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Minimal Path Length of Trees with Known Fringe
In this paper we continue the study of the path length of trees with known fringe as initiated by [1] and [2]. We compute the path length of the minimal tree with given number of leaves N and fringe ∆ for the case ∆ ≥ N/2. This complements the result of [2] that studied the case ∆ ≤ N/2. Our methods also yields a linear time algorithm for constructing the minimal tree when ∆ ≥ N/2
Finger Search in Grammar-Compressed Strings
Grammar-based compression, where one replaces a long string by a small
context-free grammar that generates the string, is a simple and powerful
paradigm that captures many popular compression schemes. Given a grammar, the
random access problem is to compactly represent the grammar while supporting
random access, that is, given a position in the original uncompressed string
report the character at that position. In this paper we study the random access
problem with the finger search property, that is, the time for a random access
query should depend on the distance between a specified index , called the
\emph{finger}, and the query index . We consider both a static variant,
where we first place a finger and subsequently access indices near the finger
efficiently, and a dynamic variant where also moving the finger such that the
time depends on the distance moved is supported.
Let be the size the grammar, and let be the size of the string. For
the static variant we give a linear space representation that supports placing
the finger in time and subsequently accessing in time,
where is the distance between the finger and the accessed index. For the
dynamic variant we give a linear space representation that supports placing the
finger in time and accessing and moving the finger in time. Compared to the best linear space solution to random
access, we improve a query bound to for the static
variant and to for the dynamic variant, while
maintaining linear space. As an application of our results we obtain an
improved solution to the longest common extension problem in grammar compressed
strings. To obtain our results, we introduce several new techniques of
independent interest, including a novel van Emde Boas style decomposition of
grammars
Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees
We prove limit theorems for sums of functions of subtrees of binary search
trees and random recursive trees. In particular, we give simple new proofs of
the fact that the number of fringe trees of size in the binary search
tree and the random recursive tree (of total size ) asymptotically has a
Poisson distribution if , and that the distribution is
asymptotically normal for . Furthermore, we prove similar
results for the number of subtrees of size with some required property , for example the number of copies of a certain fixed subtree . Using
the Cram\'er-Wold device, we show also that these random numbers for different
fixed subtrees converge jointly to a multivariate normal distribution. As an
application of the general results, we obtain a normal limit law for the number
of -protected nodes in a binary search tree or random recursive tree.
The proofs use a new version of a representation by Devroye, and Stein's
method (for both normal and Poisson approximation) together with certain
couplings
Compact routing on the Internet AS-graph
Compact routing algorithms have been presented as candidates for scalable routing in the future Internet, achieving near-shortest path routing with considerably less forwarding state than the Border Gateway Protocol. Prior analyses have shown strong performance on power-law random graphs, but to better understand the applicability of compact routing algorithms in the context of the Internet, they must be evaluated against real- world data. To this end, we present the first systematic analysis of the behaviour of the Thorup-Zwick (TZ) and Brady-Cowen (BC) compact routing algorithms on snapshots of the Internet Autonomous System graph spanning a 14 year period. Both algorithms are shown to offer consistently strong performance on the AS graph, producing small forwarding tables with low stretch for all snapshots tested. We find that the average stretch for the TZ algorithm increases slightly as the AS graph has grown, while previous results on synthetic data suggested the opposite would be true. We also present new results to show which features of the algorithms contribute to their strong performance on these graphs
Fringe trees, Crump-Mode-Jagers branching processes and -ary search trees
This survey studies asymptotics of random fringe trees and extended fringe
trees in random trees that can be constructed as family trees of a
Crump-Mode-Jagers branching process, stopped at a suitable time. This includes
random recursive trees, preferential attachment trees, fragmentation trees,
binary search trees and (more generally) -ary search trees, as well as some
other classes of random trees.
We begin with general results, mainly due to Aldous (1991) and Jagers and
Nerman (1984). The general results are applied to fringe trees and extended
fringe trees for several particular types of random trees, where the theory is
developed in detail. In particular, we consider fringe trees of -ary search
trees in detail; this seems to be new.
Various applications are given, including degree distribution, protected
nodes and maximal clades for various types of random trees. Again, we emphasise
results for -ary search trees, and give for example new results on protected
nodes in -ary search trees.
A separate section surveys results on height, saturation level, typical depth
and total path length, due to Devroye (1986), Biggins (1995, 1997) and others.
This survey contains well-known basic results together with some additional
general results as well as many new examples and applications for various
classes of random trees
Primitive Words, Free Factors and Measure Preservation
Let F_k be the free group on k generators. A word w \in F_k is called
primitive if it belongs to some basis of F_k. We investigate two criteria for
primitivity, and consider more generally, subgroups of F_k which are free
factors.
The first criterion is graph-theoretic and uses Stallings core graphs: given
subgroups of finite rank H \le J \le F_k we present a simple procedure to
determine whether H is a free factor of J. This yields, in particular, a
procedure to determine whether a given element in F_k is primitive.
Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from
the direct product of k copies of G to G), where G is an arbitrary finite
group. We call w measure preserving if given uniform measure on G x G x ... x
G, w induces uniform measure on G (for every finite G). This is the second
criterion we investigate: it is not hard to see that primitivity implies
measure preservation and it was conjectured that the two properties are
equivalent. Our combinatorial approach to primitivity allows us to make
progress on this problem and in particular prove the conjecture for k=2.
It was asked whether the primitive elements of F_k form a closed set in the
profinite topology of free groups. Our results provide a positive answer for
F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I:
A New Algorithm", and "On Primitive Words II: Measure Preservation". 42
pages, 14 figures. Some parts of the paper reorganized towards publication in
the Israel J. of Mat
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