16 research outputs found
Approximation and Streaming Algorithms for Projective Clustering via Random Projections
Let be a set of points in . In the projective
clustering problem, given and norm , we have to
compute a set of -dimensional flats such that is minimized; here
represents the (Euclidean) distance of to the closest flat in
. We let denote the minimal value and interpret
to be . When and
and , the problem corresponds to the -median, -mean and the
-center clustering problems respectively.
For every , and , we show that the
orthogonal projection of onto a randomly chosen flat of dimension
will -approximate
. This result combines the concepts of geometric coresets and
subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence,
an orthogonal projection of to an dimensional randomly chosen subspace
-approximates projective clusterings for every and
simultaneously. Note that the dimension of this subspace is independent of the
number of clusters~.
Using this dimension reduction result, we obtain new approximation and
streaming algorithms for projective clustering problems. For example, given a
stream of points, we show how to compute an -approximate
projective clustering for every and simultaneously using only
space. Compared to
standard streaming algorithms with space requirement, our approach
is a significant improvement when the number of input points and their
dimensions are of the same order of magnitude.Comment: Canadian Conference on Computational Geometry (CCCG 2015
A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching
We study the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given point sets S and R which correspond to the server and request locations; here |S|=|R|=n. All these locations are points from some metric space and the cost of matching a server to a request is given by the distance between their locations in this space. In this problem, the request points arrive one at a time. When a request arrives, we must immediately and irrevocably match it to a "free" server. The matching obtained after all the requests are processed is the online matching M. The cost of M is the sum of the cost of its edges. The performance of any online algorithm is the worst-case ratio of the cost of its online solution M to the minimum-cost matching.
We present a deterministic online algorithm for this problem. Our algorithm is the first to simultaneously achieve optimal performances in the well-known adversarial and the random arrival models. For the adversarial model, we obtain a competitive ratio of 2n-1 + o(1); it is known that no deterministic algorithm can do better than 2n-1. In the random arrival model, our algorithm obtains a competitive ratio of 2H_n - 1 + o(1); where H_n is the n-th Harmonic number. We also prove that any online algorithm will have a competitive ratio of at least 2H_n - 1-o(1) in this model.
We use a new variation of the offline primal-dual method for computing minimum cost matching to compute the online matching. Our primal-dual method is based on a relaxed linear-program. Under metric costs, this specific relaxation helps us relate the cost of the online matching with the offline matching leading to its robust properties
Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line
In the online metric bipartite matching problem, we are given a set S of server locations in a metric space. Requests arrive one at a time, and on its arrival, we need to immediately and irrevocably match it to a server at a cost which is equal to the distance between these locations. A alpha-competitive algorithm will assign requests to servers so that the total cost is at most alpha times the cost of M_{Opt} where M_{Opt} is the minimum cost matching between S and R.
We consider this problem in the adversarial model for the case where S and R are points on a line and |S|=|R|=n. We improve the analysis of the deterministic Robust Matching Algorithm (RM-Algorithm, Nayyar and Raghvendra FOCS\u2717) from O(log^2 n) to an optimal Theta(log n). Previously, only a randomized algorithm under a weaker oblivious adversary achieved a competitive ratio of O(log n) (Gupta and Lewi, ICALP\u2712). The well-known Work Function Algorithm (WFA) has a competitive ratio of O(n) and Omega(log n) for this problem. Therefore, WFA cannot achieve an asymptotically better competitive ratio than the RM-Algorithm
Polynomial-Sized Topological Approximations Using The Permutahedron
Classical methods to model topological properties of point clouds, such as
the Vietoris-Rips complex, suffer from the combinatorial explosion of complex
sizes. We propose a novel technique to approximate a multi-scale filtration of
the Rips complex with improved bounds for size: precisely, for points in
, we obtain a -approximation with at most simplices of dimension or lower. In conjunction with dimension
reduction techniques, our approach yields a -approximation of size for Rips filtrations on arbitrary metric
spaces. This result stems from high-dimensional lattice geometry and exploits
properties of the permutahedral lattice, a well-studied structure in discrete
geometry.
Building on the same geometric concept, we also present a lower bound result
on the size of an approximate filtration: we construct a point set for which
every -approximation of the \v{C}ech filtration has to contain
features, provided that for .Comment: 24 pages, 1 figur
A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings
We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time.
When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G\u27(V\u27,E\u27) of size |V\u27|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm
Improved Approximate Rips Filtrations with Shifted Integer Lattices
Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes constitutes an expensive task because of a combinatorial explosion in the complex size. For n points in R^d, we present a scheme to construct a 4.24-approximation of the multi-scale filtration of the Rips complex in the L-infinity metric, which extends to a O(d^{0.25})-approximation of the Rips filtration for the Euclidean case. The k-skeleton of the resulting approximation has a total size of n2^{O(d log k)}. The scheme is based on the integer lattice and on the barycentric subdivision of the d-cube