Low Stretch Spanning Trees

(a) Goal: find a tree-like approximation of graphs. (b) Can’t preserve all edges, maybe preserve distances. (c) Tree metric vs. graph metric. (d) Circle: any subgraph leads to a distortion of n− 1. (e) Tree plus a few edges: spanner (f) Simple spanner for unit length graphs: add edge if endpoints are more than k apart. (g) Can show: n1+O(1/k) edges. (h) Preserve distances on average: low stretch spanning trees.