ar X iv : 0 80 8 . 01 63 v 1 [ cs . D S ] 1 A ug 2 00 8 Twice - Ramanujan Sparsifiers ∗

ar X iv : 0 80 3 . 09 29 v 2 [ cs . D S ] 7 M ar 2 00 8 Graph Sparsification by Effective Resistances ∗

We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given as input a weighted graph G = (V,E,w) and a parameter ǫ > 0, we produce a weighted subgraph H = (V, Ẽ, w̃) of G such that |Ẽ| = O(n logn/ǫ) and for all vectors x ∈ R (1 − ǫ) ∑ uv∈E (x(u)− x(v))2wuv ≤ ∑ uv∈Ẽ (x(u)− x(v))2w̃uv ≤ (1 + ǫ) ∑ uv∈E (x(u)− x(v))2wuv. (1) This improves upon the spars...

ar X iv : 0 80 3 . 09 29 v 4 [ cs . D S ] 1 8 N ov 2 00 9 Graph Sparsification by Effective Resistances ∗

We present a nearly-linear time algorithm that produces high-quality spectral sparsifiers of weighted graphs. Given as input a weighted graph G = (V, E, w) and a parameter ! > 0, we produce a weighted subgraph H = (V, Ẽ, w̃) of G such that |Ẽ| = O(n log n/!) and for all vectors x ∈ R (1 − !) ∑ uv∈E (x(u) − x(v))2wuv ≤ ∑ uv∈Ẽ (x(u) − x(v))2w̃uv ≤ (1 + !) ∑ uv∈E (x(u) − x(v))2wuv. (1) This improves...

ar X iv : 0 80 3 . 44 39 v 2 [ m at h . D S ] 4 A ug 2 00 8 PERIODIC UNIQUE BETA - EXPANSIONS : THE SHARKOVSKIĬ ORDERING

Let β ∈ (1, 2). Each x ∈ [0, 1 β−1 ] can be represented in the form

ar X iv : 0 80 3 . 09 29 v 1 [ cs . D S ] 6 M ar 2 00 8 Graph Sparsification by Effective Resistances ∗

We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given as input a weighted graph G = (V,E,w) and a parameter ǫ > 0, we produce a weighted subgraph H = (V, Ẽ, w̃) of G such that |Ẽ| = O(n logn/ǫ) and for all vectors x ∈ R (1 − ǫ) ∑ uv∈E (x(u)− x(v))2wuv ≤ ∑ uv∈Ẽ (x(u)− x(v))2w̃uv ≤ (1 + ǫ) ∑ uv∈E (x(u)− x(v))2wuv. (1) This improves upon the spars...

ar X iv : 0 80 8 . 10 23 v 1 [ qu an t - ph ] 7 A ug 2 00 8 CATEGORICAL QUANTUM MECHANICS