Facebook Puzzle: sophie

March 5th 2011

This week: Facebook’s sophie puzzle. This one is “buffet” difficulty, which translates roughly to “the underlying problem is NP-complete,” which explains why I have such a hard time choosing food at sushi buffets. In any case, the problem is to find your cat in your apartment, where you know where the cat is likely to be, as well as the transit times between the various locations in your home.

I’ll document here the various bad solutions I came up with on my way to a decent one, and as a bonus: an optimized-ish version in C++!

About the math

The problem asks you to minimize the expected time to find sophie. What does this mean? Take a look at the example input (comments mine).

4
#node name    #probability sophie is there
front_door    .2
in_cabinet    .3
under_bed     .4
behind_blinds .1
5
#node x    #node y       #time between x and y
front_door under_bed     5
under_bed  behind_blinds 9
front_door behind_blinds 5
front_door in_cabinet    2
in_cabinet behind_blinds 6

This says there are four nodes and five edges between those nodes, and that 40% of the time, sophie is going to be under the bed. If sophie was under the bed 100% of the time, the optimal path to minimize the expected time to find her would be just the path that takes you to the bed in the shortest amount of time. But since some nodes are unlikely to hide sophie, you can afford to take your sweet time getting to them.

For this sample input, the optimal path is front_door, in_cabinet, under_bed, behind_blind. Note that to go from in_cabinet to under_bed, you should pass through the front_door node. The expectation for this path is 6.00 seconds, as explained from this snippet from David Eisenstat’s site:

Pr(front_door) * 0
+ Pr(in_cabinet) * Distance(front_door, in_cabinet)
+ Pr(under_bed) * (Distance(front_door, in_cabinet)
                   + Distance(in_cabinet, under_bed))
+ Pr(behind_blinds) * (Distance(front_door, in_cabinet)
                       + Distance(in_cabinet, under_bed)
                       + Distance(under_bed, behind_blinds))
    = .2 * 0 + .3 * 2 + .4 * (2 + 7) + .1 * (2 + 7 + 9) = 6.00

Building the graph

The input only includes edges between particular nodes. In order to know that, say, the distance between the cabinet and the bed is 7 (through the front door), you have to build up the shortest distances between every node in the graph. This is known as the “all pairs shortest path” problem. There exists a quite famous dynamic programming algorithm to solve this, the Floyd Warshall algorithm. Trivially implemented in Ruby:

# note that $weights[x][y] is initialized to either
# Float::MAX if there is no edge between x and y, or
# to whatever the length of the edge is if there is.
def floyd_warshall
    for k in 0..$num-1
    for i in 0..$num-1
    for j in 0..$num-1
        if $weights[i][k] + $weights[k][j] < $weights[i][j]
            $weights[i][j] = $weights[i][k] + $weights[k][j]
            $next[i][j] = k
        end
    end
    end
    end
end

# links menoizes the list of nodes you need to traverse
# between nodes i and j
def links(i, j)
    k = $next[i][j]
    return k if k.class == Set
    $next[j][i] = $next[i][j] = (k.nil? or i == j) ?
        Set.new([]) :
        (links(i, k) + Set.new([k]) + links(k, j))
end

This gives a good starting point for actually trying to start solving the problem.

BFS solution

In my hubris, I figured a breadth first search where you expand on the path with the lowest current expected time would work. Here’s what it looks like:

def solve
    queue = MinHeap.new
    queue.push 0.0, [[0], Set.new((1..$num-1).select {|n| $probs[n] > 0}), 0.0, 0.0]
    while not queue.empty?
        node, remain, time, expected = queue.pop
        if remain.empty?
            p node
            return expected
        end
        # only iterate remaining nodes that you don't need to
        # go through other remaining ndoes to reach
        remain.select {|n|
            (links(node.last, n) & remain).empty?
        }.each do |n|
            new_time = time + $weights[node.last][n]
            new_expected = expected + $probs[n] * new_time
            queue.push new_expected, [node + [n], remain - [n], new_time, new_expected]
        end
    end
end

In my defense I hadn’t yet realized that the sophie problem is a variant of the traveling salesman problem and that a BFS search would take forever on large graphs. This doesn’t work because you incrementally build all the bad paths on your way to finding the first path to complete. Complexity: proportional to the number of paths, or O(n!).

DP Solution

Hitting upon the realization that the problem is a variant of the traveling salesman problem I decided to try the canonical dynamic programming solution to TSP.

The DP solution requires that you build a structure like

C[subset][j]

where subset is some subset of all nodes, and j is a node in that subset. The value of this entry should be the minimum expected time to proceed from node 0 to node j and through all the nodes in the subset. The problem then reduces to finding the minimum of:

C[subset - [j]][i] # for all i in subset

There are some problems here, but first, some code:

def solve
    relevant = (1..$num-1).select {|n| $probs[n] > 0}
    hash = {}
    hash[[0]] = {0 => [0, 0]}
    for size in 1..relevant.size
        relevant.combination(size).each do |subset|
            subset.insert 0, 0
            hash[subset] = {0 => [Float::MAX, Float::MAX]}
            for j in subset
                next if j == 0
                reduced = subset - [j]
                hash[subset][j] = reduced.collect {|i|
                    e, t = hash[reduced][i]
                    t += $weights[i][j]
                    [e + t * $probs[j], t]
                }.min {|a,b| a.first <=> b.first}
            end
        end
    end
    hash[[0] + relevant].values.collect {|x| x.first}.min
end

This works, but isn’t fast. A 17 node graph took me about 10 minutes to finish. The problem is that since subsets aren’t ordered, there is no convenient way to represent them as array indices and you must therefore hash entire subsets. Since there are 2n subsets, and you must compute the path that ends in each node in each subset, which itself requires examining all other previous paths of the subset minus one of its elements (breathe), the complexity here is O(2nn2).

Recursive Backtracking (Pruning) Solution

Backtracking can be thought of as essentially a DFS search with fast failure. For example, say you have found a complete path that you know will give you an expected time of 30 seconds. Now you are attempting to build another path, and halfway through you know your partial expected time is already 31 seconds. You can abandon building this path, saving yourself the hassle of expanding all of that partial path’s children.

Skipping entire subtrees is known as pruning, and you can achieve some pretty massive savings depending on how well you implement it. My solution was to very conservatively estimate the remaining expectation of a partial path. For instance, if you are halfway through a path with a current expected time of 10 seconds, a path length of 20 seconds and you have covered 60% of the places where sophie can be, then even in the perfect case where the next node was 0 seconds away and had a 40% probability of hiding sophie, you would still incur an additional 20 * .4 = .8 seconds of expected time. If you have already found a minimum path length of say, 10.5, you can prune this subtree where you could not have before.

And without further ado, here’s the code.

$min = Float::MAX
def solve(node, remain, unseen, expect = 0, time = 0)
    return if expect + unseen * time >= $min
    return ($min = expect) if remain.empty?
    remain.each do |n|
        next_time  = time + $weights[node][n]
        solve n,
              remain - [n],
              unseen - $probs[n],
              expect + next_time * $probs[n],
              next_time
    end
    $min
end

This works fairly well. We can do that 17 node graph in 30 seconds now. I don’t have a good estimate of the complexity improvement here, since you can generate corner cases that can prevent any pruning from happening.

Optimizations!

Ruby is slow. At least, Cygwin’s default Ruby 1.8.7 interpreter is slow. I decided to reimplement the whole deal in C++ and see what kind of speedups I could achieve. Here is my initial implementation in C++:

double solve(int node, set<int> &remain, double unseen,
        double expect = 0.0, double time = 0.0) {
    static double min = numeric_limits<double>::max();
    if (expect + unseen * time >= min)
        return -1;
    if (remain.size() == 0) {
        min = expect;
        return -1;
    }
    for (set<int>::iterator n = remain.begin(); n != remain.end(); n++) {
        int next = *n;
        double next_time = time + weights[node][next];
        set<int> next_remain = remain;
        next_remain.erase(next);
        solve(next, next_remain, unseen - probs[next],
            expect + next_time * probs[next], next_time);
    }
    return min;
}

However, the speedup here was only a factor of two or so. Where are the bottlenecks? Turns out, a big one in both Ruby and C++ is the constant recreation of the remainder set at

set<int> next_remain = remain;

It’s much faster to just do:

set<int> next_remain = remain;
for (set<int>::iterator n = remain.begin(); n != remain.end(); n++) {
    int next = *n;
    double next_time = time + weights[node][next];
    next_remain.erase(next);
    solve(next, next_remain, unseen - probs[next],
        expect + next_time * probs[next], next_time);
    next_remain.insert(next);
}

This way, you only do one set copy per recursion, and then just pass around to all of your children. “But Vincent,” you say, “why even bother recreating the set at each recursion? Can’t you just pass alone one set of remaining nodes and add and remove from it?” Sure, but its very annoying to not invalidate iterators to a set that is constantly shrinking and growing through iteration and recursion. Here’s what I came up with:

double solve(int node, vector<node_entry> &remain, double unseen,
        double expect = 0.0, double time = 0.0) {
    static double min = numeric_limits<double>::max();
    if (expect + unseen * time >= min)
        return -1;

    bool empty = true;
    for (vector<node_entry>::iterator n = remain.begin(); n != remain.end(); n++) {
        if (!n->active)
            continue;
        empty = false;
        int next = n->index;
        double next_time = time + weights[node][next];
        n->active = false;
        solve(next, remain, unseen - probs[next],
            expect + next_time * probs[next], next_time);
        n->active = true;
    }
    if (empty) min = expect;
    return min;
}

This way we just store whether an element is active or not in the data model itself, instead of representing that information with presence in a set. This is nice because adding/removing from a set, while O(log(n)) fast, ain’t no O(1). This gets us (for our 17 node graph):

$ make && time ./sophie in_sophie5.txt
make: `sophie' is up to date.
38.20

real    0m0.220s
user    0m0.156s
sys     0m0.030s

Hooray!

Final Thoughts

I didn’t talk about any of the edge cases, but you need to check for if it’s actually possible to be sure to find sophie. In particular, if there are nodes that aren’t reachable from the first node that sophie has a chance of being in then you need to fail.

Dear submitter,

Thank you for your submission of a puzzle solution to Facebook! After running your solution to sophie (received on March 5, 2011, 8:14 pm), I have determined it to be correct. Your solution ran for 25.118 ms on its longest test case.

Also, full source and some plagiarized test cases are all on github.