7.13. Implementing Knight’s Tour

The search algorithm we will use to solve the knight’s tour problem is called depth-first search (DFS). Whereas the breadth-first search algorithm builds a search tree one level at a time, a depth-first search creates a search tree by exploring one branch of the tree as deeply as possible. In this section we will look at two algorithms that implement a depth-first search. The first algorithm we will look at specifically solves the knight’s tour problem by explicitly forbidding a node to be visited more than once. The second implementation is more general, but allows nodes to be visited more than once as the tree is constructed. The second version is used in subsequent sections to develop additional graph algorithms.

The depth-first exploration of the graph is exactly what we need in order to find a path through 64 vertices (one for each square on the chessboard) and 63 edges. We will see that when the depth-first search algorithm finds a dead end (a place in the graph where there are no more moves possible) it backs up the tree to the next deepest vertex that allows it to make a legal move.

The knight_tour function shown in Listing 3 takes four parameters: n, the current depth in the search tree; path, a list of vertices visited up to this point; u, the vertex in the graph we wish to explore; and limit, the number of nodes in the path. The knight_tour function is recursive. When the knight_tour function is called, it first checks the base case condition. If we have a path that contains 64 vertices, we return from knight_tour with a status of True, indicating that we have found a successful tour. If the path is not long enough, we continue to explore one level deeper by choosing a new vertex to explore and calling knight_tour recursively for that vertex.

Listing 3

from pythonds3.graphs import Graph


def knight_tour(n, path, u, limit):
    u.color = "gray"
    path.append(u)
    if n < limit:
        neighbors = sorted(list(u.get_neighbors()))
        i = 0
        done = False
        while i < len(neighbors) and not done:
            if neighbors[i].color == "white":
                done = knight_tour(n + 1, path, neighbors[i], limit)
            i = i + 1
        if not done:  # prepare to backtrack
            path.pop()
            u.color = "white"
    else:
        done = True
    return done

DFS also uses colors to keep track of which vertices in the graph have been visited. Unvisited vertices are colored white, and visited vertices are colored gray. If all neighbors of a particular vertex have been explored and we have not yet reached our goal length of 64 vertices, we have reached a dead end and must backtrack. Backtracking happens when we return from knight_tour with a status of False. In the breadth-first search we used a queue to keep track of which vertex to visit next. Since depth-first search is recursive, we are implicitly using a stack to help us with our backtracking. When we return from a call to knight_tour with a status of False, in line 11, we remain inside the while loop and look at the next vertex in neighbors.

Let’s look at a simple example of knight_tour (Listing 3) in action. You can refer to the figures below to follow the steps of the search. For this example we will assume that the call to the get_neighbors method on line 6 orders the nodes in alphabetical order. We begin by calling knight_tour(0, path, A, 6).

../_images/ktdfsa.png

Figure 3: Start with node A

../_images/ktdfsb.png

Figure 4: Explore B

../_images/ktdfsc.png

Figure 5: Node C is a dead end

../_images/ktdfsd.png

Figure 6: Backtrack to B

../_images/ktdfse.png

Figure 7: Explore D

../_images/ktdfsf.png

Figure 8: Explore E

../_images/ktdfsg.png

Figure 9: Explore F

../_images/ktdfsh.png

Figure 10: Finish

knight_tour starts with node A in Figure 3. The nodes adjacent to A are B and D. Since B is before D alphabetically, DFS selects B to expand next as shown in Figure 4. Exploring B happens when knight_tour is called recursively. B is adjacent to C and D, so knight_tour elects to explore C next. However, as you can see in Figure 5 node C is a dead end with no adjacent white nodes. At this point we change the color of node C back to white. The call to knight_tour returns a value of False. The return from the recursive call effectively backtracks the search to vertex B (see Figure 6). The next vertex on the list to explore is vertex D, so knight_tour makes a recursive call moving to node D (see Figure 7). From vertex D on, knight_tour can continue to make recursive calls until we get to node C again (see Figure 8, Figure 9, and Figure 10). However, this time when we get to node C the test n < limit fails so we know that we have exhausted all the nodes in the graph. At this point we can return True to indicate that we have made a successful tour of the graph. When we return the list, path has the values [A, B, D, E, F, C], which is the order we need to traverse the graph to visit each node exactly once.

Figure 11 shows you what a complete tour around an \(8 \times 8\) board looks like. There are many possible tours; some are symmetric. With some modification you can make circular tours that start and end at the same square.

../_images/completeTour.png

Figure 11: A Complete Tour of the Board Found by knight_tour

You have attempted of activities on this page