Bits of Learning

Learning sometimes happens in big jumps, but mostly in little tiny steps. I share my baby steps of learning here, mostly on topics around programming, programming languages, software engineering, and computing in general. But occasionally, even on other disciplines of engineering or even science. I mostly learn through examples and doing. And this place is a logbook of my experiences in learning something. You may find several things interesting here: little cute snippets of (hopefully useful) code, a bit of backing theory, and a lot of gyan on how learning can be so much fun.

Friday, September 02, 2016

Hisaab -- A Program to Simplify Transactions among Friends

It's always fun when your technical knowledge comes in handy while to trying to solve some day-to-day problem. you get a kick in being able to describe a problem in its core mathematical essence, and in creating an algorithm to solve the problem. The size of the kick is independent of the size and complexity of the problem.

Follows a cute example of the above.

Problem Description

Often, a group of people may transact money between each other. A situation may arise when each person may have taken some money from someone else, thus owing the other person that much amount. To settle the accounts, there might be needed a very large number of transactions needed, requiring everyone to meet everyone else.

Solution Approach

good way to simplify the settlement process would be to make one of the people involved as the bank, or the star. Everyone makes a transaction with this person only, only the net amount she owes to anyone.

Mathematical Model






The original situation can be represented as a directed graph as shown above in the left side graph. Each node represents a person, and each directed edge indicates a transaction. For example, the edge with weight 15 between A and F means that A owes Rs. 15 to F.

The solution is shown in the right side graph above. Note that all edges now involve node D. Thus every node needs to make at most one transaction with node D, which is huge simplification of the process. The accounts will now be settled with only 5 transactions as opposed to 9 in the original graph. As an added benefit, nodes which need not transact at all also get identified. For example, node B gets isolated in the right side graph, meaning that it doesn't need to transact anything with node D.


Transforms

The solution is achieving through the following two simple graph transformations.

Rerouting



Suppose we wish to make node 1 the star node. This means that we want all transactions between any pair not involving node 1 should be routed through node 1, as shown above.

Multigraph to Graph




Implementation

Follows an implementation of the above idea in OCaml. We use a purely functional style (i.e. by using a completely side-effect free style of programming). If you remove the profuse amount in-line documentation, you will notice that the code is remarkably concise (just about 25 lines of code)!

Find an OCaml and Python implementation here.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
(*
The following program is useful to simplify the transactions amongst a group of people.
Problem Description: Often, a group of people may transact money between each other. A situation may arise when each person
may have taken some money from someone else, thus owing the other person that much amount. A
good way to simplify the settlement process would be to make one of the people involved as the
bank, or the star. Everyone makes a transaction with this person only, only the net amount she
owes to anyone.

User Instruction:
1. Open the file in a editor.
2. Edit the variable g to reflect what each person owes any other.
3. Save the file
4. Open the OCaml REPL
5. enter the command: #use "hisaab.ml";;

The answer will appear on the standard output.
*)

(*
  Description: Takes an edge e = (n1, w, n2) and a node n, and returns a list of edges, to
    apply starification transform on e.

  Signature: 'a * int * 'a -> 'a -> ('a * int * 'a) list

  Example:
  1)
# reroute (1, 10, 2) 1;;
- : (int * int * int) list = [(2, -10, 1)]
# reroute (2, 10, 1) 1;;
- : (int * int * int) list = [(2, 10, 1)]
# reroute (2, 10, 3) 1;;
- : (int * int * int) list = [(2, 10, 1); (3, -10, 1)]
*)
let reroute (n1, w, n2) n =
  if      n1 = n then [(n2, -w, n1)]
  else if n2 = n then [(n1, w, n2)]
  else                [(n1, w, n); (n2, -w, n)]

(*
  Description: Takes a list of lists and returns the corresponding list

  Signature: 'a list list -> 'a list

  Example:
# flatten_list [[1;2]; [3;4;5];[6]];;
- : int list = [1; 2; 3; 4; 5; 6]
*)
let rec flatten_list = function
    [] -> []
  | h :: t -> List.append h (flatten_list t)
 
(*
  Description: Takes a graph (could be multigraph) g and and a node n and returns
  and equivalent graph g' which is starred at n, and preserves the net input/output
  at each node of g.

  Signature: ('a * int * 'a) list -> 'a -> ('a * int * 'a) list
  Example:
# starify_graph [(1, 160, 2); (2, 440, 1); (1, 300, 3); (3, 160, 2)] 1;;
- : (int * int * int) list =
[(3, 160, 1); (2, -160, 1); (3, -300, 1); (2, 440, 1); (2, -160, 1)]
*)
let starify_graph g n =
  let list_of_lists = List.map (fun g -> reroute g n) g in
    flatten_list list_of_lists

(*
  Description: Takes an edge e = (n1, w, n2) and a graph g, and returns a 
  new graph g' such that g' is same as g except that if g contains an edge
  e' with same source and target nodes as e, then e and e' are merged by
  adding their weights.

  Signature: 'a * int * 'b -> ('a * int * 'b) list -> ('a * int * 'b) list

  Assumption: g is not a multi-graph.

  Example:
# let g2 = [(3, -140, 1); (2, 120, 1)];;
val g2 : (int * int * int) list = [(3, -140, 1); (2, 120, 1)]
# merge (3, 140, 1) g2;;
- : (int * int * int) list = [(3, 0, 1); (2, 120, 1)]
# merge (3, 140, 2) g2;; 
- : (int * int * int) list = [(3, -140, 1); (2, 120, 1); (3, 140, 2)]
# merge (3, 140, 2) [];;
- : (int * int * int) list = [(3, 140, 2)]
*)
let rec merge (n1, w, n2) = function
    [] -> [(n1, w, n2)]
  | (n1', w', n2') :: t ->
      if n1 = n1' && n2 = n2' then ((n1, w + w', n2) :: t)
      else (n1', w', n2') :: (merge (n1, w, n2) t)

(*
  Description: Takes a multigraph mg and returns its corresponding graph g by merging
    all edges between common pairs of nodes.

  Signature: ('a * int * 'b) list -> ('a * int * 'b) list

  Example:
# graph_of_mgraph [(1, 160, 2); (2, 300, 1); (2, 140, 1); (1, 300, 3); (3, 160, 2)];;
- : (int * int * int) list =
[(1, 160, 2); (2, 440, 1); (1, 300, 3); (3, 160, 2)]
*)  
let graph_of_mgraph mg =
  let rec iter g = function
    [] -> g
  | e :: t -> iter (merge e g) t
  in
  iter [] mg

(*
  Description: Turns the weight of every edge of a given graph to positive value by inverting the direction of the edge if its weight is negative.

  Signature: ('a * int * 'a) list -> ('a * int * 'a) list
  
  Example:
# abs_weight [("Taru", -140, "Shraddha"); ("Shilpi", 20, "Shraddha")];;
- : (string * int * string) list =
[("Shraddha", 140, "Taru"); ("Shilpi", 20, "Shraddha")]    
*)
let abs_weight g =
  List.map (
    fun (n1, w, n2) -> if w >= 0 then (n1, w, n2) else (n2, -w, n1)
  ) g

let remove_empty_edges g =
  List.filter (fun (_, w, _) -> w <> 0) g
(*
  Example graph (Provided by Shilpi)
  node 1 -> Shilpi
  node 2 -> Shraddha
  node 3 -> Taru

  (1, 160, 2) -> Shilpi owes Rs. 160 to Shraddha
*)
let g = [("Shilpi", 160, "Shraddha"); ("Shraddha", 300, "Shilpi"); ("Shraddha", 140, "Shilpi"); ("Shilpi", 300, "Taru"); ("Taru", 160, "Shraddha")]
let star_node = "Shraddha"
let _ = remove_empty_edges (
          abs_weight (
            graph_of_mgraph (
              starify_graph g star_node)))

let e1 = [
  ("r2", 10, "r3");
  ("r1", 20, "r2");
  ("r2", 10, "r4");
  ("r4", 30, "r3");
  ("r1", 15, "r6");
  ("r6", 100,"r5");
  ("r7", 25, "r5");
  ("r5", 150,"r4");
  ("r3", 30, "r7");
  ("r7", 90, "r4");
]

let star_node = "r4"
let _ = remove_empty_edges (
          abs_weight (
            graph_of_mgraph (
              starify_graph e1 star_node)))

No comments: