The Moore Bound for Irregular Graphs

 What is the largest number of edges in a graph of order n and girth g? For d-regular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an affirmative answer to an old open problem ([4] p. 163, problem 10). Received: June 27, 2000 Final version received: July 3, 2001     

Metric Embeddings—Beyond One-Dimensional Distortion

The extensive study of metric spaces and their embeddings has so far focused on embeddings that preserve pairwise distances. A very intriguing concept introduced by Feige allows us to quantify the extent to which larger structures are preserved by a given embedding. We investigate this concept, focusing on several major graph families such as paths, trees, cubes, and expanders. We find some similarities to the regular (pairwise) distortion, as well as some striking differences.     

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Inclusion-exclusion: Exact and approximate

It is often required to find the probability of the union of givenn eventsA ₁ ,...,A _n . The answer is provided, of course, by the inclusion-exclusion formula: Pr(A _i )= _i – _i<j Pr(A _i A _j )±.... Unfortunately, this formula has exponentially many terms, and only rarely does one manage to carry out the exact calculation. From a computational point of view, finding the probability of the union is an intractable, #P-hard problem, even in very restricted cases. This state of affairs makes it reasonable to seek approximate solutions that are computationally feasible. Attempts to find such approximate solutions have a long history starting already with Boole [1]. A recent step in this direction was taken by Linial and Nisan [4] who sought approximations to the probability of the union, given the probabilities of allj-wise intersections of the events forj=1,...k. The developed a method to approximate Pr(A _i ), from the above data with an additive error of exp . In the present article we develop an expression that can be computed in polynomial time, that, given the sums _|S|=j Pr( _iS A _i ) forj=1,...k, approximates Pr(A _i ) with an additive error of exp . This error is optimal, up to the logarithmic factor implicit in the notation.The problem of enumerating satisfying assignments of a boolean formula in DNF formF=v _l ^m C _i is an instance of the general problem that had been extensively studied [7]. HereA _i is the set of assignments that satisfyC _i , and Pr( _iS A _i )=a _S /2ⁿ where _iS C _i is satisfied bya _S assignments. Judging from the general results, it is hard to expect a decent approximation ofF's number of satisfying assignments, without knowledge of the numbersa _S for, say, all cardinalities . Quite surprisingly, already the numbersa _S over |S|log(n+1)uniquely determine the number of satisfying assignments for F.We point out a connection between our work and the edge-reconstruction conjecture. Finally we discuss other special instances of the problem, e.g., computing permanents of 0,1 matrices, evaluating chromatic polynomials of graphs and for families of events whose VC dimension is bounded.Work supported in part by a grant of the Binational Israel-US Science Foundation.Work supported in part by a grant of the Binational Israel-US Science Foundation and by the Israel Science Foundation.     

Legal coloring of graphs

The following computational problem was initiated by Manber and Tompa (22nd FOCS Conference, 1981): Given a graphG=(V, E) and a real functionf:V→R which is a proposed vertex coloring. Decide whetherf is a proper vertex coloring ofG. The elementary steps are taken to be linear comparisons. Lower bounds on the complexity of this problem are derived using the chromatic polynomial ofG. It is shown how geometric parameters of a space partition associated withG influence the complexity of this problem. Existing methods for analyzing such space partitions are suggested as a powerful tool for establishing lower bounds for a variety of computational problems.     

The Linear-Array Conjecture in Communication Complexity Is False

linear array network consists of k+1 processors with links only between and (0≤i<k). It is required to compute some boolean function f(x,y) in this network, where initially x is stored at and y is stored at . Let be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, , where D(f) is the standard two-party communication complexity of f. Tiwari proved that for almost all functions and conjectured that this is true for all functions. In this paper we disprove Tiwari's conjecture, by exhibiting an infinite family of functions for which is essentially at most . Our construction also leads to progress on another major problem in this area: It is easy to bound the two-party communication complexity of any function, given the least number of monochromatic rectangles in any partition of the input space. How tight are such bounds? We exhibit certain functions, for which the (two-party) communication complexity is twice as large as the best lower bound obtainable this way. Received: March 1, 1996     

Non-Expansive Hashing

non-expansive hashing scheme, similar inputs are stored in memory locations which are close. We develop a non-expansive hashing scheme wherein any set of size from a large universe may be stored in a memory of size (any , and ), and where retrieval takes operations. We explain how to use non-expansive hashing schemes for efficient storage and retrieval of noisy data. A dynamic version of this hashing scheme is presented as well. Received: February 5, 1996     

Enumeration and randomized constructions of hypertrees

Over 30 years ago, Kalai proved a beautiful d‐dimensional analog of Cayley's formula for the number of n‐vertex trees. He enumerated d‐dimensional hypertrees weighted by the squared size of their (d ? 1)‐dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of d‐hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of d‐hypertrees. In addition, we study a random 1‐out model of d‐complexes where every (d ? 1)‐dimensional face selects a random d‐face containing it, and show that it has a negligible d‐dimensional homology.     

Shades of green in 15th century paintings: combined microanalysis of the materials using synchrotron radiation XRD, FTIR and XRF

A representative selection of green paintings from fifteenth century Catalonia and the Crown of Aragon are analyzed by a combination of synchrotron radiation microanalytical techniques including FTIR, XRD, and XRF. The green pigments themselves are found to be a mixture of copper acetates/basic copper acetates and basic copper chlorides. Nevertheless, a broader range of green shades were obtained by mixing the green pigment with yellow, white, and blue pigments and applied forming a sequence of micrometric layers. Besides the nature of the pigments themselves, degradation and reaction products, such as carboxylates, formates and oxalates were also identified. Some of the copper based compounds, such as the basic copper chloride, may be either part of the original pigment or a weathering product. The high resolution, high brilliance, and small footprint of synchrotron radiation proved to be essential for the analysis of those submillimetric paint layers made of a large variety of compounds heterogeneous in nature and distribution and present in extremely low concentrations.