Algorithm to find least number of trades between people

Gautham Rajesh :

I have been thinking about this question for a while now.

Let's say there is a list s that contains books that people are selling. I have a book in s, and I want to trade for a book that is in s (something I don't have).

Each person in this scenario is selling one book in s, and also wants one book that is in s.

Is there an algorithm to find out the least number of trades required between people to get the book that I want (and give my book to people who want it)?

Please tell me if this seems confusing (because it probably is).

Thanks for answering the question!

EDIT: List s can grow at any time.

btilly :

I would turn this into a graph as follows. Books are the nodes of the graph. You have a directed edge from book A to book B if someone in S is offering book B and wants book A. You are looking for the shortest path from the book that you have to the book that you really want.

The shortest path can be discovered by a breadth-first search. You can implement one as follows:

todo = [starting_book]
path = {starting_book: None}
while len(todo) and target_book not in path:
    book = list.pop(0)
    for next_book in neighbors(book):
        if next_book not in path:
            path[next_book] = book
            todo.append(next_book)
if target_book in path:
    solution = [target_book]
    while solution[-1] != starting_book:
        solution.append(path[solution[-1]])
else:
    solution = None

Note, the solution that I am offering here gives the set of book trades to happen, but not the people. Turning a particular book trade back into the person to talk to can be done either by searching s or with a lookup table.

This is also not an online solution - keeping track of shortest paths as things are added to/removed from s is going to be a lot of work.