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1. [10 marks] Given a set P of n points in the plane a minimum Steiner tree is a tree that

connects the points of P and has minimum total Euclidean (i.e. L2) length. For example, for 3

points p1, p2, p3 forming an acute triangle, the minimum Steiner tree has leaves p1, p2, p3 plus a

node of degree 3 in the middle of the triangle at the Fermat point where the 3 edges form

angles of 120o

. (Look at Wikipedia for examples.) Prove that there is a polynomial time 2-

approximation algorithm for the minimum Steiner tree problem in the plane. In particular,

show that the minimum spanning tree of the points (using Euclidean distances as edge weights

in the complete graph) is a 2-approximation. Hint: refresh your memory on the 2

approximation algorithm for the Travelling Salesman

Problem or see [CLRS, section 35.2]

2. [10] Given a directed graph, G, represented by its adjacency matrix. (Let’s assume there are

no loops … i.e. no edges (i,i).)

a) [6 marks] Give an efficient algorithm to determine what pairs of nodes have directed paths

of length exactly n-1. Give the runtime of your method and justify this runtime. (Hint: think

first about what nodes are connected by paths of length 2.)

b) [4 marks] We know the Hamiltonian cycle problem is NP-Complete. The Hamiltonian path

problem, of having a path go though each node exactly once, is also NP-hard. Explain this

apparent anomaly, given that you have already given an efficient algorithm to find paths of

length n-1.

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