616 research outputs found
Succinct data structures for dynamics strings
A new simple algorithm for optimal embedding of complete binary trees into hypercubes as well as a node-by-node algorithm for embedding of nearly complete binary trees into hypercubes are presented
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
Embedding cube-connected cycles graphs into faulty hypercubes
We consider the problem of embedding a cube-connected cycles graph (CCC) into a hypercube with edge faults. Our main result is an algorithm that, given a list of faulty edges, computes an embedding of the CCC that spans all of the nodes and avoids all of the faulty edges. The algorithm has optimal running time and tolerates the maximum number of faults (in a worst-case setting). Because ascend-descend algorithms can be implemented efficiently on a CCC, this embedding enables the implementation of ascend-descend algorithms, such as bitonic sort, on hypercubes with edge faults. We also present a number of related results, including an algorithm for embedding a CCC into a hypercube with edge and node faults and an algorithm for embedding a spanning torus into a hypercube with edge faults
Embedding complete ternary tree in hypercubes using AVL trees
A complete ternary tree is a tree in which every non-leaf vertex has exactly three children. We prove that a complete ternary tree of height h, TTh, is embeddable in a hypercube of dimension . This result coincides with the result of [2]. However, in this paper, the embedding utilizes the knowledge of AVL trees. We prove that a subclass of AVL trees is a subgraph of hypercube. The problem of embedding AVL trees in hypercube is an independent emerging problem
On Counting and Embedding a Subclass of Height-Balanced Trees
A height-balanced tree is a rooted binary tree in which, for every vertex v, the difference in the heights of the subtrees rooted at the left and right child of v (called the balance factor of v) is at most one. In this paper, we consider height-balanced trees in which the balance factor of every vertex beyond a level is 0. We prove that there are 22t-1 such trees and embed them into a generalized join of hypercubes
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