Lecture 8 2 Cut Sparsifier
نویسندگان
چکیده
2 Cut Sparsifier Before we state the main theorem, we remind the reader of the definition of connectivity: Definition 1 (Connectivity). The connectivity λe of edge e is the size of the smallest cut containing e. The main theorem gives a construction for a graph sparsifier which preserves cuts. Theorem 1 (Fung and Harvey [FH10], Hariharan and Panigrahi [HP10]). Consider graph G = (V,E) simple and undirected, with n = |V |. Let c be a large enough constant and ε > 0. Let Gε be a weighted subgraph where every edge e ∈ E is sampled with probability
منابع مشابه
3 Barrier Functions
In the previous lectures, we explored the problem of finding spectral sparsifiers and saw both deterministic and randomized algorithms that given a graph G with n vertices finds a spectral sparsifier with O ( n logn 2 ) edges. In this lecture, we improve upon this result to find linear-sized spectral sparsifiers, or more concretely, spectral sparsifiers with O ( n 2 ) edges. This is an interest...
متن کاملSixth Cargese Workshop on Combinatorial Optimization Lecture 3: Sparsifiers
In this last lecture we will discuss graph sparsification: approximating a graph by weighted subgraphs of itself. Sparsifications techniques have been used to design fast algorithms for combinatorial or linear algebraic problems, as a rounding technique in approximation algorithms, and have motivated startling results in pure mathematics. 1 Spectral Sparsifiers Let G = (V,E) be a graph with V =...
متن کاملRefined Vertex Sparsifiers of Planar Graphs
We study the following version of cut sparsification. Given a large edge-weighted network G with k terminal vertices, compress it into a small network H with the same terminals, such that the minimum cut in H between every bipartition of the terminals approximates the corresponding one in G within factor q ≥ 1, called the quality. We provide two new insights about the structure of cut sparsifie...
متن کاملOrie 6334
Then LH (1 + ε)LG, ⇐⇒ (1 + ε)LG − LH 0⇒ (1 + ε)xLGx− xLHx ≥ 0⇒ (1 + ε)|δG(S)| ≥ w(δH(S)). Similarly, if LH (1− ε)LG, we can show that w(δH(S)) ≥ (1− ε)|δG(S)|. We use cut sparsifiers in algorithms that find cuts (sparse cuts, min s-t cuts) in order to replace m in the runtime with O((n log n)/ε2). Of course, this is only useful if we can find the cut sparsifier in the first place in time that’s...
متن کاملLecture 2: Matrix Chernoff bounds
The purpose of my second and third lectures is to discuss spectral sparsifiers, which are the second key ingredient in most of the fast Laplacian solvers. In this lecture we will discuss concentration bounds for sums of random matrices, which are an important technical tool underlying the simplest sparsifier construction.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2015