There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

An approach to complexity theory which offers a means of analysing algorithms in terms of their tractability. The authors consider the problem in terms of parameterized languages and taking "k-slices" of the language, thus introducing readers to new classes of algorithms which may be analysed more precisely than was the case until now. The book is as self-contained as possible and includes a great deal of background material. As a result, computer scientists, mathematicians, and graduate students interested in the design and analysis of algorithms will find much of interest.

Parameterized Complexity

PART I: FOUNDATIONS 1. Introduction to Fixed-Parameter Algorithms 2. Preliminaries and Agreements 3. Parameterized Complexity Theory - A Primer 4. Vertex Cover - An Illustrative Example 5. The Art of Problem Parameterization 6. Summary and Concluding Remarks PART II: ALGORITHMIC METHODS 7. Data Reduction and Problem Kernels 8. Depth-Bounded Search Trees 9. Dynamic Programming 10. Tree Decompositions of Graphs 11. Further Advanced Techniques 12. Summary and Concluding Remarks PART III: SOME THEORY, SOME CASE STUDIES 13. Parameterized Complexity Theory 14. Connections to Approximation Algorithms 15. Selected Case Studies 16. Zukunftsmusik References Index

Invitation to fixed-parameter algorithms

This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.

Parameterized Algorithms

We consider the indexable dictionary problem which consists in storing a set S ⊆ {0,…, m - 1} for some integer m, while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x e S, and -1 otherwise; and select(i) which returns the i-th smallest element in S.We give a structure that supports both operations in O(1) time on the RAM model and requires B(n,m) + o(n) + O(lg lg m) bits to store a set of size n, where B(n,m) = ⌈lg (nm)⌉ is the minimum number of bits required to store any n-element subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh.We also present two applications of our dictionary structure:• An information-theoretically optimal representation for k-ary cardinal trees (aka k-ary tries). Our structure uses C(n,k) + o(n + lg k) bits to store a k-ary tree with n nodes and can support parent, i-th child, child labeled i, and the degree of a node in constant time, where C(n,k) is the minimum number of bits to store any n-node k-ary tree. Previous space efficient representations for cardinal k-ary trees required C(n,k) + Ω(n) bits.• An optimal representation for multisets where (appropriate generalisations of) the select and rank operations can be supported in O(1) time. Our structure uses B(n, m + n) + o(n) + O(lg lg m) bits to represent a multiset of size n from an m element set; the first term is the minimum number of bits required to represent such a multiset.

Succinct indexable dictionaries with applications to encoding k-ary trees and multisets

We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,…,m − 1} for some integer m while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i), which returns the ith smallest element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n, m) p o(n) p O(lg lg m) bits to store a set of size n, where B(n, m) e ⌊lg (m/n)⌋ is the minimum number of bits required to store any n-element subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen [1995] and Pagh [2001]. We present extensions and applications of our indexable dictionary data structure, including: —an information-theoretically optimal representation of a k-ary cardinal tree that supports standard operations in constant time; —a representation of a multiset of size n from {0,…,m − 1} in B(n, m p n) p o(n) bits that supports (appropriate generalizations of) rank and select operations in constant time; and p O(lg lg m) —a representation of a sequence of n nonnegative integers summing up to m in B(n, m p n) p o(n) bits that supports prefix sum queries in constant time.

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Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets

We consider the implementation of abstract data types for the static objects: binary tree, rooted ordered tree, and a balanced sequence of parentheses. Our representations use an amount of space within a lower order term of the information theoretic minimum and support, in constant time, a richer set of navigational operations than has previously been considered in similar work. In the case of binary trees, for instance, we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned. The approach is applied to produce a succinct representation of planar graphs in which one can test adjacency in constant time.

Succinct Representation of Balanced Parentheses and Static Trees

This paper focuses on space efficient representations of rooted trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (or k-ary tries), where each node has k slots, labelled {1,...,k}, each of which may have a reference to a child, and ordinal trees, where the children of each node are simply ordered. Our representations use a number of bits close to the information theoretic lower bound and support operations in constant time. For ordinal trees we support the operations of finding the degree, parent, ith child, and subtree size. For cardinal trees the structure also supports finding the child labelled i of a given node apart from the ordinal tree operations. These representations also provide a mapping from the n nodes of the tree onto the integers {1, ..., n}, giving unique labels to the nodes of the tree. This labelling can be used to store satellite information with the nodes efficiently.

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Representing Trees of Higher Degree

In this paper we investigate the parameterized complexity of the problems MaxSat and MaxCut using the framework developed by Downey and Fellows. LetGbe an arbitrary graph havingnvertices andmedges, and letfbe an arbitrary CNF formula withmclauses onnvariables. We improve Cai and Chen'sO(22ckcm) time algorithm for determining if at leastkclauses of ac-CNF formulafcan be satisfied; our algorithm runs inO(|f|+k2?k) time for arbitrary formulae and inO(cm+ck?k) time forc-CNF formulae, where ? is the golden ratio(1+5)/2. We also give an algorithm for finding a cut of size at leastk; our algorithm runs inO(m+n+k4k) time. We then argue that the standard parameterization of these problems is unsuitable, because nontrivial situations arise only for large parameter values (k??m/2?), in which range the fixed-parameter tractable algorithms are infeasible. A more meaningful question in the parameterized setting is to ask whether ?m/2?+kclauses can be satisfied, or ?m/2?+kedges can be placed in a cut. We show that these problems remain fixed-parameter tractable even under this parameterization. Furthermore, for up to logarithmic values of the parameter, our algorithms for these versions also run in polynomial time.

Venkatesh Raman

Papers

Succinct indexable dictionaries with applications to encoding k-ary trees and multisets

Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets

Succinct Representation of Balanced Parentheses and Static Trees

Representing Trees of Higher Degree

Parametrizing Above Guaranteed Values: MaxSat and MaxCut