1. Let G = (V,E) be a c-edge connected graph. In other words, assume that the size of minimum cut in G is at least c. Construct a graph G′(V,E ′) by sampling each edge of G with probability p independently at random and reweighing each edge with weight 1/p. Suppose c > log n, and is such that log(n) c 2 ≤ 1. Show that as long as p ≥ log(n) c 2 , with high probability the size of every cut in G′...

In each iteration i of this algorithm there is a ow fi and a residual graph Gfi is constructed by for every edge e ∈ E decreasing the capacity of e by fi(e) and adding an edge in the reverse direction of e with capacity fi(e). A ow in this residual graph, gi, is computed and we update our ow fi by essentially adding gi. As we saw the residual graph was constructed precisely so that the set of f...

There are two main topics we cover in this lecture. First, we perform a slightly more formal analysis of the algorithms we provided last week and discuss how they yield provably e cient algorithms for all the problems we have considered so far: global minimum cut, computing connected components, bipartiteness, articulation points, etc. Second, we introduce a new problem, the minimum spanning tr...

~CP-PACS Collaboration! Center for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan ~Received 27 November 2002; published 1...

i) Graduate School ofScience and 71echnology, Chiba Uhiversity, 648 Matsudo, Matsudo, Chiba 271-851O, Japan 2} Ndtional Institute ofAgrobiotogical Resources, 21 -2 Kannondai, Tsukuba, Ibaraki 305-8602, Japan 3) 77te institute of the Society for 71echno-innovation ofAgricutture, forestT:y and Fisheries (S7Aif), 446-1 Ipp Kamiyokoba, Tsukuba, Ibaraki 305-0854, Japan 4) IVationalAgriculturResearch...

Solution: Let G be a disconnected graph in which case we can decompose it into k connected components C1, C2, . . . , Ck. We want to show that G is connected i.e. there is a path between any u and v in G. In the case that u and v are in different components we know that there exist an edge (a path of length one) between them in G. In the case that u and v are in the same component, say Ci, we c...

1. Let G = (V,E) be a c-edge connected graph. In other words, assume that the size of minimum cut in G is at least c. Construct a graph G′(V,E ′) by sampling each edge of G with probability p independently at random and reweighing each edge with weight 1/p. Suppose c > log n, and is such that 10 log(n) c 2 ≤ 1. Show that as long as p ≥ 10 log(n) c 2 , with high probability the size of every cut...

“⇒”: assume G has an Eulerian circuit. Clearly, G must be connected, otherwise we will be unable to traverse all the edges in a closed walk. Suppose that an Eulerian circuit starts at v1; note that every time the walk enters vertex v 6= v1, it must leave it by traversing a new edge. Hence, every time that the walk visits v it traverses two edges of v. Thus dv is even. The same is true for v1 ex...

Solution: Let G(V,E) denote the graph in question; we construct a new graph H(W,F ) in which an ordinary single particle random walk corresponds to the two particle random walk on G. Take W = V × V = {(v1, v2)|v1, v2 ∈ V } and F = {((v1, v2), (w1, w2))|(v1, w1), (v2, w2) ∈ E}. The idea here is that a uniform random walk on H encodes the state of a two particle random walk on G (in the same way ...

PURPOSE We designed and synthesized CUDC-305, an HSP90 inhibitor of the novel imidazopyridine class. Here, we report its unique pharmacologic properties and antitumor activities in a variety of tumor types. EXPERIMENTAL DESIGN The potency of the compound was analyzed by fluorescence polarization competition binding assay. Its antiproliferative activities were assessed in 40 human cancer cell ...