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\begin{document}

\lecture{2 --- 14 February, 2007}{Spring 2007}{Prof.\ Erik
  Demaine}{Christopher Moh, Aditya Rathnam}

\section{Overview}

In the last lecture we covered linked lists and some results regarding
the move-to-front heuristic.  We discovered that although
move-to-front was competitive in the single finger model, it was not
competitive in a multi-finger model, when compared to the
Order-By-Next-Request (OBNR) algorithm.  In today's lecture we will
talk about Binary Search Trees (BSTs) instead of lists, and in
particular, splay trees. 

BSTs store elements in internal nodes, and every node satisfies the
following \emph{symmetric} property:  for every non-leaf node, the
value of its left 
child (if present) is less than or equal to the value of the node, and
the value of the right child (if present) is more than or equal to the
value of the node. 

We keep a pointer to the current element, initially at the root, and
allow the following operations:  move the pointer to the left, to the
right, or to rotate the current node.  Note that rotations are
unambiguous as each node has at most one sibling.  We can perform accesses by
simply moving down the left or right pointers depending on how the value at
the node compares with the search value.  Unless
stated otherwise (notably in the finger theorems stated below), we
assume that the pointer starts at the root for every access.

\subsection{One Extra Claim from Last Lecture}
Before we start, we will make the following claim:

\begin{claim}
The amortized cost of $OBNR(x)$, that is, the cost OBNR achieves when
accessing element $x$ is less than or equal to $1 + 4 \lceil \log(w_{x})
\rceil$ where $w_{x}$ is the number of distinct elements accessed
since the last access to $x$, including $x$ itself.
\end{claim}

The above bound is known as a \emph{working set bound}:  The running
time of OBNR when all accesses are fairly close to each other is
logarithmic in the size of this so-called working set, and not in the
size of the entire set of elements.

\section{Cost Models for BSTs}
We will examine the basic cost models but first some quick notation.
A general  access sequence is of the form $\langle x_{1}, x_{2}, x_{3}, \dots, x_{m} \rangle$;
where $m$ denotes the total number of acesses. We use $n$ to denote the number of elements
in the BST.

For the static and dynamic models, we will examine the
idea of competitiveness.  In general, an online algorithm is said to
be $\alpha$-competitive if for all access sequences $x$, the cost of
the algorithm on sequence $x$, $COST(x) \leq \alpha{}OPT(x)$ where
$OPT(x)$ is the minimum possible cost of accessing sequence $x$.  Note
that under certain models, $OPT(x)$ can be computed offline, that is,
if all accesses are known in advance.  We say that an online algorithm
is optimal if it is $O(1)$-competitive.

\subsection{Worst case}
In the worst case, the cost per access is $\Omega(\lg{n})$ for a balanced BST like a red-black tree or an AVL tree. However,
we hope to do better than that. We begin by presenting preliminary examples
of sequences for which we can achieve $o(\lg n)$ bounds. For instance,
if the sequence 
is $\langle 1, 2,..., n \rangle\ (m=n)$, we can achieve $O(1)$ cost
per access by 
starting with a linear, sorted tree and rotating it after each access:

\begin{center}
\scalebox{.5}{\includegraphics{figure1.jpg}}
\end{center}

\subsection{Stochastic and Static models}

In the stochastic model, the $x_j$'s in the access sequence are
i.i.d.~random variables; 
element $i \in [n]$ is accessed with probability $p_{i}$.  In a
degenerate case where $\exists i: p_{i}\approx 1$, storing element $i$ at the
root allows us to achieve roughly $O(1)$ time per access.  

In the static model, the $x_j$'s are not necessarily independent.
This minor change has significant implications:  For example, we will
see below that the Transpose algorithm is stochastically optimal but
not statically optimal.

In general,
the setup is similar to a Huffman tree, which is a data structure
that stores data only at the leaves, and attempts to minimize the expected
depth of an element chosen according to the given distribution.  This
expected depth matches the entropy of the distribution, up to an
additive $1$. The entropy is defined as:

\begin{displaymath}
H = \sum_{i} p_{i} \lg{\frac{1}{p_{j}}} + O(1)
\end{displaymath}

The Huffman tree achieves the entropy bound $O(H)$, which from
information theory is the optimal bound.  BSTs are more constrained in
that they must arrange nodes in symmetric order in the tree. However,
we will see that it is still possible to match the entropy bound (up
to an additive factor).  We also note that it is possible for BSTs to
slightly beat the entropy bound for some access sequences because
unlike Huffman trees (which achieve the entropy bound) they can store
elements in internal nodes.  

In the stochastic and static models, we disallow rotations; that is,
the tree is static and cannot be modified from its initial
configuration.  Under this model, we can find 
an optimal BST via dynamic programming (DP) and in this way for any
sequence $x$ calculate $OPT(x)$.  

First observe that only the frequency of accessing each element
matters; because the tree is 
static, temporal correlations are inconsequential.  If we have a
deterministic sequence of accesses, $p_{i}$ 
is the number of appearances of $i$ in our sequence divided by $m$
(the empirical probability).  The DP looks at
each element $r$, and tries to place $r$ at the root. For each $r$, we
have two subproblems: finding an optimal left subtree and an optimal
right subtree. 

\begin{displaymath}
Cost(r) = 1 + Pr(left)*Cost(left) + Pr(right)*Cost(right)
\end{displaymath}

In general, the subproblems are finding the optimal
tree for an interval of the elements, so there are $O(n^2)$
subproblems. Trying all roots for each subproblem gives a dynamic program
with run time $O(n^3)$. Knuth improved the DP to run in $O(n^{2})$ \cite{knuth}. 
Whether we can achieve a better result is an open problem (3-SUM
hard?). In this static case, the optimal cost per access is bounded by:

\begin{displaymath}
H-\lg{H}-\lg{e}+1 \leq \frac{OPT}{access} \leq H+3
\end{displaymath}

From information theory, $O(H)$ is the statically optimal bound. This
means that this (offline) BST is $O(1)$-competitive against any
static tree. Here we have to know the $p_i$'s
 in advance and thus the algorithm is offline. As we
 will see later, splay trees give us an online statically optimal algorithm.

\subsection{Dynamic model}
In linked lists, the constant-finger model proved to be better than
the one-finger model.  For trees however, we simply examine the
one-finger model as the constant-finger model does not provide much
added benefit. Building on the Sleator and Tarjan cost model from the
last lecture, we pay 1 unit for each standard rotation we perform
using a node x and its parent. We use the notation Rotate(x) to
specify that we are rotating x with its parent. In the subsequent two
sections, we will consider some algorithms for self-organizing trees
and conclude by detailing the properties of splay trees.

There is no known algorithm to calculate the cost of the optimal dynamic binary
search tree for all access sequences.  In addition, it is not known
whether any online BST is $O(1)$-competitive.

\section{BST Access Algorithms}
\subsection{Transpose}
The Transpose algorithm for BSTs takes the search element x and performs a 
standard rotation to move x one level closer to the root everytime it is 
requested. Unfortunately, this algorithm is bad for uniform searches
because it generates all binary trees with equal probability
\cite{AM78}. This gives us an average search cost of $O(\sqrt n)$, which is worse than a balanced binary tree.

\subsection{Move-to-root}
The Move-to-root algorithm extends the Transpose idea by repeatedly rotating the
search element x all the way to the root, every time it is requested. The repeated
rotations result in better search performance - the search cost is within a factor of
$2ln2 \approx 1.38$ of the statically optimal tree in the stochastic model. This is still
not statically optimal however, as repeated rotations cause many nodes to get too deep 
into the tree, causing a performance hit when they are requested.  

\section{Splay Trees}

Splay trees were first introduced by Sleator and Tarjan \cite{ST85}
and are an elegant self-organizing binary search tree. Splay trees search for
an element x and then always bring x to the root. First, we locate x by walking
down the tree and comparing with the elements. Then, depending on the placement
of x, we perform one of two special rotations shown below until we bring x to the
root. At the end, we may need to perform a regular rotation if x becomes the child 
of the root. Splay trees do not maintain any balance or auxiliary information
but nonetheless, achieve a remarkable set of properties including static 
optimality and are conjectured to satisfy dynamic optimality as well. 

The zig-zig case:

\begin{center}
\scalebox{.5}{\includegraphics{figure2.jpg}}
\end{center}

The zig-zag case:

\begin{center}
\scalebox{.5}{\includegraphics{figure3.jpg}}
\end{center}

\subsection{Access Lemma}
We state a fundamental result about Splay trees without proof.  For
details, consult \cite{ST85}.

\begin{lemma}
For each element $x_{i}$ in the tree, we give it an arbitrary positive
weight $w_{i}$.  For any node $x$, let $W_{x}$ be $\sum_{i \in
  subtree(x)} w_{i}$.  Let the potential $\Phi$ be
denoted as $\sum_{x} \log W_{x}$.  Then the amortized cost of
a single rotation (zig-zig or zig-zag) is at most $3(\log W^{new}_{x}
- \log W^{old}_{x})$, where the superscripts old and new denote the
values of $W_{x}$ before and after the rotation.
\end{lemma}

\begin{corollary}
From telescoping the sum above through all rotations needed in a splay
step, the amortized cost of access$(x)$ is at most $O(\log W_{tree} -
\log W^{old}_{x}) + 1$, where $W_{tree}$ be the value of
$W_{x}$ when $x$ is the root of the tree.
\end{corollary}

The above \emph{access lemma} can be used to prove many theorems about
splay trees, as we will see below.  It counts rotations, instead of
the path length needed to access the element from the root, but this
is simply a constant factor difference, as the splay step rotates the
accessed element all the way back to the root.  To see that the
potential function makes sense, consider a perfectly balanced tree and
a perfectly unbalanced tree (i.e. a linked list), and assume that
$w_{i} = 1$ for all elements.  The potential of the perfectly balanced
tree is $O(n)$, while that of the linked list is $O(n \log n)$.  This
confirms our intuition that unbalanced trees should have a higher
potential.

Finally, we note that the above lemma only holds when the number of
accesses is $\Omega(n \log n)$.  This is because unlike standard
amortized analysis, the initial potential is non-zero.  We can only
ignore this initial potential (so-called ``startup cost'') if the
number of accesses is large enough.  Since $\Phi_{max} - \Phi_{min} =
O(n \log n)$, where $\Phi_{max}$ and $\Phi_{min}$ denote the maximum
and minimum possible potential, we need the number of accesses to be
$\Omega(n \log n)$.

\section{Results and Conjectures about Splay trees}

\subsection{Results following from the Access Lemma}

We will give four results that follow directly (with some ingenuity)
from the access lemma.

\subsubsection{Logarithmic Access Cost}

For every element in the tree, we give it weight $w_{i} = 1$.  Then
$W_{tree} = n$ and the cost of access is $O(\log n - \log 1) = O(\log
n)$.  Intuitively, this lemma is pleasing because it confirms that
splay trees do no worse than balanced BSTs.

\subsubsection{Static Optimality Theorem}

Let each element $i$ have a fixed probability of access
$p_{i} = \frac{f_{i}}{m}$, where $m$ is the total number of accesses and
$f_{i}$ the frequency of accessing element $i$.  Set $w_{i}$ =
$p_{i}$.  Since the sum of probabilities is $1$, then $W_{tree} = 1$,
and the cost of accessing element $i$ is $O(\log 1 - \log p_{i}) =
O(-\log p_{i})$.

This result shows that splay trees achieve the entropy bound and thus
do as well as the best offline static tree.  The most amazing thing
about this is that splay trees are completely online and have no idea
what $p_{i}$ is, yet they still achieve this bound.  Although we used
$p_{i}$ in the analysis, the splay tree is completely ignorant of it.

\subsubsection{Static Finger Theorem}

Suppose we fix one of the elements $f$ in the tree as a ``finger''.
That is to say, all searches start from this ``finger'' instead of
from the root.
Set $w_{i} = \frac{1}{1 + (i-f)^{2}} < \frac{1}{(i-f)^{2}}$ when $i
\ne f$.  Then the weight of the tree $W_{tree} \leq 2 \sum_{d=1}^{n+1}
\frac{1}{d^{2}} \leq 2 \sum_{d=1}^{\infty} \frac{1}{d^{2}} =
\frac{\pi^{2}}{3} = O(1)$.  Then the cost of access is $O(\log O(1) -
\log \frac{1}{1 + (x-f)^{2}}) = O(1 + \log(1 + (x-f)^{2})) = O(\log[2
  + |x-f|])$.

This theorem shows that
regardless of what finger is chosen, a splay tree does as well as a
BST with that finger, and runs in time logarithmic to the distance
of the element from the finger.  Thus, if you access some element near the finger,
the amortized cost of accessing that element is cheap.  This is
remarkable because splay trees have no idea where the finger is.

\subsubsection{Working Set Theorem}
Let $t_{i}(x)$ be the number of distinct elements accessed since the
last access of element $x$ before time $i$.  We set the potential of a
key $w_{x}$ at time $i$ to be $\frac{1}{t_{i}(x)^{2}}$.  We can show,
using analysis similar to that of the static finger theorem, that
$W_{tree} \leq \frac{\pi^{2}}{6} = O(1)$.  After applying the access
lemma, the cost of accessing element $x$ is $O(1 + \log t_{i}(x_{i})$.
This shows that the cost of accessing elements within a working set is
logarithmic in the size of that set and not in the size of the entire
set of elements.  Equivalently, it says that if you accessed something
you accessed recently, the cost of accessing it is cheap.

The observant reader might note something fishy here:  We are changing
the potential after every access!  In particular, after accessing
element $x$, $w_{x}$ increases to $1$.  If during the reweighting of
the entire tree, the total potential increases, then we have obtained
``free'' potential from the reweighting and the above analysis is
flawed.

However, we will show that this is not the case.  Note that for all
elements $y \ne x$, $w_{y}$ cannot increase:  It either decreases, if
$x$ was accessed for the first time since $y$ was last accessed, and
stays the same otherwise.  Hence the only potential increase that can
come from the reweighting is from the increase in $w_{x}$.  But after
each access, $x$ goes to the root, and therefore the only subtree
where the potential increases is the root, and the only $W_{i}$ value
affected is $W_{tree}$.  But we already know that $W_{tree}$ is upper
bounded, and we have used this upper bound, so the potential can never
increase after re-weighting.  Therefore, this potential change is
justified and our analysis holds.

\subsection{Other Results}

The results here are cited without proof.

\subsubsection{Scanning Theorem}
The Scanning Theorem, demonstrated by Tarjan \cite{Tar85}, states the
following:

\begin{theorem}
If elements of a splay tree are accessed sequentially in order, then
the total running time is linear, regardless of the initial structure
of the splay tree.
\end{theorem}

\subsubsection{Dynamic Finger Theorem}
The Dynamic Finger Theorem (formerly conjecture) which was first
conjectured in \cite{ST85} and finally proven by Cole et al.\cite{CMSS},
\cite{Cole} after a lengthy analysis, states:

\begin{theorem}
The amortized cost of accessing the $i^{th}$ element $x_{i}$ is
$O(\log [1 + |x_{i} - x_{i-1}|])$.
\end{theorem}

The intuition behind this theorem is the static finger theorem:
suppose instead of having a fixed finger, the finger could move
after every access (we have to pay a cost for moving the
finger around).  In this case, it is always optimal to move the finger to the
next element accessed:  since we have to touch the next element
anyway, why not take the fastest route to it, instead of returning to
the original finger (or root) position?  This theorem therefore shows
that splay trees run within a constant factor of this optimal bound.

\subsection{Conjectures}

In this section, we show results that are conjectured to be true about
splay trees but have not been proved.

\subsubsection{Dynamic Optimality Conjecture}

The Dynamic Optimality conjecture is one of the biggest open problems
in the field of self-adjusting data structures.  It was first stated
in \cite{ST85}:

\begin{conjecture}
Consider an optimal offline algorithm OPT that is allowed to modify
the tree via rotations between accesses.  The cost of access is the same
i.e. $1$ plus the depth of the accessed node.  In addition, each
rotation costs $O(1)$.  The conjecture is that the run time of splay
trees is within a constant factor of OPT for any access sequence.
\end{conjecture}

If the dynamic optimality conjecture is correct, then splay trees are
$O(1)$-competitive even in a general dynamic model.  This is a very
powerful result:  splay trees are as good as any other tree on any
given request sequence, even if the other tree is specially designed
for the given request sequence.  We note as an aside that splay trees
can be easily shown to be at least $O(\log n)$-competitive dynamically.  

It is an open question as to whether
there exists \emph{any} BST (not just splay trees) that satisfy the
dynamic optimality conjecture.  In the next few lectures, we will see
recent results on this problem that have reduced the competitive gap
from $O(\log n)$ to $O(\log \log n)$ \cite{demaine}.

\subsubsection{The Unified Conjecture}

In the lecture, we have shown several properties of splay trees, but
how do they all compare to each other?  We note the following:

\begin{itemize}
\item The Working set theorem implies the static optimality theorem, which
  in turn implies the static finger theorem.
\item The Dynamic finger theorem implies the static finger theorem.
\end{itemize}

In addition, Iacono \cite{Iac01} has proposed the following
\emph{unified conjecture}:

\begin{conjecture}
The amortized cost to access element $x_{i}$ is upper bounded by:
\begin{align*}
O(\min_{y} \log [t_{i}(y) + |x_{i}-y| + 2])
\end{align*}
\end{conjecture}
Intuitively, this suggests that element $x_{i}$ is cheap to access if there
exists some different element $y$ that is both spatially (in terms of
the structure of the tree) and temporally (in terms of access time)
close to it.  The unified conjecture, if true, would imply both the
working set and dynamic finger theorems.  It is not known whether the unified
conjecture holds for a BST, but Badoiu and Demaine \cite{BD} have
shown that it can be done on a pointer machine.

\subsubsection{Traversal conjecture}

\begin{conjecture}
Supposing we have two splay trees, each with the same elements.  Take
a preorder traversal of one of the splay trees to obtain an
access sequence.  The conjecture is that the cost of accessing
elements in this access sequence in order in the other splay tree is
$O(n)$ \cite{ST85}.
\end{conjecture}

\subsubsection{Deque conjecture}
Supposing now we have two fingers, and we can choose to start our
search pointer from either of them during an access.  After each
access, we are allowed to move either finger by one element, thus
simulating a double-ended queue (deque).  We now have the following
conjecture: 

\begin{conjecture}
The cost of $m$ accesses in splay tree of $n$ elements is $O(m+n)$
\cite{Tar85}. 
\end{conjecture}

As an aside, it is known that the cost is at most
$O((m+n)\alpha{}(m+n))$ \cite{Sun}.

\subsubsection{Removing elements from the access sequence}

Finally, an unpublished but simple conjecture states:

\begin{conjecture}
Consider a sequence of accesses on a splay tree.  If any of these
accesses is removed, the resulting sequence of accesses is cheaper
than the original sequence of accesses.
\end{conjecture}

This conjecture seems obvious, but has so far remained resistant to attacks.

\bibliographystyle{alpha}

\begin{thebibliography}{99}

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\bibitem{BD}
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\bibitem{demaine}
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\bibitem{CMSS}
R. Cole, B. Mishra, J. Schmidt, and A. Siegel, \emph{On the dynamic
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\bibitem{Cole}
R. Cole, \emph{On the dynamic finger conjecture for splay trees.  Part
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\bibitem{Iac01}
J. Iacono, \emph{Alternatives to Splay Trees with $o(\log n)$ worst
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\bibitem{knuth} 
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\bibitem{ST85}
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\bibitem{Sun}
R. Sundar, \emph{On the Deque conjecture for the splay algorithm},
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\bibitem{Tar85}
R. Tarjan, \emph{Sequential access in splay trees takes linear time},
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\end{thebibliography}

\end{document}