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Journal ArticleDOI

Applications of Ramsey's theorem to decision tree complexity

TL;DR: All existing lower bounds for comparison-based algorithms are valid for general k-bounded decision trees, where k is a constant, and are shown to hold for nondeterministic and probabilistic decision trees as well.
Abstract: Combinatorial techniques for extending lower bound results for decision trees to general types of queries are presented. Problems that are defined by simple inequalities between inputs, called order invariant problems, are considered. A decision tree is called k-bounded if each query depends on at most k variables. No further assumptions on the type of queries are made. It is proved that one can replace the queries of any k-bounded decision tree that solves an order-invariant problem over a large enough input domain with k-bounded queries whose outcome depends only on the relative order of the inputs. As a consequence, all existing lower bounds for comparison-based algorithms are valid for general k-bounded decision trees, where k is a constant.An O(n log n) lower bound for the element uniqueness problem and several other problems for any k-bounded decision tree, such that k = O(nc) and c

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Citations
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Journal ArticleDOI
TL;DR: The problem of electing a leader in a synchronous ring of n processors is considered and positive and negative results are obtained.
Abstract: The problem of electing a leader in a synchronous ring of n processors is considered. Both positive and negative results are obtained. On the one hand, if processor IDS are chosen from some countable set, then there is an algorithm that uses only O(n) messages in the worst case. On the other hand, any algorithm that is restricted to use only comparisons of IDs requires O(n log n) messages in the worst case. Alternatively, if the number of rounds is required to be bounded by some t in the worst case, and IDs are chosen from any set having at least ƒ(n, t) elements, for a certain very fast-growing function ƒ, then any algorithm requires O(n log n) messages in the worst case.

196 citations

Proceedings ArticleDOI
01 Jun 1993
TL;DR: A study of computation that can be done locally in a distributed network, where \locally" means within time (or distance) independent of the size of the network, and results include Locally Checkable Labeling problems, where the legality of a labeling can be checked locally.
Abstract: The purpose of this paper is a study of computation that can be done locally in a distributed network, where "locally" means within time (or distance) independent of the size of the network. Locally checkable labeling (LCL) problems are considered, where the legality of a labeling can be checked locally (e.g., coloring). The results include the following: There are nontrivial LCL problems that have local algorithms. There is a variant of the dining philosophers problem that can be solved locally. Randomization cannot make an LCL problem local; i.e., if a problem has a local randomized algorithm then it has a local deterministic algorithm. It is undecidable, in general, whether a given LCL has a local algorithm. However, it is decidable whether a given LCL has an algorithm that operates in a given time $t$. Any LCL problem that has a local algorithm has one that is order-invariant (the algorithm depends only on the order of the processor IDs).

145 citations

Proceedings ArticleDOI
01 Feb 1995
TL;DR: The happy coloring and orientation problem is introduced and it is shown that it yields a robust local solution to the (d,m)-dining philosophers problem of Naor and Stockmeyer and the amount of initial symmetry-breaking needed to solve certain problems locally is investigated.
Abstract: The purpose of this paper is a study of computation that can be done locally in a distributed network. By locally we mean within time (or distance) independent of the size of the network. In particular we are interested in algorithms that ore robust, i.e., perform well even if the underlying graph is not stable and links continuously fail and come-up. We introduce and study the happy coloring and orientation problem and show that it yields a robust local solution to the (d,m)-dining philosophers problem of Naor and Stockmeyer [17]. This problem is similar to the usual dining philosophers problem, except that each philosopher has access to d forks but needs only m of them to eat. We give a robust local solution if m/spl les/[d/2] (necessity of this inequality for any local solution was known previously). Two other problems we investigate are: (1) the amount of initial symmetry-breaking needed to solve certain problems locally (for example, our algorithms need considerably less symmetry-breaking than having a unique ID on each node), and (2) the single-step color reduction problem: given a coloring with c colors of the nodes of a graph, what is the smallest number of colors c' such that every node can recolor itself with one of c' colors as a function of its immediate neighborhood only. >

65 citations


Cites background from "Applications of Ramsey's theorem to..."

  • ..., [15, 24, 33], Naor and Stockmeyer [28] show that if an LCL can be solved by a local algorithm in time t, then it can be solved by...

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Journal ArticleDOI
TL;DR: The first nontrivial lower bounds on time-space trade-offs for the selection problem are established, and deterministic lower bounds for I/O-efficient algorithms as well are got.
Abstract: We establish the first nontrivial lower bounds on time-space trade-offs for the selection problem. We prove that any comparison-based randomized algorithm for finding the median requires Ω(nlog logSn) expected time in the RAM model (or more generally in the comparison branching program model), if we have S bits of extra space besides the read-only input array. This bound is tight for all S > log n, and remains true even if the array is given in a random order. Our result thus answers a 16-year-old question of Munro and Raman l1996r, and also complements recent lower bounds that are restricted to sequential access, as in the multipass streaming model lChakrabarti et al. 2008br.We also prove that any comparison-based, deterministic, multipass streaming algorithm for finding the median requires Ω(nloga(n/s)+ nlogsn) worst-case time (in scanning plus comparisons), if we have s cells of space. This bound is also tight for all s >log2n. We get deterministic lower bounds for I/O-efficient algorithms as well.The proofs in this article are self-contained and do not rely on communication complexity techniques.

56 citations

Journal ArticleDOI
TL;DR: There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science as discussed by the authors.
Abstract: There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramsey-type theorems to various fields in mathematics are well documented in published books and monographs. The main objective of this survey is to list applications mostly in theoretical computer science of the last two decades not contained in these.

52 citations


Cites methods from "Applications of Ramsey's theorem to..."

  • ...Any assignment of values bi to the input variables xi defines a unique computation path from S the electronic journal of combinatorics (Dec 2004), #DS13 19 to a sink, leaving each vertex labeled xi through the edge labeled bi....

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References
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Journal ArticleDOI
TL;DR: This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula.
Abstract: This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula*. But in the course of this investigation it is necessary to use certain theorems on combinations which have an independent interest and are most conveniently set out by themselves beforehand.

2,223 citations

Journal ArticleDOI
TL;DR: The problem of finding all maximal elements of V with respect to the partial ordering is considered and the computational com- plexity of the problem is defined to be the number of required comparisons of two components and is denoted by Cd(n).
Abstract: H. T. KUNG Carnegze-Mellon Un~verszty, P2ttsburgh, Pennsylvanza F. LUCCIO Unwerszht d~ P~sa, P~sa, Italy F. P. PREPARATA University of Ilhno~s, Urbana, Illinois ASSTRACT. Let U1 , U2, . . . , Ud be totally ordered sets and let V be a set of n d-dimensional vectors In U~ X Us. . X Ud . A partial ordering is defined on V in a natural way The problem of finding all maximal elements of V with respect to the partial ordering ~s considered The computational com- plexity of the problem is defined to be the number of required comparisons of two components and is denoted by Cd(n). It is tnwal that C~(n) = n - 1 and C,~(n) _ flog2 n!l for d _> 2

856 citations

Proceedings ArticleDOI
01 Dec 1983
TL;DR: All the apparently known lower bounds for linear decision trees are extended to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20].
Abstract: A topological method is given for obtaining lower bounds for the height of algebraic computation trees, and algebraic decision trees. Using this method we are able to generalize, and present in a uniform and easy way, almost all the known nonlinear lower bounds for algebraic computations. Applying the method to decision trees we extend all the apparently known lower bounds for linear decision trees to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20]. We also show how this new method can be used to establish lower bounds on the complexity of constructions with ruler and compass in plane Euclidean geometry.

584 citations

Journal ArticleDOI
TL;DR: It is shown that, in a rather general model including al1 the commonly-used schemes, $\lceil $ lg(n+l) $\rceil$ probes to the table are needed in the worst case, provided the key space is sufficiently large.
Abstract: We examine optimality questions in the following information retrieval problem: Given a set S of n keys, store them so that queries of the form "Is x $\in$ S?" can be answered quickly. It is shown that, in a rather general model including al1 the commonly-used schemes, $\lceil$ lg(n+l) $\rceil$ probes to the table are needed in the worst case, provided the key space is sufficiently large. The effects of smaller key space and arbitrary encoding are also explored.

405 citations

Proceedings ArticleDOI
05 May 1975
TL;DR: An effort is made to recast classical theorems into a useful computational form and analogies are developed between constructibility questions in Euclidean geometry and computability questions in modern computational complexity.
Abstract: The complexity of a number of fundamental problems in computational geometry is examined and a number of new fast algorithms are presented and analyzed. General methods for obtaining results in geometric complexity are given and upper and lower bounds are obtained for problems involving sets of points, lines, and polygons in the plane. An effort is made to recast classical theorems into a useful computational form and analogies are developed between constructibility questions in Euclidean geometry and computability questions in modern computational complexity.

287 citations