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123
Statistical Region Merging
 IEEE Trans. on Pattern Analysis and Machine Intelligence
, 2004
"... This paper explores a statistical basis for a process often described in computer vision: image segmentation by region merging following a particular order in the choice of regions. We exhibit a particular blend of algorithmics and statistics whose segmentation error is, as we show, limited from b ..."
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Cited by 73 (8 self)
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This paper explores a statistical basis for a process often described in computer vision: image segmentation by region merging following a particular order in the choice of regions. We exhibit a particular blend of algorithmics and statistics whose segmentation error is, as we show, limited from both the qualitative and quantitative standpoints. This approach can be efficiently approximated in linear time/space, leading to a fast segmentation algorithm tailored to processing images described using most common numerical pixel attribute spaces. The conceptual simplicity of the approach makes it simple to modify and cope with hard noise corruption, handle occlusion, authorize the control of the segmentation scale, and process unconventional data such as spherical images. Experiments on graylevel and color images, obtained with a short readily available Ccode, display the quality of the segmentations obtained.
ON THE COVERINGS OF GRAPHS
, 1980
"... Let p(n) denote the smallest integer with the property that any graph with n vertices can be covered by p(n) complete bipartite subgraphs. We prove a conjecture of J.C. Bermond by showing p(n) = n + o(n 11’14+c) for any positive E. ..."
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Cited by 69 (6 self)
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Let p(n) denote the smallest integer with the property that any graph with n vertices can be covered by p(n) complete bipartite subgraphs. We prove a conjecture of J.C. Bermond by showing p(n) = n + o(n 11’14+c) for any positive E.
BALANCED ALLOCATIONS: THE HEAVILY LOADED CASE
, 2006
"... We investigate ballsintobins processes allocating m balls into n bins based on the multiplechoice paradigm. In the classical singlechoice variant each ball is placed into a bin selected uniformly at random. In a multiplechoice process each ball can be placed into one out of d ≥ 2 randomly selec ..."
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Cited by 58 (7 self)
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We investigate ballsintobins processes allocating m balls into n bins based on the multiplechoice paradigm. In the classical singlechoice variant each ball is placed into a bin selected uniformly at random. In a multiplechoice process each ball can be placed into one out of d ≥ 2 randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when m ≈ n. In this paper we present the first tight analysis in the heavily loaded case, that is, when m ≫ n rather than m ≈ n. The best previously known results for the multiplechoice processes in the heavily loaded case were obtained using majorization by the singlechoice process. This yields an upper bound of the maximum load of bins of m/n + O ( √ m ln n/n) with high probability. We show, however, that the multiplechoice processes are fundamentally different from the singlechoice variant in that they have “short memory. ” The great consequence of this property is that the deviation of the multiplechoice processes from the optimal allocation (that is, the allocation in which each bin has either ⌊m/n ⌋ or ⌈m/n ⌉ balls) does not increase with the number of balls as in the case of the singlechoice process. In particular, we investigate the allocation obtained by two different multiplechoice allocation schemes,
On the bias of traceroute sampling: or, powerlaw degree distributions in regular graphs
 In ACM STOC
, 2005
"... Understanding the graph structure of the Internet is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining this graph structure can be a surprisingly difficult task, as edges cannot be explicitly queried. For instance, empiri ..."
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Cited by 55 (1 self)
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Understanding the graph structure of the Internet is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining this graph structure can be a surprisingly difficult task, as edges cannot be explicitly queried. For instance, empirical studies of the network of Internet Protocol (IP) addresses typically rely on indirect methods like traceroute to build what are approximately singlesource, alldestinations, shortestpath trees. These trees only sample a fraction of the network’s edges, and a recent paper by Lakhina et al. found empirically that the resulting sample is intrinsically biased. Further, in simulations, they observed that the degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is Poisson. In this paper, we study the bias of traceroute sampling mathematically and, for a very general class of underlying degree distributions, explicitly calculate the distribution that will be observed. As example applications of our machinery, we prove that traceroute sampling finds powerlaw degree distributions in both δregular and Poissondistributed random graphs. Thus, our work puts the observations of Lakhina et al. on a rigorous footing, and extends them to nearly arbitrary degree distributions.
Fast Concurrent Access to Parallel Disks
 In 11th ACMSIAM Symposium on Discrete Algorithms
, 1999
"... High performance applications involving large data sets require the efficient and flexible use of multiple disks. In an external memory machine with D parallel, independent disks, only one block can be accessed on each disk in one I/O step. This restriction leads to a load balancing problem that is ..."
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Cited by 52 (11 self)
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High performance applications involving large data sets require the efficient and flexible use of multiple disks. In an external memory machine with D parallel, independent disks, only one block can be accessed on each disk in one I/O step. This restriction leads to a load balancing problem that is perhaps the main inhibitor for adapting singledisk external memory algorithms to multiple disks. This paper shows that this problem can be solved efficiently using a combination of randomized placement, redundancy and an optimal scheduling algorithm. A buffer of O(D) blocks suffices to support efficient writing of arbitrary blocks if blocks are distributed uniformly at random to the disks (e.g., by hashing). If two randomly allocated copies of each block exist, N arbitrary blocks can be read within dN=De + 1 I/O steps with high probability. In addition, the redundancy can be reduced from 2 to 1 + 1=r for any integer r. These results can be used to emulate the simple and powerful "singledisk multihead" model of external computing [1] on the physically more realistic independent disk model [33] with small constant overhead. This is faster than a lower bound for deterministic emulation [3].
Eigenvalues of random power law graphs
 Annals of Combinatorics
, 2003
"... 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 de ..."
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Cited by 44 (7 self)
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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 kth largest eigenvalue is almost surely (1 + o(1)) √ m k where mk is the kth 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. 1
Concentration inequalities and martingale inequalities – a survey
 Internet Math
"... Abstract. We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and web graphs. 1. ..."
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Cited by 43 (1 self)
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Abstract. We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and web graphs. 1.
D.X.: Shannon sampling and function reconstruction from point values
 Bull. Am. Math. Soc
, 2004
"... then came to the University of Chicago, where I was starting my job as instructor for the fall of 1956. He, Suzanne, Clara and I became good friends and saw much of each other for many decades, especially at IHES in Paris. Thom’s encouragement and support were important for me, especially in my firs ..."
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Cited by 32 (8 self)
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then came to the University of Chicago, where I was starting my job as instructor for the fall of 1956. He, Suzanne, Clara and I became good friends and saw much of each other for many decades, especially at IHES in Paris. Thom’s encouragement and support were important for me, especially in my first years after my Ph.D. I studied his work in cobordism, singularities of maps, and transversality, gaining many insights. I also enjoyed listening to his provocations, for example his disparaging remarks on complex analysis, 19th century mathematics, and Bourbaki. There was also a stormy side in our relationship. Neither of us could hide the pain that our public conflicts over “catastrophe theory ” caused. René Thom was a great mathematician, leaving his impact on a wide part of mathematics. I will always treasure my memories of him.
A Geometric Preferential Attachment Model of Networks
 In Algorithms and Models for the WebGraph: Third International Workshop, WAW 2004
, 2004
"... We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with powerlaw degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generat ..."
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Cited by 32 (2 self)
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We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with powerlaw degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2,..., xn chosen uniformly at random from the unit sphere in R 3. After generating xt, we randomly connect it to m points from those points in x1, x2,..., xt−1. 1
A Probabilistic and RIPless Theory of Compressed Sensing
, 2010
"... This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models — e.g. Gaussian, frequency measurements — discussed in the literature, ..."
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Cited by 32 (1 self)
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This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models — e.g. Gaussian, frequency measurements — discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) — they make use of a much weaker notion — or a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.