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Splay trees, DavenportSchinzel sequences, and the deque conjecture
, 2007
"... We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct ..."
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Cited by 15 (5 self)
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We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan’s deque conjecture and prove that n deque operations require O(nα ∗ (n)) time, where α ∗ (n) is the minimum number of applications of the inverseAckermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.
Engineering an External Memory Minimum Spanning Tree Algorithm
 IN PROC. 3RD IFIP INTL. CONF. ON THEORETICAL COMPUTER SCIENCE
, 2004
"... We develop an external memory algorithm for computing minimum spanning trees. The algorithm is considerably simpler than previously known external memory algorithms for this problem and needs a factor of at least four less I/Os for realistic inputs. Our implementation indicates that this algorithm ..."
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Cited by 14 (3 self)
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We develop an external memory algorithm for computing minimum spanning trees. The algorithm is considerably simpler than previously known external memory algorithms for this problem and needs a factor of at least four less I/Os for realistic inputs. Our implementation indicates that this algorithm processes graphs only limited by the disk capacity of most current machines in time no more than a factor 2–5 of a good internal algorithm with sufficient memory space.
Applications of forbidden 01 matrices to search tree and path compression based data structures
, 2009
"... In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (nongeometric) algorithms and data s ..."
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Cited by 6 (5 self)
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In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (nongeometric) algorithms and data structures, namely, to bound the complexity of an object via its forbidden substructures. To analyze an algorithm or data structure in the forbidden substructure framework one proceeds in three discrete steps. First, one transcribes the behavior of the algorithm as some combinatorial object M; for example, M may be a graph, sequence, permutation, matrix, set system, or tree. (The size of M should ideally be linear in the running time.) Second, one shows that M excludes some forbidden substructure P, and third, one bounds the size of any object avoiding this substructure. The power of this framework derives from the fact that M lies in a more pristine environment and that upper bounds on the size of a Pfree object M may be reused in different contexts. All of our proofs begin by transcribing the individual operations of a dynamic data structure
Origins of nonlinearity in DavenportSchinzel sequences. Manuscript in submission. Avalabile from http://www.eecs.umich.edu/~pettie
, 2009
"... A generalized DavenportSchinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Expσ, nq be the maximum length of a sequence ..."
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Cited by 6 (5 self)
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A generalized DavenportSchinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Expσ, nq be the maximum length of a sequence over an alphabet of size n excluding subsequences isomorphic to σ. It has been proved that for every σ, Expσ, nq is either linear or very close to linear. In particular it is Opn2 αpnqOp1q q, where α is the inverseAckermann function and Op1q depends on σ. In much the same way that the complete graphs K5 and K3,3 represent the minimal causes of nonplanarity, there must exist a set ΦNonlin of minimal nonlinear forbidden subsequences. Very little is known about the size or membership of ΦNonlin. In this paper we construct an infinite antichain of nonlinear forbidden subsequences which, we argue, strongly supports the conjecture that ΦNonlin is itself infinite. Perhaps the most novel contribution of this paper is a succinct, humanly readable code for expressing the structure of forbidden subsequences. 1
An Algorithm for Enumerating All Spanning Trees of a Directed Graph
 Algorithmica
, 2000
"... We present an O(NV +V ) time algorithm for enumerating all spanning trees of a directed graph. This improves the previous best known bound of O(NE +V +E) ([1]) when V = o(N ), which will be true for most graphs. Here, N refers to the number of spanning trees of a graph having V vertices and ..."
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Cited by 5 (0 self)
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We present an O(NV +V ) time algorithm for enumerating all spanning trees of a directed graph. This improves the previous best known bound of O(NE +V +E) ([1]) when V = o(N ), which will be true for most graphs. Here, N refers to the number of spanning trees of a graph having V vertices and E edges. The algorithm is based on the technique of obtaining one spanning tree from another by a series of edge swaps. This result complements the result in the companion paper ([2]) which enumerates all spanning trees in an undirected graph in O(N + V +E) time.
Derivation of a Minimum Spanning Tree Algorithm
, 1997
"... View of a Graph As the components of the graph will have to be accessed by several modules it is worth encapsulating them into an abstract machine that should be shared by the other modules. We prefer the sharing mechanism provided by the SEES clause since it is a fullhiding one, supporting indepe ..."
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View of a Graph As the components of the graph will have to be accessed by several modules it is worth encapsulating them into an abstract machine that should be shared by the other modules. We prefer the sharing mechanism provided by the SEES clause since it is a fullhiding one, supporting independent reønement of the seen and seeing machines. In our case, this means that the seeing components do not have to depend on a particular representation of the graph, like an adjacency matrix or adjacency lists. Instead of committing from this early stage to such a representation, we shall postpone the choice of the most convenient data structure to the implementation level. At the speciøcation level we simply model the graph as a ønite nonempty set Nodes , a relation on this set Edges 2 Nodes $ Nodes and a weight function weight 2 Edges ! NAT . All three components should be declared as abstract constants 1 since they belong to the static part of the speciøcation and they are supposed to...
Provably Efficient NonPreemptive Task Scheduling with Cilk
"... We consider the problem of scheduling static task graphs by using Cilk, a Cbased runtime system for multithreaded parallel programming. We assume no preemption of task execution and no prior knowledge of the task execution times. Given a task graph G, the output of the scheduling algorithm is a Ci ..."
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We consider the problem of scheduling static task graphs by using Cilk, a Cbased runtime system for multithreaded parallel programming. We assume no preemption of task execution and no prior knowledge of the task execution times. Given a task graph G, the output of the scheduling algorithm is a Cilk program P which, when executed, initiates the tasks in consistence with the precedence requirements of G. We show that the Cilk model has restrictions in implementing optimal schedules for certain types of task graphs; however, the restriction does not fundamentally hinder the practical applications of Cilk, as it is possible to produce reasonably good quality schedules (in the sense of expected execution time). Our algorithm identifies a minimal number of stages, assigns tasks to these stages, and bundles parallel tasks of the same stage into one Cilk procedure. By using Tarjan's algorithm (for set operations) to implement the bundling process, we demonstrate that the parallel schedule c...
TimeSpace Tradeoffs for Graph st Connectivity
, 1992
"... TimeSpace Tradeoffs for Graph st Connectivity by Gregory Barnes Chairperson of Supervisory Committee: Walter L. Ruzzo Department of Computer Science and Engineering The problem of graph st connectivity, determining whether two vertices s and t in a graph are in the same connected component, ..."
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TimeSpace Tradeoffs for Graph st Connectivity by Gregory Barnes Chairperson of Supervisory Committee: Walter L. Ruzzo Department of Computer Science and Engineering The problem of graph st connectivity, determining whether two vertices s and t in a graph are in the same connected component, is a fundamental problem in computational complexity theory. Determining the space complexity of st connectivity either for directed graphs (stcon) or for undirected graphs (ustcon) would tell us a great deal about the relationships among deterministic, nondeterministic, and probabilistic logarithmic space bounded complexity classes. A fruitful intermediate step to determining the space complexity of stcon and ustcon is to explore timespace tradeoffs for the problems: the simultaneous time and space requirements of algorithms for connectivity. Prior to this work, all deterministic connectivity algorithms that used less than linear space (the space bound for wellknown algorithms such as ...
Sharp Bounds on DavenportSchinzel Sequences of Every Order ∗
"... One of the oldest unresolved problems in extremal combinatorics is to determine the maximum length of DavenportSchinzel sequences, where an orders DS sequence is defined to be one over an nletter alphabet that avoids alternating subsequences of the form a · · · b · · · a · · · b · · · wit ..."
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One of the oldest unresolved problems in extremal combinatorics is to determine the maximum length of DavenportSchinzel sequences, where an orders DS sequence is defined to be one over an nletter alphabet that avoids alternating subsequences of the form a · · · b · · · a · · · b · · · with length s + 2. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since become an indispensable tool in computational geometry and the analysis of discrete geometric structures. Let λs(n) be the extremal function for such sequences. What is λs asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch) when s is even or s ≤ 3. However, since the work of Agarwal, Sharir, and Shor in the 1980s there has been a persistent gap in our understanding of the odd orders, a gap that is just as much qualitative as quantitative. In this paper we establish the following bounds on λs(n) for every order s. n s = 1 2n − 1 s = 2 ⎪ ⎨ 2nα(n) + O(n) s = 3 λs(n) = Θ(n2 α(n) ) s = 4 Θ(nα(n)2 α(n) ) s = 5 n2 (1+o(1))αt (n)/t! s ≥ 6, t = ⌊ s−2 2 ⌋ These results refute a conjecture of Alon, Kaplan, Nivasch, Sharir, and Smorodinsky and run counter to common sense. When s is odd, λs behaves essentially like λs−1.