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For any graph H, we de ne distH (u v) as the distance between u and v in the graph H. Given a graph G = (V E) a t-spanner is a spanning subgraph G0 = (V E 0 ) of G with the property that for all pairs of vertices u and v, distG0 (u v) t distG (u v): In other words, we wish to compute a subgraph that provides approximate shortest paths between each pair of vertices. Clearly, our goal is to minimize the size and weight of G0, given a xed value of t (also called the stretch factor). The size of G0 refers to the number of edges in G0, and the weight of G0 refers to the total weight of the edges in G0.

If it is a perfect matching, according to the theorem above, we are done. Let S = the set of free nodes in X. Grow hungarian trees from each node in S. Let T = all nodes in Y encountered in the search for an augmenting path from nodes in S. Add all nodes from X that are encountered in the search to S. Step 5: Revise the labeling, l, adding edges to Gl until an augmenting path is found, adding vertices to S and T as they are encountered in the search, as described above. Augment along this path and increase the size of the matching.

Is added at the top of GD , and every node in GD corresponding to a face that has an edge on the top of G, is connected to s? Node t? is similarly placed and every node in GD , corresponding to a face that has an edge on the bottom of G, is connected to t? The resulting graph is denoted GD? If some edges form an s ; t cut in G, the corresponding edges in GD? form a path from s? to t? In D? G , the weight of each edge is equal to the capacity of the corresponding edge in G, so the min-cut in G corresponds to a shortest path between s?

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