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CSC B63                 Lecture Summary for Week 11             Winter 2007
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Minimum Spanning Tree algorithms (continued from last week)
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[ finished MST notes from week 10 posting ]

Running time of MST algorithms:

  - Kruskal (using disjoint sets with "union-by-rank" and path compression ):
      O(m log n)

  - Prim (using heap): O(m log n)

  - Prim (using binomial heap): O(m + n log n)

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Approximation algorithms [ chapter 35.2, not on exam ]
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[ most of this is a preview of courses like C73 or for interest,
  but it's a nice example of using our data structures and algorithms
  to solve a hard problem ]

Some problems are hard to solve.
  eg. NP-complete problems: no one knows how to solve in worst-case
      polynomial time.

  - Perhaps a "good" answer (near-optimal) answer found efficiently
    is good enough.

  - "approximation ratio": an algorithm has approximation ratio r(n)
    if for any input of size n,
      C_alg / C_opt <= r(n)
    where C_alg is the "cost" of the solution found by the algorithm
    and C_opt is the "cost" of the optimal solution.
      - in some cases, ratio does not depend on n (constant approximation ratio)

  - "r(n)-approximation algorithm": an algorithm (usually polynomial time)
    that achieves a r(n) approximation ratio


Travelling Salesperson Problem (TSP):

  - Input: complete graph G=(V,E) with non-negative (integer) edge costs c(e)
  - Output: a tour of G with minimum cost

  - a tour is a Hamiltonian cycle, that is, a simple cycle that visits every
    vertex in G
  - cost of a tour is the sum of edge costs in the tour
  - TSP is NP-complete in general
  - TSP is also hard to approximate

  - "triangle inequality": one side of a triangle is no more than the sum of
    the other two sides
	c(u,w) <= c(u,v) + c(v,w)
      - common when considering Euclidean or metric spaces

  - TSP with triangle inequality: also NP-complete, but often interesting
      - can approximate the solution within factor of 2

TSP with triangle inequality:

  - example: vertices are locations on a 6x5 grid

      vertices: a at (1,4), b at (2,1), c at (0,1), d at (3,4),
      		e at (4,3), f at (3,2), g at (5,2), h at (2,0).
      edge costs: distance in Euclidean plane

  - lower bound: cost of a minimum spanning tree
      - an optimal tour minus one edge is a spanning tree
  - upper bound: 2*cost of a minimum spanning tree
      - we can traverse each edge of the MST twice to visit all the vertices
      - need to make this more precise to prove

  - example: an MST of example above is {(b,c), (b,h), (a,d), (d,e),
  	(e,f), (e,g), (b,f)}

  - algorithm:
      pick a root vertex r
      compute a MST T from the root r
      L <- order of vertices in preorder walk of T
      return Hamiltonian cycle H produced by visiting vertices in order L
      (ignoring duplicated vertices)

  - running time: clearly polynomial time

  - Theorem: This is a 2-approximation algorithm for the TSP with triangle ineq.
    Proof: Let Opt be an optimal tour, let T be a MST.
      Deleting an edge from Opt yields a spanning tree, thus
        c(T) <= c(Opt).
      A full walk W of T lists vertices each time it is enountered in a
      preorder traversal of T.
        eg., abcbhbadefegeda
      The walk traverses every edge of T twice, so
        c(W) = 2*c(T).
      Combining with above inequality,
        c(W) <= 2*c(Opt).
      In general, W is not a tour, since vertices may appear multiple times.
      However, we can delete repeated vertices from W without increasing the
      cost (by triangle inequality).
      Repeat deleting repeated vertices until a valid tour H remains.
      Then c(H) <= c(W) <= 2*c(Opt), showing our approximation ratio.