Showing papers by "Van Vu published in 2003"

Spectra of random graphs with given expected degrees

Abstract: In the study of the spectra of power-law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semicircle law, whereas the other predicts that the eigenvalues follow a power-law distribution. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power-law graph follow the semicircle law, whereas the spectrum of the adjacency matrix of a power-law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent β, where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as, for example, randomized algorithms that involve rapidly mixing Markov chains.

Eigenvalues of Random Power law Graphs

Abstract: Many graphs arising in various information networks exhibit the "power law" behavior — the number of vertices of degree k is proportional to k −β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1+ o(1)) √ m where m is the maximum degree. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power law distribution with exponent 2β − 1 if the maximum degree is sufficiently large, where k is a function depending on β, m and d, the average degree. When 2< β < 2.5, the largest eigenvalue is heavily concentrated at cm 3−β for some constant c depending on β and the average degree. This result follows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and ˜ d, the weighted average of the squares of the expected degrees. We show that the k-th largest eigenvalue is almost surely (1+ o(1)) √ m k where mk is the k-th largest expected degree provided mk is large enough. These results have implications on the usage of spectral techniques in many areas related to pattern detection and information retrieval.

Generating random regular graphs

Abstract: Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [9] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d=O(n1/3-e), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd2). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author.Besides being perhaps the only algorithm which works for relatively large d in practical time, our result also has a significant theoretical value, as it can be used to derive many properties of uniform random regular graphs.

Turán's theorem in sparse random graphs

Abstract: We prove the analogue of Turan's Theorem in random graphs with edge probability p(n) > n-1/(k-1.5). With probability 1 - o(1), one needs to delete approximately 1/k-1-fraction of the edges in a random graph in order to destroy all cliques of size k.

Covering codes with improved density

Abstract: We prove a general recursive inequality concerning /spl mu//sup */(R), the asymptotic (least) density of the best binary covering codes of radius R. In particular, this inequality implies that /spl mu//sup */(R)/spl les/e/spl middot/(RlogR+logR+loglogR+2), which significantly improves the best known density 2/sup R/R/sup R/(R+1)/R!. Our inequality also holds for covering codes over arbitrary alphabets.

Multirate rearrangeable clos networks and a generalized edge coloring problem on bipartite graphs

Abstract: Chung and Ross (SIAM J. Comput., 20, 1991) conjectured that the minimum number m(n, r) of middle-state switches for the symmetric 3-stage Clos network C(n, m(n, r), r) to be rearrangeable in the multirate enviroment is at most 2n -- 1. This problem is equivalent to a generalized version of the biparite graph edge coloring problem. The best bounds known so far on the function m(n, r) is 11n/9 ≤ m(n, r) ≤ 41n/16 + O(1), for n, r ≥ 2, derived by Du-Gao-Hwang-Kim (SIAM J. Comput., 28, 1999). In this paper, we make several contributions. Firstly, we give evidence to show that even a stronger result might hold. In particular, we give a coloring algorithm to show that m(n, r) ≤ [(r + 1)n/2], which implies m(n, 2) ≤ [3n/2] - stronger than the conjectured value of 2n -- 1. Secondly, we derive that m(2, r) = 3 by an elegant argument. Lastly, we improve both the best upper and lower bounds given above: [5n/4] ≤ m(n, r) ≤ 2n -- 1 + [(r -- 1)/2], where the upper bound is an improvement over 41n/16 when r is relatively small compared to n. We also conjecture that m(n, r) ≤ [2n (1 -- 1/2(r)].

Multirate rearrangeable Clos networks and a generalized edge-coloring problem on bipartite graphs

Abstract: Chung and Ross [SIAM J. Comput., 20 (1991), pp. 726--736] conjectured that the minimum number m(n,r) of middle-stage switches for the symmetric 3-stage Clos network C(n,m(n,r),r) to be rearrangeable in the multirate environment is at most 2n-1. This problem is equivalent to a generalized version of the bipartite graph edge-coloring problem. The best bounds known so far on this function m(n,r) are $11n/9 \leq m(n,r) \leq 41n/16 + O(1)$, for $n, r \geq 2$, derived by Du et al. [SIAM J. Comput., 28 (1999), pp. 464--471]. In this paper, we make several contributions. First, we give evidence to show that even a stronger result might hold. In particular, we give a coloring algorithm to show that $m(n,r) \leq \lceil (r+1)n/2 \rceil$, which implies $m(n,2) \leq \lceil 3n/2 \rceil$---stronger than the conjectured value of 2n-1. Second, we derive that m(2,r) = 3 by an elegant argument. Last, we improve both the best upper and lower bounds given above: $\lceil 5n/4 \rceil \leq m(n,r) \leq 2n-1+\lceil (r-1)/2 \rceil$...

Distinct distances in homogeneous sets

Abstract: We show that the number of distinct distances in a well-distributed set of n points in Rd is O (n2/d-1/d2) which is not far from the best known upper bound O(n2/d).