CS519 HW9 - Graph Algorithms (part 2), DP (part 4) Solved

Textbooks for References:
[1] CLRS Ch. 22 (graph), Ch. 15 (DP)
[2] my DP tutorial (up to page 16):
    http://web.engr.oregonstate.edu/~huanlian/slides/COLING-tutorial-anim.pdf
[3] DPV Ch. 3, 4.2, 4.4, 4.7, 6 (Dasgupta, Papadimitriou, Vazirani)
    https://www.cs.berkeley.edu/~vazirani/algorithms/chap3.pdf
    https://www.cs.berkeley.edu/~vazirani/algorithms/chap4.pdf
    https://www.cs.berkeley.edu/~vazirani/algorithms/chap6.pdf
[4] KT Ch. 6 (DP)
    http://www.aw-bc.com/info/kleinberg/assets/downloads/ch6.pdf
[5] KT slides: Greedy II (Dijkstra)
    http://www.cs.princeton.edu/~wayne/kleinberg-tardos/
[6] Wikipedia: Traveling Salesman Problem
[7] Wikipedia: Held-Karp Algorithm (1962) for TSP


***Please answer time/space complexities for each problem in report.txt.

1. [WILL BE GRADED]
   Dijkstra (see CLRS 24.3 and DPV 4.4)
   
   Given an undirected graph, find the shortest path from source (node 0)
   to target (node n-1). 
   
   Edge weights are guaranteed to be non-negative, since Dijkstra doesn't work
   with negative weights, e.g.
 
       3
   0 ------ 1   
     \    /
    2 \  / -2
       \/
       2
   
   in this example, Dijkstra would return length 2 (path 0-2), 
   but path 0-1-2 is better (length 1).

   For example (return a pair of shortest-distance and shortest-path):
   
       1
   0 ------ 1   
     \    /  \
    5 \  /1   \6
       \/   2  \
       2 ------ 3
            
   >>> shortest(4, [(0,1,1), (0,2,5), (1,2,1), (2,3,2), (1,3,6)])
   (4, [0,1,2,3])

   [UPDATE] Tiebreaking: arbitrary. Any shortest path would do.

   Filename: dijkstra.py


2. Traveling Salesman Problem (TSP).
   
   Given an undirected graph of n nodes (0..n-1) representing a road network,
   the traveling salesman has to start from city 0 and visit each city
   once and only once, and return to city 0. Find the minimum-length tour (cycle)
   that satisifies these conditions (this is also called "Hamiltonian Cycle").
   
   Write the subproblem definition, recurrence relation, and space/time complexities in report.txt.

   Input: same as Dijkstra
   Output: (cycle_length, cycle_list)
   Tiebreaking: arbitrary

   e.g., for the above example in Dijkstra, one possible best cycle is 0-1-3-2-0, with a cost of 14.

   >>> tsp(4, [(0,1,1), (0,2,5), (1,2,1), (2,3,2), (1,3,6)])
   (14, [0,1,3,2,0])

   If we add an edge (3,0,1), then the best cycle cost reduces to 5:
   
   >>> tsp(4, [(0,1,1), (0,2,5), (1,2,1), (2,3,2), (1,3,6), (3,0,1)])
   (5, [0,1,2,3,0])   

   Note: This problem can be solved by either Viterbi (recommended) or Dijkstra. 
         The classical Edmonds-Karp TSP algorithm is an instance of the former.

   Additional real-world examples: # from a map of germany: https://stackoverflow.com/questions/11007355/data-for-simple-tsp

   >>> tsp(11, [(0,1,29),(0,2,20),(0,3,21),(0,4,16),(0,5,31),(0,6,100),(0,7,12),(0,8,4),(0,9,31),(0,10,18),
                (1,2,15),(1,3,29),(1,4,28),(1,5,40),(1,6,72),(1,7,21),(1,8,29),(1,9,41),(1,10,12),
                (2,3,15),(2,4,14),(2,5,25),(2,6,81),(2,7,9),(2,8,23),(2,9,27),(2,10,13),
                (3,4,4),(3,5,12),(3,6,92),(3,7,12),(3,8,25),(3,9,13),(3,10,25),
                (4,5,16),(4,6,94),(4,7,9),(4,8,20),(4,9,16),(4,10,22),
                (5,6,95),(5,7,24),(5,8,36),(5,9,3),(5,10,37),
                (6,7,90),(6,8,101),(6,9,99),(6,10,84),
                (7,8,15),(7,9,25),(7,10,13),
                (8,9,35),(8,10,18),
                (9,10,38)]))

   (253, [0, 8, 10, 1, 6, 2, 5, 9, 3, 4, 7, 0])

   (Viterbi: 0.0s; Dijkstra: 0.3s)

   Random examples:

   >>> tsp(16, [(1, 2, 0), (11, 5, 5), (9, 8, 4), (6, 1, 4), (5, 13, 5), (12, 11, 4), (14, 8, 0), (0, 11, 3), (10, 12, 3), (5, 5, 1), (7, 0, 1), (10, 5, 1), (11, 5, 3), (13, 11, 4), (11, 11, 3), (5, 12, 5), (14, 7, 3), (8, 15, 4), (11, 14, 3), (11, 14, 3), (7, 10, 5), (5, 8, 3), (9, 9, 5), (13, 9, 5), (6, 15, 4), (11, 2, 2), (0, 6, 5), (3, 1, 4), (1, 8, 4), (7, 3, 4), (4, 8, 1), (6, 1, 3), (1, 1, 2), (11, 5, 1), (0, 2, 0), (2, 0, 0), (0, 11, 2), (4, 5, 5), (5, 0, 3), (1, 7, 1), (1, 0, 2), (3, 9, 2), (15, 0, 2), (14, 1, 2), (12, 4, 3), (7, 2, 5), (10, 3, 0), (14, 4, 4), (12, 15, 4), (10, 4, 2), (8, 8, 4), (13, 0, 5), (4, 1, 2), (1, 4, 1), (5, 3, 3), (7, 1, 1), (7, 14, 0), (8, 2, 4), (7, 11, 2), (13, 8, 4), (0, 4, 0), (12, 13, 1), (3, 2, 1), (3, 3, 0), (5, 7, 0), (6, 0, 4), (14, 14, 2), (12, 6, 5), (6, 13, 3), (0, 1, 3), (5, 3, 5), (15, 11, 0), (3, 11, 2), (11, 9, 0), (13, 3, 0), (9, 6, 5), (0, 14, 0), (13, 15, 3), (6, 2, 0), (9, 0, 2), (9, 2, 1), (15, 6, 0), (11, 12, 5), (14, 4, 2), (12, 3, 2), (3, 3, 0), (10, 12, 1), (3, 0, 4), (15, 1, 5), (15, 9, 2), (14, 4, 2), (8, 15, 4), (15, 13, 3), (9, 12, 1), (5, 15, 4), (8, 13, 5), (2, 3, 0), (11, 5, 4), (4, 13, 0), (2, 1, 1)])
   
   (6, [0, 4, 8, 14, 7, 5, 10, 3, 13, 12, 9, 11, 15, 6, 2, 1, 0])

   (Viterbi: 2.1s, Dijkstra: 0.9s)


   Filename: tsp.py

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