******* UWSpace has been experiencing unusually long wait times during the depositing process. If you are a graduate student depositing a thesis, it is recommended that while the browser is loading that you do not try to close the connection. If you receive an error or a timeout message, please logout and then log back in. Please do not continually try to resubmit, or recreate and resend a new thesis deposit. In most cases, despite the error message, your deposit has successfully been sent to be reviewed. You can verify this by checking under the ‘deposits being reviewed' page after logging back in, or by contacting UWSpace. We apologize for the inconvenience. We are working hard to resolve this issue quickly. *******

Show simple item record

dc.contributor.authorFarzan, Arash
dc.date.accessioned2009-10-29 19:54:06 (GMT)
dc.date.available2009-10-29 19:54:06 (GMT)
dc.date.issued2009-10-29T19:54:06Z
dc.date.submitted2009
dc.identifier.urihttp://hdl.handle.net/10012/4832
dc.description.abstractIn this thesis, we study succinct representations of trees and graphs. A succinct representation of a combinatorial object is a space efficient representation that supports a reasonable set of operations and queries on the object in constant or near constant time on the RAM with logarithmic word size. The storage requirement of a succinct representation is intended to be optimal to within lower order terms. We first propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decompositions of trees into subtrees. The approach simplifies the existing representation of ordinal trees while allowing the full set of navigational operations and queries. The approach applied to cardinal (i.e., k-ary) trees yields a space-optimal succinct representation allowing cardinal-type operations (e.g., determining child labeled i) as well as the full set of ordinal-type operations (e.g., reporting the number of siblings to the left of a node). Previous space-optimal succinct representations had not been able to support both types of operations efficiently. We demonstrate how the approach can be applied to obtain a space-optimal succinct representation for the family of free trees where the order of children is insignificant. Furthermore, we show that our approach can be used to obtain entropy-based succinct representations. The approach adapts to match the degree-distribution entropy suggested by Jansson et al. We discuss that our approach can be made adaptive to various other entropy measures. Next, we focus on ordinal trees, and present a novel universal succinct representation. Our new representation is able to simultaneously emulate previous ordinal tree representations of the balanced parenthesis (BP), depth first unary degree sequence (DFUDS) and partitioned representations using a single instance of the data structure. They not only support the union of all the ordinal tree operations supported by these representations, but will also automatically inherit any new operations supported by these representations in the future; hence the universality title we attributed to the representation. We then move to more general graphs rather than trees, and consider the problem of encoding a graph with $n$ vertices and m edges compactly supporting adjacency, neighborhood and degree queries in constant time. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of edges incident to a given vertex. The representation is to achieve the optimal space requirement as a function of n and m to within lower order terms. We prove a lower bound in the cell probe model that it is impossible to achieve the information theoretic lower bound to within lower order terms unless the graph is too sparse (namely, $m=o(n^\delta)$ for any constant \delta > 0) or too dense (namely m = \littleOmega{n^{2-\delta}}) for any constant \delta > 0). We also present a succinct encoding for graphs for all values of n,m supporting queries in constant time. The space requirement of the representation is always within a multiplicative 1+\epsilon factor of the information-theory lower bound for any constant $\epsilon > 0$. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly to within lower order terms when the graph is sparse (m=o(n^\delta) for any constant \delta > 0), or very dense (m = \littleOmega (n^2/(\sqrt{\log n})).en
dc.language.isoenen
dc.publisherUniversity of Waterlooen
dc.subjectTree, Graphen
dc.subjectSuccincten
dc.titleSuccinct Representation of Trees and Graphsen
dc.typeDoctoral Thesisen
dc.pendingfalseen
dc.subject.programComputer Scienceen
uws-etd.degree.departmentSchool of Computer Scienceen
uws-etd.degreeDoctor of Philosophyen
uws.typeOfResourceTexten
uws.peerReviewStatusUnrevieweden
uws.scholarLevelGraduateen


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record


UWSpace

University of Waterloo Library
200 University Avenue West
Waterloo, Ontario, Canada N2L 3G1
519 888 4883

All items in UWSpace are protected by copyright, with all rights reserved.

DSpace software

Service outages