CS 514, Algorithms, Fall 2022
HW8 - Graphs (part I); DP (part III)

Due on Monday Nov 14, 11:59pm.
No late submission will be accepted.

Include in your submission: report.txt, topol.py, viterbi.py.
viterbi.py will be graded for correctness (1%).

To submit:
flip $ /nfs/farm/classes/eecs/fall2022/cs514-001/submit hw8 report.txt {topol,viterbi}.py
(You can submit each file separately, or submit them together.)

To see your best results so far:
flip $ /nfs/farm/classes/eecs/fall2022/cs514-001/query hw8

Textbooks for References:
[1] CLRS Ch. 23 (Elementary Graph Algorithms)
[2] KT Ch. 3 (graphs), or Ch. 2 in this earlier version:
    http://cs.furman.edu/~chealy/cs361/kleinbergbook.pdf
[3] KT slides (highly recommend!):
    https://www.cs.princeton.edu/~wayne/kleinberg-tardos/pdf/03Graphs.pdf
[4] Jeff Erickson: Ch. 5 (Basic Graph Algorithms):
    http://jeffe.cs.illinois.edu/teaching/algorithms/book/05-graphs.pdf
[5] DPV Ch. 3, 4.2, 4.4, 4.7 (Dasgupta, Papadimitriou, Vazirani)
    https://www.cs.berkeley.edu/~vazirani/algorithms/chap3.pdf (decomposition of graphs)
    https://www.cs.berkeley.edu/~vazirani/algorithms/chap4.pdf (paths, shortest paths)
[6] my advanced DP tutorial (up to page 16):
    http://web.engr.oregonstate.edu/~huanlian/slides/COLING-tutorial-anim.pdf

Please answer non-coding questions in report.txt.

0. For the following graphs, decide whether they are
   (1) directed or undirected, (2) dense or sparse, and (3) cyclic or acyclic:

   (a) Facebook
   (b) Twitter
   (c) a family
   (d) V=airports, E=direct_flights
   (e) a mesh
   (f) V=courses, E=prerequisites
   (g) a tree
   (h) V=linux_software_packages, E=dependencies
   (i) DP subproblems for 0-1 knapsack

   Can you name a very big dense graph?

1. Topological Sort
   
   For a given directed graph, output a topological order if it exists.
   
   Tie-breaking: ARBITRARY tie-breaking. This will make the code 
   and time complexity analysis a lot easier. 

   e.g., for the following example:

     0 --> 2 --> 3 --> 5 --> 6
        /    \   |  /    \
       /      \  v /      \
     1         > 4         > 7

   >>> order(8, [(0,2), (1,2), (2,3), (2,4), (3,4), (3,5), (4,5), (5,6), (5,7)])
   [0, 1, 2, 3, 4, 5, 6, 7]

   Note that order() takes two arguments, n and list_of_edges, 
   where n specifies that the nodes are named 0..(n-1).

   If we flip the (3,4) edge:

   >>> order(8, [(0,2), (1,2), (2,3), (2,4), (4,3), (3,5), (4,5), (5,6), (5,7)])
   [0, 1, 2, 4, 3, 5, 6, 7]

   If there is a cycle, return None

   >>> order(4, [(0,1), (1,2), (2,1), (2,3)])
   None

   Other cases:

   >>> order(5, [(0,1), (1,2), (2,3), (3,4)])
   [0, 1, 2, 3, 4]

   >>> order(5, [])
   [0, 1, 2, 3, 4]  # could be any order   

   >>> order(3, [(1,2), (2,1)])
   None

   >>> order(1, [(0,0)]) # self-loop
   None

   Tie-breaking: arbitrary (any valid topological order is fine).
   
   filename: topol.py 

   questions: 
   (a) did you realize that bottom-up implementations of DP use (implicit) topological orderings?
       e.g., what is the topological ordering in your (or my) bottom-up bounded knapsack code?
   (b) what about top-down implementations? what order do they use to traverse the graph?
   (c) does that suggest there is a top-down solution for topological sort as well? 

2. [WILL BE GRADED]
   Viterbi Algorithm For Longest Path in DAG (see DPV 4.7, [2], CLRS problem 15-1)
   
   Recall that the Viterbi algorithm has just two steps:
   a) get a topological order (use problem 1 above)
   b) follow that order, and do either forward or backward updates

   This algorithm captures all DP problems on DAGs, for example,
   longest path, shortest path, number of paths, etc.

   In this problem, given a DAG (guaranteed acyclic!), output a pair (l, p) 
   where l is the length of the longest path (number of edges), and p is the path. (you can think of each edge being unit cost)

   e.g., for the above example:

   >>> longest(8, [(0,2), (1,2), (2,3), (2,4), (3,4), (3,5), (4,5), (5,6), (5,7)])
   (5, [0, 2, 3, 4, 5, 6])

   >>> longest(8, [(0,2), (1,2), (2,3), (2,4), (4,3), (3,5), (4,5), (5,6), (5,7)])
   (5, [0, 2, 4, 3, 5, 6]) 

   >>> longest(8, [(0,1), (0,2), (1,2), (2,3), (2,4), (4,3), (3,5), (4,5), (5,6), (5,7), (6,7)])
   (7, [0, 1, 2, 4, 3, 5, 6, 7])  # unique answer

   Note that longest() takes two arguments, n and list_of_edges, 
   where n specifies that the nodes are named 0..(n-1).

   Tie-breaking: arbitrary. any longest path is fine.   

   Filename: viterbi.py

   Note: you can use this program to solve MIS, knapsacks, coins, etc.


Debriefing (required!): --------------------------

0. What's your name?
1. Approximately how many hours did you spend on this assignment?
2. Would you rate it as easy, moderate, or difficult?
3. Did you work on it mostly alone, or mostly with other people?
4. How deeply do you feel you understand the material it covers (0%-100%)?
5. Take a moment to reflect on your midterm performance; separate the data structures and DP parts.
   Now, do you understand all the problems you didn't solve correctly?
6. Any other comments?

This section is intended to help us calibrate the homework assignments. 
Your answers to this section will *not* affect your grade; however, skipping it
will certainly do.