We present new approaches to constructing graph sparsifiers --- weighted
subgraphs for which every cut has the same value as the original graph, up to a
factor of (1±ϵ). Our first approach independently samples each
edge uv with probability inversely proportional to the edge-connectivity
between u and v. The fact that this approach produces a sparsifier resolves
a question posed by Bencz\'ur and Karger (2002). Concurrent work of Hariharan
and Panigrahi also resolves this question. Our second approach constructs a
sparsifier by forming the union of several uniformly random spanning trees.
Both of our approaches produce sparsifiers with O(nlog2(n)/ϵ2)
edges. Our proofs are based on extensions of Karger's contraction algorithm,
which may be of independent interest