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Searching 4 Solutions

Once a problem has been properly formulated, as outlined in the preceding 3 sections, it is now possible to solve the problem. The solution to the problem is an action sequence. Therefore, search algorithms work by considering the various possible action sequences. The first possible action sequence begins at the intial state of a search tree, with the intial state being called the root. The branches in the tree are actions, while the nodes correspond to the states, which are defined as the state space of the problem. And the state space are all of the possible decisions that can be made during the process of solving the problem. Figure Agent-6 shows the first few steps in growing the search tree for finding a route from Arad to Bucharest:

Arad-Bucharest Partial

Figure Agent-6: Partial search trees for finding a route from Arad to Bucharest. Nodes that have been expanded are shaded; Nodes that have been generated but not yet expanded are outlined in bold; nodes that have not yet been generated are shown in faint dashed lines.


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In the example we will be using in this guide, the root node of the tree which corresponds to the initial state will be referred to as In(Arad). The first step is to test whether this is a goal state. (In this case it is not, however, it should always be the first step in the algorithm, so that we don't take any unnecessary steps, if we are already at the goal). The second step is to consider the various actions that are available to us. This is done by expanding the current state (or node). Expanding the node, means creating all possible actions from our current position, and generating all the legal possible destinations for the next step. Once the root node is expanded, it becomes the parent node. The parent node: In(Arad) leads to three new child nodes: In(Sibiu), In(Timisoara), and In(Zerind). The next step is to choose which of these 3 possibilities are to be expanded in the search of our goal.

This is the essence of the search: Follow up on one option now, and put the others aside for later. assuming the first choice has not already led to a solution. Suppose we choose Sibiu first. First, check to see if it is the goal state: Is Sibiu == Bucharest? Looks like it isn't, I guess we have to expand it, giving us: In(Arad), In(Fagaras), In(Oradea), and In(Rimnicu Vilcea). We now have six possibilities: The four from expanding Sibiu and the 2 left over from the original expansion of Arad. These six nodes are referred to as leaf nodes, that is, they don't have any children yet. The set of all leaf nodes available for expansion at any given point is called the frontier. In Figure Agent-6, above, the frontier of the tree consists of the nodes with bold outlines.

The process of choosing and expanding nodes in the frontier continues until either a solution is found or there are no more states to be expanded. The general Tree - Search algorithm is shown in Figure Agent-7. Search algorithms all share this basic structure. The differences are primarily how they choose which state to expand next, also known as the Search Strategy.

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Function Tree-Search(problem) returns a solution, or failure
  Initialize the frontier using the initial state of the problem
  loop do
    If the frontier is empty then return failure
    choose a leaf node and remove it from the frontier
    If the node contains a goal state then return the corresponding solution
    Expand the chosen node, adding the resuting nodes to the frontier.

Function Graph-Search(problem) returns a solution or a failure
  intialize the frontier using the initial state of problem
  initialize the explored set to be empty
  loop do
    If the frontier is empty then return failure
    choose a leaf node and remove it from the frontier
    If the node contains a goal state then return the corresponding solution
    add the node to the explored set
    expand the chosen node, adding the resulting nodes to the frontier
      only if not in the frontier or explored set
Figure Agent-7: An informal description of the general tree-search and graph-search algorithms. The parts of Graph-Search marked in bold italic are the additions needed to handle repeated states.


Looking back at Figure Agent-6, we can see quite a dilemma in that it is possible to return to Arad, thus potentially making this a very long journey, indeed. The official term for this is that In(Arad) is a repeated state in the search tree, generated by a loopy path, or unofficially: our case of Budweiser is going to run out long before our trip is over. Considering such loopy paths means that the complete search tree for Romania is infinite, because you can get stuck running around in circles, if you're not careful. On the other hand, the state space as they call it, or you might just say then number of towns on the map, has only 20 states. So, since this initial algorithm allows us to repeat our path, we obviously need to make some minor modifications. Fortunately, since we are only trying to cut down our trip distance, we don't need to ever consider returning to a place we have already visited.

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Loopy paths are a special case of the more general concept of redundant paths, which exist whenever there is more than one way to get from one state to another. Consider the paths Arad-Sibiu, about 140 km, and Arad-Zerind-Oradea-Sibiu, about 297 km.


Romania 1

Figure Agent-8: Two possible paths - One good - One not so good.



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Officially we would say that the second path is redundant, unofficially, we might say: We better have our navigator cut back on the Bud a little bit. If you are concerned about reaching the goal, there's never any reason to keep around more than one path to any given state, because any goal state that is reachable by extending the other.

In some cases, it is possible to define the problem itself so as to eliminate redundant paths. For example, if we formulate the 8-Queens problem so that a Queen can be placed in any column, then each state with n queens can be reached by n! different paths; but if we reformulate the problem so that each new queen is placed in the leftmost empty column, then each state can be reached only through one path.

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Last Updated 7-27-2010

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Normalized Information Distance