Shirokuro: A Backtracking ApproachBenjamin Bush

Faculty Advisors: Dr. Russ Abbott, Dr. Gary BrookfieldDepartment of Computer Science, Department of MathematicsCollege of Engineering, Computer Science, and Technology,

College of Natural and Social SciencesCalifornia State University, Los Angeles

Funding provided by National Science Foundation grant #HRD-0331537 and the CSU Chancellor's Office

IntroductionWe strive to develop an algorithm that can find the solutions to a given Shirokuro puzzle in a reasonable amount of time. We employ a simple yet powerful technique known as “backtracking,” which incrementally builds candidate solutions and abandons each partial candidate (backtracks) as soon as it determines that the candidate cannot possibly be extended to create a valid solution.

Dead End DetectionBelow are two nontrivial Shirokuro configurations that our backtracking algorithm currently detects as dead ends.

This above configuration, known as the diamond, leads to the separation of at least 1 pair of disks. To see this, note that any connection that is made between the two black disks invariably forms a loop which completely encapsulates one of the white disks, making any connection between the two white disks impossible.

How To Play ShirokuroFill in every empty grid space so that the following conditions are satisfied:

No two-by-two region of the grid contains four disks of the same color.

Each pair of like-colored disks is connected via a chain which travels horizontally or vertically through disks of the same color as the pair.example solution

Backtracking essentially amounts to a depth first search through the tree of all partial candidate solutions to a problem. Each time a partial candidate solution is determined to be a dead end, a portion of the tree is pruned, resulting in a smaller search space. Good backtracking algorithms can efficiently detect dead ends at nodes that are close to the root, so that the pruned subtrees are as large as possible.

If two black disks lie along the boundary of the grid, they must be connected by a chain of black disks, all of which also lie along the boundary. To see why, consider the alternative: suppose a pair of black disks and a pair of white disks are arranged along the boundary so that such a chain would be impossible, as illustrated above (left). Then any connection drawn between the two black disks would bisect the grid, isolating the pair of white discs from one another (right).

Detecting Isolated GroupsSince no colored disk may become isolated from others of it’s kind, the early and efficient detection of isolated groups of disks is essential for achieving a reasonable runtime. To this end, we have developed a technique which involves building and maintaining a pair of spanning trees.

S represents the current state of the Shirokuro solving process. Swhite and Sblack are a modified version of S formed by filling every empty cell with white or black disks, respectively. The tree shown in Swhite spans the region that the white disks in S may potentially connect with.

SSwhite Sblack

Theorem: A graph G is connected if and only if it has a spanning tree.

Thus, if our algorithm finds that there does not exist a tree spanning the white disks in Swhite, we may conclude that there are regions of the grid that are isolated from each other, as far as the white disks are concerned. A similar situation holds for Sblack.

One of the advantages of using spanning trees is the ease with which they can be restructured when disturbed. In figure above, the spanning tree connecting the black disks is broken into 3 pieces when one of it’s nodes is removed. The smaller pieces are searched and found to contain nodes that can be reconnected (starred). These nodes become the new roots of their respective subtrees in the final configuration.

Dynamic Variable OrderingIn most backtracking algorithms, the order in which variables are instantiated is not fixed, but instead changes dynamically by following several variable ordering heuristics. One important heuristic, the degree heuristic, states that those variables that are involved in a larger number of constraints should be instantiated first.

In our future work, we will be exploring the role that our spanning trees may have in dynamic variable ordering.

In the figure to the left, the cell in the center should be instantiated first because it is involved in 4 “two-by-two” constraints, 1 “boundary” constraint, and 1 global constraint for a total of 5 constraints.

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