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54
Space Efficient Hash Tables With Worst Case Constant Access Time
 In STACS
, 2003
"... We generalize Cuckoo Hashing [23] to dary Cuckoo Hashing and show how this yields a simple hash table data structure that stores n elements in (1 + ffl) n memory cells, for any constant ffl ? 0. Assuming uniform hashing, accessing or deleting table entries takes at most d = O(ln ffl ) probes ..."
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Cited by 47 (4 self)
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We generalize Cuckoo Hashing [23] to dary Cuckoo Hashing and show how this yields a simple hash table data structure that stores n elements in (1 + ffl) n memory cells, for any constant ffl ? 0. Assuming uniform hashing, accessing or deleting table entries takes at most d = O(ln ffl ) probes and the expected amortized insertion time is constant. This is the first dictionary that has worst case constant access time and expected constant update time, works with (1 + ffl) n space, and supports satellite information. Experiments indicate that d = 4 choices suffice for ffl 0:03. We also describe variants of the data structure that allow the use of hash functions that can be evaluted in constant time.
Lower Bounds for Local Search by Quantum Arguments
"... The problem of finding a local minimum of a blackbox function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1} n (, we show a lower bound of Ω 2 n/4) /n on the number of queries needed by a quantum computer to solve this ..."
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Cited by 33 (2 self)
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The problem of finding a local minimum of a blackbox function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1} n (, we show a lower bound of Ω 2 n/4) /n on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis’s quantum ( adversary method, also yields a lower bound of Ω 2 n/2 /n 2 on the problem’s classical randomized query complexity. This improves and simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, we give the first nontrivial lower bounds for finding local minima on grids of constant dimension d ≥ 3. 1.
On Infinite Cycles I
"... We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, a ..."
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Cited by 29 (11 self)
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We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, and that the spanning trees whose fundamental cycles generate this cycle space are precisely the endfaithful spanning trees. We also generalize Euler's theorem by showing that a locally finite connected graph with ends contains a closed topological curve traversing every edge exactly once if and only if its entire edge set lies in this cycle space.
Testing versus estimation of graph properties
 Proc. of STOC 2005
, 2005
"... In the course of the proof we develop a framework for extending Szemer'edi's Regularity Lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. In particular ..."
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Cited by 29 (7 self)
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In the course of the proof we develop a framework for extending Szemer'edi's Regularity Lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. In particular, we construct a probabilistic algorithm that finds the parameters of a regular partition of an input graph using a constant number of queries, and an algorithm to find a regular partition of a graph using a TC0 circuit. This, in some ways, strengthens the results of [1].
The Cycle Space of an Infinite Graph
 COMB., PROBAB. COMPUT
, 2004
"... Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph togethe ..."
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Cited by 26 (9 self)
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Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach
Topological Paths, Cycles and Spanning Trees in Infinite Graphs
"... We study topological versions of paths, cycles and spanning trees in infinite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consisting of the graph a ..."
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Cited by 26 (13 self)
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We study topological versions of paths, cycles and spanning trees in infinite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consisting of the graph and all its ends, but for one where only its topological ends are added as new points, while rays from other ends are made to converge to certain vertices.
On Network Design Problems: Fixed Cost Flows and the Covering Steiner Problem
, 2001
"... Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edgecostow problems. An edgecost ow problem is a mincost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow. ..."
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Cited by 23 (3 self)
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Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edgecostow problems. An edgecost ow problem is a mincost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow.
Approximation algorithms for constrained node weighted Steiner tree problems
 In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing
, 2001
"... We consider a class of optimization problems, where the input is an undirected graph with two weight functions defined for each node, namely the node’s profit and its cost. The goal is to find a connected set of nodes of low cost and high profit. We present approximation algorithms for three natural ..."
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Cited by 17 (0 self)
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We consider a class of optimization problems, where the input is an undirected graph with two weight functions defined for each node, namely the node’s profit and its cost. The goal is to find a connected set of nodes of low cost and high profit. We present approximation algorithms for three natural optimization criteria that arise in this context, all of which are NPhard. The budget problem asks for maximizing the profit of the set subject to a budget constraint on its cost. The quota problem requires minimizing the cost of the set subject to a quota constraint on its profit. Finally, the prize collecting problem calls for minimizing the cost of the set plus the profit (here interpreted as a penalty) of the complement set. For all three problems, our algorithms give an approximation guarantee of, where is the number of nodes. To the best of our knowledge, these are the first approximation results for the quota problem and for the prize collecting problem, both of which are at least as hard to approximate as set cover. For the budget problem, our results improve on a previous result of Guha, Moss, Naor, and Schieber. Our methods involve new theorems relating tree packings to (node) cut conditions. We also show similar theorems (with better bounds) using edge cut conditions. These imply bounds for the analogous budget and quota problems with edge costs which are comparable to known (constant factor) bounds. 1
MacLane's Planarity Criterion for Locally Finite Graphs
, 2003
"... MacLane's planarity criterion states that a finite graph is planar if and only if its cycle space has a basis B such that every edge is contained in at most two members of B. Solving a problem of Wagner (1970), we show that the cycle space introduced recently by Diestel and Kühn allows a verbatim ge ..."
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Cited by 16 (2 self)
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MacLane's planarity criterion states that a finite graph is planar if and only if its cycle space has a basis B such that every edge is contained in at most two members of B. Solving a problem of Wagner (1970), we show that the cycle space introduced recently by Diestel and Kühn allows a verbatim generalization of MacLane's criterion to locally finite graphs.
On Infinite Cycles II
"... We adapt the cycle space of a finite or locally graph to graphs with vertices of infinite degree, using as cycles the homeomorphic images of the unit circle S in the graph together with its ends. We characterize the spanning trees whose fundamental cycles generate this cycle space, and prove ..."
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Cited by 14 (0 self)
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We adapt the cycle space of a finite or locally graph to graphs with vertices of infinite degree, using as cycles the homeomorphic images of the unit circle S in the graph together with its ends. We characterize the spanning trees whose fundamental cycles generate this cycle space, and prove infinite analogues to the standard characterizations of finite cycle spaces in terms of edgedecomposition into single cycles and orthogonality to cuts.