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Exhaustive Search

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Some problems involve searching a vast number of potential solutions to find an answer, and do not seem to be amenable to a solution by efficient algorithms.

What is an efficient algorithm? We are used to thinking in terms of O(N), O(N log N), and O(N3/2) for example. We are used to O(N3), for example, being labelled as inefficient.

For the kind of problem we are about to discuss, O(N50) would be delightful (at least from a theoretical standpoint)!

Suppose we have an algorithm that runs O(2N).

Let’s adopt the naive assumption that it would take 2N CPU cycles to solve this problem (an overly-optimistic and simplistic assumption).

Further, assume that we have a ‘super-fast’ 3GHz machine capable of 3E9 cycles per second.

Thus, for a problem of size N=50, this would take:

250 cycles = 1.259E15 cycles

Our CPU would then require 1.259E15/3E9 = 419667 sec.

That’s 116 hours of computing.

So our problem of N=50 takes 116 hours of computing.

But now consider the problem N=51. This will take 208 hours.

So we’ve approximately doubled the computing effort but have solved the problem for only one more element in the problem set (50=>51).

Now to compute the problem N=59, this would take six years using our ‘super-fast’ CPU!

This is definitely not to our advantage. Even if we came up with a computer a million times the speed of the current one, it would take 13 million years to finish the computation for N=100.

The most famous problem of this type is the Traveling Salesman Problem: given N cities, find the shortest route connecting them all, with no city visited twice.

It is unthinkable to solve an arbitrary instance of this problem for N=1000 for example.

The problem is difficult because there does not seem to be a way to avoid having to check the length of a very large number of possible tours. Checking each possible tour is exhaustive search.

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We can model the Traveling Salesman Problem with graphs: given a weighted (and possibly directed) graph, we want to find the shortest simple cycle that connects all the nodes.

An instance of the Traveling Salesman Problem

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This brings up a another problem that would seem to be easier: given an undirected graph, is there any way to connect all the nodes with a simple cycle? That is, starting at some node, can we visit all the other nodes and return to the original node, visiting every node in the graph exactly once? This is known as the Hamilton Cycle Problem.

As it turns out, the Hamilton Cycle Problem is technically equivalent to the Traveling Salesman Problem.

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DFS visits the nodes in the graph below in this order:

A B C E F D G H I K J L M

assuming an adjacency matrix or sorted adjacency-list representation. This is not a simple cycle, so to dins a Hamilton cycle we have to try another way to visit the nodes. As it turns out, we can try all possibilities systematically with a simple modification of the DFS visit function.

void visit(int* val, int k, int & id)

{

int t;

cout << "visit: visiting node " << k << endl;

val[k] = ++id;

t = adj[k];

for (t=0; t<N; t++) {

if (a[k][t])

if (val[t]==0)

visit(val, t, id);

id--; val[k] = 0;

}

}

The only marked nodes are those for which visit hasn’t completed, and they correspond to a simple path of length id in the graph, from the initial node to the one currently being visited.

To visit a node, we simply visit all unmarked adjacent nodes (marked ones would not correspond to a simple path). The recursive procedure checks all simple paths in the graph that start at the initial node.

Self-cleanup code on return.

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Each node in the tree corresponds to a call to visit.

Each path is traversed twice (i.e. in both directions)

For this example, there is only one Hamilton cycle. There are other paths of length V that do not make up a Hamilton cycle.

Note carefully that in the regular DFS, nodes are marked as visited and remain marked after they are visited, whereas in exhaustive search nodes are visited many times. The unmarking of the nodes makes exhaustive search different from DFS in an essential way.

Our DFS modification, so far, explores all possible paths.But we need to detect a Hamilton cycle if we come across one! Recall that id is the length of the path being tried, and val[k] is the position of node k on that path. Thus we can make the visit function test the existence of a Hamilton cycle by testing whether there is an edge from k to 1 when val[k] = V;

We can use the same technique described above to solve the Traveling Salesman Problem, by keeping track of the length of the current path in the val array, then keeping track of the minimum of the lengths of the Hamilton cycles found.

The time taken to do exhaustive search is proportional to the number of calls to visit, which is the number of nodes in the exhaustive search tree. For large graphs, this will be very large. (cf. if this is a complete graph, with every node connected to every other node, then there are V! simple cycles, one corresponding to each arrangement of the nodes).

We need to add tests to visit to discover that recursive calls should not be made for certain nodes. This corresponds to pruning the exhaustive search tree: cutting certain branches and deleting everything connected to them.

One important technique is the removal of symmertries. In our exhaustive search, each cycle is traversed in both directions. We can try to ensure that we find each cycle just once, by insisting that three particular nodes appear in particular order. For example, if we insist on the order “C after A but before B”, then we do not call visit for B unless C is already on the path.

This is not always a possible technique. Suppose we’re trying to find a path (not necessarily a cycle) connecting all vertices. Now this scheme can not be used since we don’t know in advance whether a path will lead to a cycle or not.

Each time we cut off the search at a node, we avoid searching the entire

subtree below that node. For very large trees, this is very substantial savings.

Those savings are so significant that it is worthwhile to do as much as possible within visit to avoid making recursive calls.

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Example: some paths might divide the graph in such a way that the unmarked nodes aren’t connected, so no cycle can be found. Take for instance the path starting with ABCE.

Applying this rule to the pruned tree with symmetry heuristics applied leads to the reduced tree above. (This costs us one DFS to discover the heuristic, but saves us many otherwise wasted calls to visit.)

When searching for the best path (for instance Traveling Salesman Problem), another important pruning technique is available:

Suppose a path of cost x through the graph has been found. Then it’s useless to continue along any path for which the cost so far is greater than x.

This can be implemented by making no recursive calls to visit if the cost of the current partial path is greater than the cost of the best path found so far. We clearly won’t miss the minimum-cost path by adopting such a policy.

The pruning is more effective if a low-cost path is found early in the search. One way to make this likely is to visit the nodes adjacent to the current node in order of increasing cost.

But we can do better: often, we can compute a bound on the cost of all full paths that begin with a general partial path: add the cost of the minimum spanning tree of the unmarked nodes.

This general technique of calculating bounds on partial solutions in order to limit the number of full solutions needing to be examined is sometimes called branch-and-bound. It applies whenever costs are attached to paths.

The general procedure just described, of solving a problem by systematically generating all possible solutions is called backtracking.

Whenever partial solutions to a problem can be successfully augmented in many ways to produce a complete solution, a recursive implementation may be appropriate.

The process can be described by an exhaustive search whose tree nodes correspond to a partial solution. Going down the tree corresponds to progress towards more complete solutions. Going up the tree corresponds to “backtracking” to some previously generated partial solution, from which point it might be worthwhile to proceed forward again.

Backtracking and branch-and-bound are widely applicable as general problem-solving techniques. For example, they form the basis for many programs that play chess or checkers.

In this case, a partial solution is some legal positioning of all the pieces on the board, and the descendant of a node in the exhaustive search tree is a position that can be the result of some legal move.

Ideally, we want an exhaustive search. Realistically, we need to prune (in quite a sophisticated way).

However sophisticated the criteria, it is generally true that the running time for backtracking algorithms remains exponential.

If each node in the search tree has a children, on the average, and the length of the solution path is N, then we expect the number of nodes in the tree to be proportional to aN.

Different backtracking rules correspond to reducing the value of a, the number of choices to try at each node. This is worthwhile because this increases the size of the problem that can be solved.

For example, an algorithm that runs O(1.1N) can solve a problem perhaps eight times as large as one which runs in time O(2N).

On the other hand, neither will do well with very large problems.

Sometimes the line between “easy” and “hard” problems is fine. Recall our solution to the problem: “find the shortest path from vertex x to vertex y”. But if we ask “what is the longest path (without cycles) from x to y?” we have a real problem on our hands.

Sometimes the fine line is even more striking when you consider similar problems that ask for only yes/no questions:

Easy: is there a path from x to y with weight <= M?

Hard: is there a path from x to y with weight >= M?

BFS gives a solution for the first question in linear time. All known solutions to the second could take exponential time1.

Without loss of generality, exponential time here means 2N. The 2 can be replaced with anything > 1 and the argument still holds.

Let’s develop a simple formal model to distinguish between those “efficient” algorithms that we’ve been studying, and the brute-force “exponential” algorithms that we’ve discussed recently.

In this model, the efficiency of an algorithm is a function of the number of bits used to encode the input, using a “reasonable” encoding scheme. For example you would expect a number M to be represented with logM bits not M bits. In any case, we are merely interested in identifying algorithms guaranteed to run in time proportional to some polynomial in the number of bits of input.

P: the set of all problems that can be solved by deterministic algorithms in polynomial time.

Deterministic? - at any point in time, the next step in a program can be predicted.

Let’s now empower our computer with the power of nondeterminism: when an algorithm is faced with a choice of several options, it has the power to “guess” the right one.

NP: the set of all problems that can be solved by nondeterministic algorithms in polynomial time.

Obviously, and problem in P is also in NP. But it seems that there should be many more problems in NP.

Let’s now empower our computer with the power of nondeterminism: when an algorithm is faced with a choice of several options, it has the power to “guess” the right one.

NP: the set of all problems that can be solved by nondeterministic algorithms in polynomial time.

Obviously, any problem in P is also in NP. But it seems that there should be many more problems in NP.

Why bother considering an imaginary tool that makes difficult problems seem trivial?

No one has been able to find a single problem that can be proven to be in NP but not in P (or even prove that one exists). i.e., we do not know whether NP=P or not.

This is quite frustrating, since if we determine that a particular problem is in NP and not in P, we could abandon the search for an efficient solution to it.

In the absence of such proof, there is the possibility that some efficient algorithms have gone undiscovered.

In fact, given what we know so far, there could be an efficient algorithm for every problem in NP => many efficient algorithms have gone undiscovered.

Virtually no-one believes that P=NP, but this is an outstanding research question.

Another example of a problem in NP is the satisfiability problem. Given a logical formula of the form:

(x1 + x3 +x5) * (x1 + !x2 + x4) * (!x3 + x4 + x5) * (x2 + !x3 + x5)

where xi represents a boolean variable, ‘+’ represents or, ‘*’ represents and, and ‘!’ represents not.

The satisfiability problem is to determine whether there is an assignment of truth values to the variables that makes the formula true (‘satisfies’ it).

-- hold that thought, we’re going to need satisfiability in a little bit.

Let’s look at some problems known to belong to NP but might or might not belong to P. That is, they are easy to solve on a non-deterministic machine, but no-one has been able to find an efficient algorithm for them on a conventional machine.

These problems have an added property: if any of them can be solved in polynomial time on a conventional machine, then so can all of them (i.e. P=NP). Such problems are said to be NP-Complete.

The primary tool used to prove that problems are NP-Complete employs the idea of polynomial reducibility: transform any instance of the known NP-Complete problem to an instance of the new problem, solve the problem using the given algorithm, then transform the solution back to a solution of the NP-Complete problem.

Polynmially reducible means that the transformation can be done in polynomial time.

For example, to prove that a problem in NP is NP-Complete, we need only show that some known NP-Complete problem is polynomially reducible to it: that is, that a polynomial time algorithm for the new problem can be used to solve the NP-Complete proble, and then can, in turn, be used to solve all problems in NP.

Example:

Traveling Salesman Problem: Given a set of cities and distances between all pairs, find a tour of all cities of distance less than M.

Hamilton Cycle: given a graph, find a simple cycle that includes all the vertices.

The Hamilton Cycle problem reduces to the Traveling Salesman Problem (assuming we know the Hamilton Cycle problem is NP-Complete)

The Hamilton Cycle problem reduces to the Traveling Salesman Problem (assuming we know the Hamilton Cycle problem is NP-Complete): that was relatively easy to illustrate.

Reducibility can sometimes be very hard indeed to prove!

There are literally thousands of problems that have been shown to be NP-Complete.

Reduction uses one NP-Complete problem to imply another. But how was the first problem found?

S. A. Cook gave a direct proof in 1971 that satisfiability is NP-Complete, i.e. that if there is a polynomial time algorithm for satisfiability, then all problems in NP can be solved in polynomial time.

The proof is extremely complicated and involves laying down the specification of a Turing Machine with the added property of nondeterminism.

The solution of the satisfiability problem is essentially a simulation of the machine running the given program on the given input to produce a solution to an instance of the given problem.

Fortunately, this proof had to be done only once. We can use reducibility to show that other problems are NP-Complete.

Partition: given a set of integers, can they be divided into two sets whose sum is equal?

Integer Linear Programming: given a linear problem, is there a solution in integers?

Multiprocessor Scheduling: given a deadline and a set of tasks of varying lengths to be performed on two identical processors, can the tasks be arranged so that the deadline is met?

Vertex Cover: given a graph and an N, is there a set of fewer than N vertices which touches all the edges?

Molecular Energy: given a molecule with a set of rotatable bonds, what is the minimum energy of the molecule if it is allowed to freely rotate its bonds?