{b+1}. \cr}

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 makes no difference.

If we want to avoid division, we can use double-precision multiplication instead:

\setslot{\hbox{~$\0x_{d-1}$}}
\def\.{\[]}
\def\z{\[\0^b\1]}
\def\dd{\slot{$\cdots$}}
\def\vd{\slot{$\vdots$}}
\def\rule{\noalign{\medskip\nointerlineskip}$\hrulefill$\cr\noalign{\nointerlineskip\medskip}}

\alik{
\[\0x_{d-1}] \dd \[\0x_2] \[\0x_1] \[\0x_0] \cr
*~~ \z \dd \z\z\z \cr
\rule
\[x_{d-1}] \dd \[x_2] \[x_1] \[x_0] \cr
\[x_{d-1}] \[x_{d-2}] \dd \[x_1] \[x_0] \. \cr
\[x_{d-1}] \[x_{d-2}] \[x_{d-3}] \dd \[x_0] \. \. \cr
\vd\vd\vd\vd\.\.\.\cr
\[x_{d-1}] \dd \[x_2]\[x_1]\[x_0] \. \. \. \. \cr
\rule
\[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:
$s_k=\sum_{i=0}^{k-1}x_i$, $r_k=\sum_{i=k}^{d-1}x_i$.

\:$\<Cmp>(x,y)$ --- element-wise comparison of~vectors ${\bf x}$ and~${\bf y}$,
i.e., a~vector ${\bf z}$ such that $z_i=1$ 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
carries from propagating, so the fields do not interact with each other:

\setslot{\vbox{\hbox{~$x_{d-1}$}\hbox{~$y_{d-1}$}}}
\def\9{\rack{\0}{\hss ?\hss}}
\alik{
   \1 \[x_{d-1}] \1 \[x_{d-2}] \[\cdots] \1 \[x_1] \1 \[x_0]  \cr
-~ \0 \[y_{d-1}] \0 \[y_{d-2}] \[\cdots] \0 \[y_1] \0 \[y_0]  \cr
\rule
   \9 \[\ldots]  \9 \[\ldots]  \[\cdots] \9 \[\ldots] \9 \[\ldots] \cr
}

It only remains to shift the separator bits to the right positions, negate them
and mask out all other bits.

\:$\<Rank>(x,\alpha)$ --- returns 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
}

\:$\<Insert>(x,\alpha)$ --- inserts~$\alpha$ into a~sorted vector $\bf x$:

We calculate the rank of~$\alpha$ in~$x$ first, then we insert~$\alpha$ as the $k$-th
field of~$\bf x$ using masking operations and shifts.

\algo
\:$k\=\<Rank>(x,\alpha)$.
\:$\ell\=x \band \1^{(b+1)(n-k-1)}\0^{(b+1)(k+1)}$. \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

\:$\<Unpack>(\alpha)$ --- creates a~vector whose elements are the bits of~$\(\alpha)_d$.
In other words, inserts blocks~$\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 $\<Cmp>(z,y)$.
\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.

\:$\<Unpack>_\pi(\alpha)$ --- like \<Unpack>, but changes 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 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,
it crosses out the $\0^b$ blocks).

We interpret the~$x$ as an~encoding of a~vector with elements one bit shorter
and we sum these elements. For example, when $n=4$ and~$b=4$:

\setslot{\hbox{$x_3$}}
\def\z{\[\0]}
\def\|{\hskip1pt\vrule height 10pt depth 4pt\hskip1pt}
\def\.{\hphantom{\|}}

\alik{
\|\z\.\z\.\z\.\z\.\[x_3]\|\z\.\z\.\z\.\z\.\[x_2]\|\z\.\z\.\z\.\z\[x_1]\|\z\.\z\.\z\.\z\.\[x_0]\|\cr
\|\z\.\z\.\z\.\z\|\[x_3]\.\z\.\z\.\z\|\z\.\[x_2]\.\z\.\z\|\z\.\z\[x_1]\.\z\|\z\.\z\.\z\.\[x_0]\|\cr
}

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> 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 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
of $b$~bits each.

\algn{Integer operations in quadratic workspace}\id{lsbmsb}

\itemize\ibull

\:$\<Weight>(\alpha)$ --- computes the Hamming weight of~$\alpha$, i.e., the number of ones in~$\(\alpha)$.

We perform \<Unpack> and then \<Sum>.

\:$\<Permute>_\pi(\alpha)$ --- shuffles the bits of~$\alpha$ according
to a~fixed permutation~$\pi$.

We perform $\<Unpack>_\pi$ and \<Pack> back.

\:$\<LSB>(\alpha)$ --- finds 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
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-1&=                      \9\9\9\9\9\0\1\1\1\1\cr
\alpha\bxor(\alpha-1)&=         \0\9\9\9\0\1\1\1\1\1\cr
}

Then we calculate the \<Weight> of the result and subtract~1.

\:$\<MSB>(\alpha)$ --- finds the most significant bit of~$\alpha$ (the position
of the highest bit set).

Reverse the bits of the number~$\alpha$ first by calling \<Permute>, then apply \<LSB>
and subtract the result from~$b-1$.

\endlist

\rem
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
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.

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}
\:$\ell\=b$. \cmt{the number of blocks is the same}
\:$x\=(\alpha \band (\0\1^b)^\ell) \bor (\alpha \band (\1\0^b)^\ell)$.
\hfill\break
\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
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}
\:$p\=\<LSB>(\beta)$. \cmt{the index of the lowest non-zero block}
\:$\gamma\=(\alpha \shr bp) \band \1^b$. \cmt{the contents of that block}
\:$q\=\<LSB>(\gamma)$. \cmt{the lowest bit set there}
\algout $\<LSB>(\alpha) = bp+q$.
\endalgo

\rem
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,
$(\0^b\1)^d = \1^{(b+1)d} / \1^{b+1} = (2^{(b+1)d}-1)/(2^{b+1}-1)$. The only exceptions
are the~$w$ and~$b$ in the LSB algorithm \ref{lsb}, which we are unable to produce
in constant time. In practice we use the ``bit tricks'' as frequently called subroutines
in an~encompassing algorithm, so we usually can spend a~lot of time on the precalculation
of constants performed once during algorithm startup.

%--------------------------------------------------------------------------------

\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
and that the huge constants involved everywhere make these heaps useless
in practical algorithms. Many of the tricks used however prove themselves
useful even in real-life implementations.

Spending the time on reprocessing makes it possible to precompute tables for
almost arbitrary functions and then assume that they 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

\para
We will first show an~auxiliary construction based on tries and then derive
the real definition of the Q-heap from it.

\nota
Let us introduce some notation first:
\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 the 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 from the trie by removing the vertices of outdegree~1
except for the root and marked vertices.
Whereever 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 contatenation
of the original edge's 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 and for every other vertex, the
corresponding prefix 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 have 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 ; x\in X \}$.

\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 inorder 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

\para
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 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
$g_1,\ldots,g_n$ 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 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 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
In the Q-heap 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 together with a~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.

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

\para
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, 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}\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 arbitrarily increased 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 $\vert B\vert \le k$.

\proof
See Fredman and Willard \cite{fw:transdich}.
\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

\rem
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 useful construction, sometimes called the \df{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