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Course: STOR 891, Fall 2008
School: UNC
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Word Count: 385

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Network STOR891: Algorithms Assignment 1. Using the definition of a spanning tree on vertex set N to be a connected graph on N with no cycles, prove: (a) Every tree has at least one vertex that is adjacent to a single edge. (b) Every tree has exactly |N | - 1 edges. (Hint: Use (a) plus induction.) (c) There is exactly one path between every pair of vertices in a tree. (d) Adding an edge to a tree creates exactly...

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Network STOR891: Algorithms Assignment 1. Using the definition of a spanning tree on vertex set N to be a connected graph on N with no cycles, prove: (a) Every tree has at least one vertex that is adjacent to a single edge. (b) Every tree has exactly |N | - 1 edges. (Hint: Use (a) plus induction.) (c) There is exactly one path between every pair of vertices in a tree. (d) Adding an edge to a tree creates exactly one cycle. 2. Let G be an (s, t)-planar graph, that is, G can be drawn on the plane with no crossing edges and with s and t on the boundary. Consider solving the max flow problem on G by always choosing the augmenting path to correspond to the topmost directed (s, t)-path in the residual network. Show: (a) Every augmentation will be done along a directed (s, t)-path in G, that is, there are never any backward arcs in the augmenting path. (b) The maximum flow will always be reached in at most m augmentations. 3. Breadth-first search is an efficient way of labeling all vertices in a network that can be reached from some root node s by a directed path. It is performed as follows: Start by putting s onto the list L. while L = i Take the node x out of L that was the first node put on L is that still remaining on the list. ii Label x, and put onto the list L all unlabeled nodes y such that (x, y) is an arc. Show that this procedure labels the nodes in exactly the same order as Dijkstra's algorithm, when applied to the cardinality (cij = 1) problem. Show how to assign the shortest path distances. 4. Consider the problem of pushing 80 units of flow through the following network, with edges marked by uij , \$cij : 2 20,\$2 45,\$1 5 90,\$9 10,\$8 7 50,\$3 10,\$2 50,\$2 3 10,\$1 8 1 30,\$6 15,\$8 10,\$3 10,\$1 60,\$5 80,\$4 10,\$1 10,\$3 11 15,\$2 15,\$8 4 6 20,\$2 9 10 10,\$3 20,\$3 Starting with the following arc flows and node distances: 3 2 14 10 0 5 5 10 0 7 50 0 1 40 5 3 0 13 8 10 40 0 5 17 11 ...

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