11.2 Sequential Search Algorithms
The most suitable sequential search algorithm to apply to a state space depends on whether the space forms a graph or a tree. In a tree, each new successor leads to an unexplored part of the search space. An example of this is the 0/1 integer-programming problem. In a graph, however, a state can be reached along multiple paths. An example of such a problem is the 8-puzzle. For such problems, whenever a state is generated, it is necessary to check if the state has already been generated. If this check is not performed, then effectively the search graph is unfolded into a tree in which a state is repeated for every path that leads to it (Figure 11.3).
For many problems (for example, the 8-puzzle), unfolding increases the size of the search space by a small factor. For some problems, however, unfolded graphs are much larger than the original graphs. Figure 11.3(b) illustrates a graph whose corresponding tree has an exponentially higher number of states. In this section, we present an overview of various sequential algorithms used to solve DOPs that are formulated as tree or graph search problems.
11.2.1 Depth-First Search Algorithms
Depth-first search (DFS) algorithms solve DOPs that can be formulated as tree-search problems. DFS begins by expanding the initial node and generating its successors. In each subsequent step, DFS expands one of the most recently generated nodes. If this node has no successors (or cannot lead to any solutions), then DFS backtracks and expands a different node. In some DFS algorithms, successors of a node are expanded in an order determined by their heuristic values. A major advantage of DFS is that its storage requirement is linear in the depth of the state space being searched. The following sections discuss three algorithms based on depth-first search.
Simple backtracking is a depth-first search method that terminates upon finding the first solution. Thus, it is not guaranteed to find a minimum-cost solution. Simple backtracking uses no heuristic information to order the successors of an expanded node. A variant, ordered backtracking, does use heuristics to order the successors of an expanded node.
Depth-first branch-and-bound (DFBB) exhaustively searches the state space; that is, it continues to search even after finding a solution path. Whenever it finds a new solution path, it updates the current best solution path. DFBB discards inferior partial solution paths (that is, partial solution paths whose extensions are guaranteed to be worse than the current best solution path). Upon termination, the current best solution is a globally optimal solution.
Iterative Deepening A*
Trees corresponding to DOPs can be very deep. Thus, a DFS algorithm may get stuck searching a deep part of the search space when a solution exists higher up on another branch. For such trees, we impose a bound on the depth to which the DFS algorithm searches. If the node to be expanded is beyond the depth bound, then the node is not expanded and the algorithm backtracks. If a solution is not found, then the entire state space is searched again using a larger depth bound. This technique is called iterative deepening depth-first search (ID-DFS). Note that this method is guaranteed to find a solution path with the fewest edges. However, it is not guaranteed to find a least-cost path.
Iterative deepening A* (IDA*) is a variant of ID-DFS. IDA* uses the l-values of nodes to bound depth (recall from Section 11.1 that for node x, l(x) = g(x) + h(x)). IDA* repeatedly performs cost-bounded DFS over the search space. In each iteration, IDA* expands nodes depth-first. If the l-value of the node to be expanded is greater than the cost bound, then IDA* backtracks. If a solution is not found within the current cost bound, then IDA* repeats the entire depth-first search using a higher cost bound. In the first iteration, the cost bound is set to the l-value of the initial state s. Note that since g(s) is zero, l(s) is equal to h(s). In each subsequent iteration, the cost bound is increased. The new cost bound is equal to the minimum l-value of the nodes that were generated but could not be expanded in the previous iteration. The algorithm terminates when a goal node is expanded. IDA* is guaranteed to find an optimal solution if the heuristic function is admissible. It may appear that IDA* performs a lot of redundant work across iterations. However, for many problems the redundant work performed by IDA* is minimal, because most of the work is done deep in the search space.
Example 11.5 Depth-first search: the 8-puzzle
Figure 11.4 shows the execution of depth-first search for solving the 8-puzzle problem. The search starts at the initial configuration. Successors of this state are generated by applying possible moves. During each step of the search algorithm a new state is selected, and its successors are generated. The DFS algorithm expands the deepest node in the tree. In step 1, the initial state A generates states B and C. One of these is selected according to a predetermined criterion. In the example, we order successors by applicable moves as follows: up, down, left, and right. In step 2, the DFS algorithm selects state B and generates states D, E, and F. Note that the state D can be discarded, as it is a duplicate of the parent of B. In step 3, state E is expanded to generate states G and H. Again G can be discarded because it is a duplicate of B. The search proceeds in this way until the algorithm backtracks or the final configuration is generated.
Figure 11.4. States resulting from the first three steps of depth-first search applied to an instance of the 8-puzzle.
In each step of the DFS algorithm, untried alternatives must be stored. For example, in the 8-puzzle problem, up to three untried alternatives are stored at each step. In general, if m is the amount of storage required to store a state, and d is the maximum depth, then the total space requirement of the DFS algorithm is O (md). The state-space tree searched by parallel DFS can be efficiently represented as a stack. Since the depth of the stack increases linearly with the depth of the tree, the memory requirements of a stack representation are low.
There are two ways of storing untried alternatives using a stack. In the first representation, untried alternates are pushed on the stack at each step. The ancestors of a state are not represented on the stack. Figure 11.5(b) illustrates this representation for the tree shown in Figure 11.5(a). In the second representation, shown in Figure 11.5(c), untried alternatives are stored along with their parent state. It is necessary to use the second representation if the sequence of transformations from the initial state to the goal state is required as a part of the solution. Furthermore, if the state space is a graph in which it is possible to generate an ancestor state by applying a sequence of transformations to the current state, then it is desirable to use the second representation, because it allows us to check for duplication of ancestor states and thus remove any cycles from the state-space graph. The second representation is useful for problems such as the 8-puzzle. In Example 11.5, using the second representation allows the algorithm to detect that nodes D and G should be discarded.
Figure 11.5. Representing a DFS tree: (a) the DFS tree; successor nodes shown with dashed lines have already been explored; (b) the stack storing untried alternatives only; and (c) the stack storing untried alternatives along with their parent. The shaded blocks represent the parent state and the block to the right represents successor states that have not been explored.
11.2.2 Best-First Search Algorithms
Best-first search (BFS) algorithms can search both graphs and trees. These algorithms use heuristics to direct the search to portions of the search space likely to yield solutions. Smaller heuristic values are assigned to more promising nodes. BFS maintains two lists: open and closed. At the beginning, the initial node is placed on the open list. This list is sorted according to a heuristic evaluation function that measures how likely each node is to yield a solution. In each step of the search, the most promising node from the open list is removed. If this node is a goal node, then the algorithm terminates. Otherwise, the node is expanded. The expanded node is placed on the closed list. The successors of the newly expanded node are placed on the open list under one of the following circumstances: (1) the successor is not already on the open or closed lists, and (2) the successor is already on the open or closed list but has a lower heuristic value. In the second case, the node with the higher heuristic value is deleted.
A common BFS technique is the A* algorithm. The A* algorithm uses the lower bound function l as a heuristic evaluation function. Recall from Section 11.1 that for each node x , l(x) is the sum of g(x) and h(x). Nodes in the open list are ordered according to the value of the l function. At each step, the node with the smallest l-value (that is, the best node) is removed from the open list and expanded. Its successors are inserted into the open list at the proper positions and the node itself is inserted into the closed list. For an admissible heuristic function, A* finds an optimal solution.
The main drawback of any BFS algorithm is that its memory requirement is linear in the size of the search space explored. For many problems, the size of the search space is exponential in the depth of the tree expanded. For problems with large search spaces, memory becomes a limitation.
Example 11.6 Best-first search: the 8-puzzle
Consider the 8-puzzle problem from Examples 11.2 and 11.4. Figure 11.6 illustrates four steps of best-first search on the 8-puzzle. At each step, a state x with the minimum l-value (l(x) = g(x) + h(x)) is selected for expansion. Ties are broken arbitrarily. BFS can check for a duplicate nodes, since all previously generated nodes are kept on either the open or closed list.
Figure 11.6. Applying best-first search to the 8-puzzle: (a) initial configuration; (b) final configuration; and (c) states resulting from the first four steps of best-first search. Each state is labeled with its h-value (that is, the Manhattan distance from the state to the final state).