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Overhang [article]

Mike Paterson, Uri Zwick
2007 arXiv   pre-print
How far off the edge of the table can we reach by stacking n identical, homogeneous, frictionless blocks of length 1? A classical solution achieves an overhang of 1/2 H_n, where H_n n is the nth harmonic number. This solution is widely believed to be optimal. We show, however, that it is, in fact, exponentially far from optimality by constructing simple n-block stacks that achieve an overhang of c n^1/3, for some constant c>0.

Maximum overhang [article]

Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, Uri Zwick
2007 arXiv   pre-print
Recently, Paterson and Zwick constructed n-block stacks with overhangs of order n^1/3, exponentially better than previously thought possible.  ...  The Paterson-Zwick construction Paterson and Zwick [PZ2006] describe a family of balanced n-block stacks that achieve an overhang of about (3n/16) 1/3 0.57n 1/3 .  ...  In Section 3 we briefly review the Paterson-Zwick construction of stacks that achieve an overhang of order n 1/3 .  ...

Hollow Heaps [article]

Thomas Dueholm Hansen, Haim Kaplan, Robert E. Tarjan, Uri Zwick
2015 arXiv   pre-print
We introduce the hollow heap, a very simple data structure with the same amortized efficiency as the classical Fibonacci heap. All heap operations except delete and delete-min take O(1) time, worst case as well as amortized; delete and delete-min take O( n) amortized time on a heap of n items. Hollow heaps are by far the simplest structure to achieve this. Hollow heaps combine two novel ideas: the use of lazy deletion and re-insertion to do decrease-key operations, and the use of a dag
more » ... e of a dag (directed acyclic graph) instead of a tree or set of trees to represent a heap. Lazy deletion produces hollow nodes (nodes without items), giving the data structure its name.

Fibonacci Heaps Revisited [article]

Haim Kaplan and Robert E. Tarjan and Uri Zwick
2014 arXiv   pre-print
The Fibonacci heap is a classic data structure that supports deletions in logarithmic amortized time and all other heap operations in O(1) amortized time. We explore the design space of this data structure. We propose a version with the following improvements over the original: (i) Each heap is represented by a single heap-ordered tree, instead of a set of trees. (ii) Each decrease-key operation does only one cut and a cascade of rank changes, instead of doing a cascade of cuts. (iii) The
more » ... ts. (iii) The outcomes of all comparisons done by the algorithm are explicitly represented in the data structure, so none are wasted. We also give an example to show that without cascading cuts or rank changes, both the original data structure and the new version fail to have the desired efficiency, solving an open problem of Fredman. Finally, we illustrate the richness of the design space by proposing several alternative ways to do cascading rank changes, including a randomized strategy related to one previously proposed by Karger. We leave the analysis of these alternatives as intriguing open problems.

Overhang

Mike Paterson, Uri Zwick
2009 The American mathematical monthly
How far off the edge of the table can we reach by stacking n identical blocks of length 1? A classical solution achieves an overhang of 1 2 H n , where H n = n i=1 1 i ∼ ln n is the n th harmonic number, by stacking all the blocks one on top of another with the i th block from the top displaced by 1 2i beyond the block below. This solution is widely believed to be optimal. We show that it is exponentially far from optimal by giving explicit constructions with an overhang of Ω(n 1/3 ). We also
more » ... (n 1/3 ). We also prove some upper bounds on the overhang that can be achieved. The stability of a given stack of blocks corresponds to the feasibility of a linear program and so can be efficiently determined.

Overhang

Mike Paterson, Uri Zwick
2009 The American mathematical monthly
How far off the edge of the table can we reach by stacking n identical blocks of length 1? A classical solution achieves an overhang of 1 2 H n , where H n = n i=1 1 i ∼ ln n is the n th harmonic number, by stacking all the blocks one on top of another with the i th block from the top displaced by 1 2i beyond the block below. This solution is widely believed to be optimal. We show that it is exponentially far from optimal by giving explicit constructions with an overhang of Ω(n 1/3 ). We also
more » ... (n 1/3 ). We also prove some upper bounds on the overhang that can be achieved. The stability of a given stack of blocks corresponds to the feasibility of a linear program and so can be efficiently determined.

A forward-backward single-source shortest paths algorithm [article]

David B. Wilson, Uri Zwick
2014 arXiv   pre-print
We describe a new forward-backward variant of Dijkstra's and Spira's Single-Source Shortest Paths (SSSP) algorithms. While essentially all SSSP algorithm only scan edges forward, the new algorithm scans some edges backward. The new algorithm assumes that edges in the outgoing and incoming adjacency lists of the vertices appear in non-decreasing order of weight. (Spira's algorithm makes the same assumption about the outgoing adjacency lists, but does not use incoming adjacency lists.) The
more » ... lists.) The running time of the algorithm on a complete directed graph on n vertices with independent exponential edge weights is O(n), with very high probability. This improves on the previously best result of O(n n), which is best possible if only forward scans are allowed, exhibiting an interesting separation between forward-only and forward-backward SSSP algorithms. As a consequence, we also get a new all-pairs shortest paths algorithm. The expected running time of the algorithm on complete graphs with independent exponential edge weights is O(n^2), matching a recent algorithm of Demetrescu and Italiano as analyzed by Peres et al. Furthermore, the probability that the new algorithm requires more than O(n^2) time is exponentially small, improving on the O(n^-1/26) probability bound obtained by Peres et al.

On Dynamic Shortest Paths Problems

Liam Roditty, Uri Zwick
2010 Algorithmica
We obtain the following results related to dynamic versions of the shortest-paths problem: (i) Reductions that show that the incremental and decremental singlesource shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems. (ii) A randomized fully-dynamic algorithm for the all-pairs shortestpaths problem in directed
more » ... problem in directed unweighted graphs with an amortized update time ofÕ(m √ n) and a worst case query time is O(n 3/4 ). (iii) A deterministic O(n 2 log n) time algorithm for constructing a (log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.

Coloring k-colorable graphs using relatively small palettes [article]

Eran Halperin, Ram Nathaniel, Uri Zwick
2001 arXiv   pre-print
We obtain the following new coloring results: * A 3-colorable graph on n vertices with maximum degree Δ can be colored, in polynomial time, using O((ΔΔ)^1/3·n) colors. This slightly improves an O((Δ^1/3^1/2Δ)·n) bound given by Karger, Motwani and Sudan. More generally, k-colorable graphs with maximum degree Δ can be colored, in polynomial time, using O((Δ^1-2/k^1/kΔ) ·n) colors. * A 4-colorable graph on n vertices can be colored, in polynomial time, using (n^7/19) colors. This improves an
more » ... s improves an (n^2/5) bound given again by Karger, Motwani and Sudan. More generally, k-colorable graphs on n vertices can be colored, in polynomial time, using (n^α_k) colors, where α_5=97/207, α_6=43/79, α_7=1391/2315, α_8=175/271, ... The first result is obtained by a slightly more refined probabilistic analysis of the semidefinite programming based coloring algorithm of Karger, Motwani and Sudan. The second result is obtained by combining the coloring algorithm of Karger, Motwani and Sudan, the combinatorial coloring algorithms of Blum and an extension of a technique of Alon and Kahale (which is based on the Karger, Motwani and Sudan algorithm) for finding relatively large independent sets in graphs that are guaranteed to have very large independent sets. The extension of the Alon and Kahale result may be of independent interest.

The memory game

Uri Zwick, Michael S Paterson
1993 Theoretical Computer Science
Zwick, U. and MS. Paterson, The memory game, Theoretical Computer Science 110 (1993) I69-196.  ...

Color-coding

Noga Alon, Raphael Yuster, Uri Zwick
1995 Journal of the ACM
We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V, E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions. Using the color-coding method we obtain, in particular, the following new results: • For every fixed k, if a graph G = (V, E) contains a simple cycle of size exactly k, then such a cycle can be
more » ... a cycle can be found in either 376 is the exponent of matrix multiplication. (Here and in what follows we use V and E instead of |V | and |E| whenever no confusion may arise.) • For every fixed k, if a planar graph G = (V, E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ) expected time or O(V log V ) worst-case time. The same algorithm applies, in fact, not only to planar graphs, but to any minor closed family of graphs which is not the family of all graphs. • If a graph G = (V, E) contains a subgraph isomorphic to a bounded tree-width graph H = (V H , E H ) where |V H | = O(log V ), then such a copy of H can be found in polynomial time. This was not previously known even if H were just a path of length O(log V ). These results improve upon previous results of many authors. The third result resolves in the affirmative a conjecture of Papadimitriou and Yannakakis that the LOG PATH problem is in P. We can show that it is even in NC.

Fast Sparse Matrix Multiplication [chapter]

Raphael Yuster, Uri Zwick
2004 Lecture Notes in Computer Science
. , 1997 Yuster and Zwick 2004] , for finding small cliques and other small subgraphs [Nešetřil and Poljak 1985] , for finding shortest paths [Seidel 1995; Shoshan and Zwick 1999; Zwick 2002] , for obtaining  ...  improved dynamic reachability algorithms [Demetrescu and Italiano 2000; Roditty and Zwick 2002] , and for matching problems [Mulmuley et al. 1987; Rabin and Vazirani 1989; Cheriyan 1997; Sankowski 2004a  ...

Multicriteria Global Minimum Cuts [chapter]

Amitai Armon, Uri Zwick
2004 Lecture Notes in Computer Science
We consider two multicriteria versions of the global minimum cut problem in undirected graphs. In the k-criteria setting, each edge of the input graph has k non-negative costs associated with it. These costs are measured in separate, non interchangeable, units. In the AND-version of the problem, purchasing an edge requires the payment of all the k costs associated with it. In the OR-version, an edge can be purchased by paying any one of the k-costs associated with it. Given k bounds b1, b2, . .
more » ... bounds b1, b2, . . . , b k , the basic multicriteria decision problem is whether there exists a cut C of the graph that can be purchased using a budget of bi units of the i-th criterion, for 1 ≤ i ≤ k. We show that the AND-version of the multicriteria global minimum cut problem is polynomial for any fixed number k of criteria. The ORversion of the problem, on the other hand, is NP-hard even for k = 2, but can be solved in pseudo-polynomial time for any fixed number k of criteria. It also admits an FPTAS. Further extensions, some applications, and multicriteria versions of two other optimization problems are also discussed.

Approximate distance oracles

Mikkel Thorup, Uri Zwick
2005 Journal of the ACM
Cohen and Zwick [CZ01] have shown that a stretch 3 oracle that uses O(n 2 ) space can be constructed in O(n 2 log n) time.  ...  For stretches 2, 7/3, and 3, Cohen and Zwick [CZ01] have shown that APSP can be solved in timeÕ(m 1/2 n 3/2 ),Õ(n 7/3 ), and O(n 2 ), respectively.  ...

All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication [article]

Uri Zwick
2000 arXiv   pre-print
(See also Shoshan and Zwick [SZ99] .)  ...  The algorithms of Cohen and Zwick [CZ97] use ideas obtained by Aingworth, Chekuri, Indyk and Motwani [ACIM99] and by Dor, Halperin and Zwick [DHZ00] who designed algorithms that approximate distances  ...
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