Results 1  10
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12
Verification and Sensitivity Analysis Of Minimum Spanning Trees In Linear Time
 SIAM J. Comput
, 1992
"... . Koml'os has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. We describe a lineartime algorithm for verif ..."
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Cited by 54 (2 self)
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. Koml'os has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. We describe a lineartime algorithm for verifying a minimum spanning tree. Our algorithm combines the result of Koml'os with a preprocessing and table lookup method for small subproblems and with a previously known almostlineartime algorithm. Additionally, we present an optimal deterministic algorithm and a lineartime randomized algorithm for sensitivity analysis of minimum spanning trees. 1. Introduction. Suppose we wish to solve some problem for which we know in advance the size of the input data, using an algorithm from some welldefined class of algorithms. For example, consider sorting n numbers, when n is fixed in advance, using a binary comparison tree. Given a sufficient amount of preprocessing time and storage space, we ca...
Matching Nuts and Bolts
, 1994
"... ) Noga Alon Manuel Blum y Amos Fiat z Sampath Kannan x Moni Naor  Rafail Ostrovsky k Abstract We describe a procedure which may be helpful to any disorganized carpenter who has a mixed pile of bolts and nuts and wants to find the corresponding pairs of bolts and nuts. The procedure use ..."
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Cited by 8 (2 self)
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) Noga Alon Manuel Blum y Amos Fiat z Sampath Kannan x Moni Naor  Rafail Ostrovsky k Abstract We describe a procedure which may be helpful to any disorganized carpenter who has a mixed pile of bolts and nuts and wants to find the corresponding pairs of bolts and nuts. The procedure uses our (and the carpenter's) ability to construct efficiently highly expanding graphs. The problem considered is given a collection of n bolts of distinct widths and n nuts such that there is a 11 correspondence between the nuts and bolts. The goal is to find for each bolt its corresponding nut by comparing nuts to bolts but not nuts to nuts or bolts to bolts. Our objective is to minimize the number of operations of this kind (as well as the total running time). The problem has a randomized algorithm similar to Quicksort. Our main result is an n(log n) O(1)  Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel and AT & T...
Bounds on the Chromatic Polynomial and on the Number of Acyclic Orientations of a Graph
 Combinatorica
, 1996
"... An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided. A lower bound on the numbe ..."
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Cited by 7 (0 self)
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An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided. A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This argument can take advantage of structural properties of the graph: it is shown how to obtain stronger bounds for smalldegree graphs of girth at least five, than are possible for arbitrary graphs. A simpler proof of the known lower bound for arbitrary graphs is also obtained. Both the upper and lower bounds are shown to extend to the general problem of bounding the chromatic polynomial from above and below along the negative real axis. 1 XEROX Palo Alto Research Center, 3333 Coyote Hill Road, CA 94304. Partially supported by the NSF under grant CCR9404113. Most of this research was done while th...
Minimizing Randomness in Minimum Spanning Tree, Parallel Connectivity, and Set Maxima Algorithms
 In Proc. 13th Annual ACMSIAM Symposium on Discrete Algorithms (SODA'02
, 2001
"... There are several fundamental problems whose deterministic complexity remains unresolved, but for which there exist randomized algorithms whose complexity is equal to known lower bounds. Among such problems are the minimum spanning tree problem, the set maxima problem, the problem of computing conne ..."
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Cited by 7 (4 self)
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There are several fundamental problems whose deterministic complexity remains unresolved, but for which there exist randomized algorithms whose complexity is equal to known lower bounds. Among such problems are the minimum spanning tree problem, the set maxima problem, the problem of computing connected components and (minimum) spanning trees in parallel, and the problem of performing sensitivity analysis on shortest path trees and minimum spanning trees. However, while each of these problems has a randomized algorithm whose performance meets a known lower bound, all of these randomized algorithms use a number of random bits which is linear in the number of operations they perform. We address the issue of reducing the number of random bits used in these randomized algorithms. For each of the problems listed above, we present randomized algorithms that have optimal performance but use only a polylogarithmic number of random bits; for some of the problems our optimal algorithms use only log n random bits. Our results represent an exponential savings in the amount of randomness used to achieve the same optimal performance as in the earlier algorithms. Our techniques are general and could likely be applied to other problems.
Matching Nuts and Bolts Faster
 In ISAAC
, 1995
"... . The problem of matching nuts and bolts is the following : Given a collection of n nuts of distinct sizes and n bolts such that there is a oneto one correspondence between the nuts and the bolts, find for each nut its corresponding bolt. We can only compare nuts to bolts. That is we can neither co ..."
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Cited by 5 (0 self)
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. The problem of matching nuts and bolts is the following : Given a collection of n nuts of distinct sizes and n bolts such that there is a oneto one correspondence between the nuts and the bolts, find for each nut its corresponding bolt. We can only compare nuts to bolts. That is we can neither compare nuts to nuts, nor bolts to bolts. This humble restriction on the comparisons appears to make this problem very hard to solve. In fact, the best deterministic solution to date is due to Alon et al . [1] and takes \Theta(n log 4 n) time. Their solution uses (efficient) graph expanders. In this paper, we give a simpler O(n log 2 n) time algorithm which uses only a simple (and not so efficient) expander. 1 Introduction In [7], page 293, Rawlins posed the following interesting problem : We wish to sort a bag of n nuts and n bolts by size in the dark. We can compare the sizes of a nut and a bolt by attempting to screw one into the other. This operation tells us that either the nut is b...
An InverseAckermann Style Lower Bound for Online Minimum Spanning Tree Verification
 Combinatorica
"... 1 Introduction The minimum spanning tree (MST) problem has seen a flurry of activity lately, driven largely by the success of a new approach to the problem. The recent MST algorithms [20, 8, 29, 28], despite their superficial differences, are all based on the idea of progressively improving an appro ..."
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Cited by 3 (2 self)
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1 Introduction The minimum spanning tree (MST) problem has seen a flurry of activity lately, driven largely by the success of a new approach to the problem. The recent MST algorithms [20, 8, 29, 28], despite their superficial differences, are all based on the idea of progressively improving an approximately minimum solution, until the actual minimum spanning tree is found. It is still likely that this progressive improvement approach will bear fruit. However, the current
Randomized Minimum Spanning Tree Algorithms Using Exponentially Fewer Random Bits
"... For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel mi ..."
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Cited by 2 (0 self)
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For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel minimum spanning tree problems, and the local sorting and set maxima problems. (For the first two problems there are provably optimal deterministic algorithms with unknown, and possibly superlinear running times.) One downside of the randomized methods for solving these problems is that they use a number of random bits linear in the size of the input. In this paper we develop some general methods for reducing exponentially the consumption of random bits in comparison based algorithms. In some cases we are able to reduce the number of random bits from linear to nearly constant without affecting the expected running time. Most of our results are obtained by adjusting or reorganizing existing randomized algorithms to work well with a pairwise or O(1)wise independent sampler. The prominent exception — and the main focus of this paper — is a lineartime randomized minimum spanning tree algorithm that is not derived from the well known KargerKleinTarjan algorithm. In many ways it resembles more closely the deterministic minimum spanning tree algorithms based on Soft Heaps. Further,
The SetMaxima Problem: an Overview
, 1998
"... Sorting problems have long been one of the foundations of theoretical computer science. Sorting problems attempt to learn properties of an unknown total order of a known set. We test the order by comparing pairs of elements, and through repeated tests deduce some order structure on the set. The set ..."
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Cited by 1 (0 self)
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Sorting problems have long been one of the foundations of theoretical computer science. Sorting problems attempt to learn properties of an unknown total order of a known set. We test the order by comparing pairs of elements, and through repeated tests deduce some order structure on the set. The setmaxima problem is: given a family S of subsets of a set X, produce the maximal element of each element of S. Local sorting is a subproblem of setmaxima, when S ` i X 2 j , i.e. there exists a graph G with vertex set X and edge set S: We compare algorithms by estimating the number of comparisons needed, as a function of n = jXj; and m = jSj. In this paper, we review the informationtheory lower bounds for the setmaxima and localsorting problems. We review deterministic algorithms which have optimally solved the setmaxima problem, as a function of m;n, in settings where extra assumptions about S have been made. Also, we review randomized algorithms for local sorting and setmaxima ...
Seminar on Advanvced Topics in Data Structures
, 2001
"... We shall focus on the following topics in data structures ..."