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 ...