Snake-in-the-box

The snake-in-the-box problem in graph theory and computer science deals with finding a certain kind of path along the edges of a hypercube. This path starts at one corner and travels along the edges to as many corners as it can reach. After it gets to a new corner, the previous corner and all of its neighbors must be marked as unusable. The path should never travel to a corner which has been marked unusable.

Drawing of a snake in a three-dimensional hypercube.

In other words, a snake is a connected open path in the hypercube where each node connected with path, with the exception of the head (start) and the tail (finish), it has exactly two neighbors that are also in the snake. The head and the tail each have only one neighbor in the snake. The rule for generating a snake is that a node in the hypercube may be visited if it is connected to the current node and it is not a neighbor of any previously visited node in the snake, other than the current node.

In graph theory terminology, this is called finding the longest possible induced path in a hypercube; it can be viewed as a special case of the induced subgraph isomorphism problem. There is a similar problem of finding long induced cycles in hypercubes, called the coil-in-the-box problem.

The snake-in-the-box problem was first described by Kautz (1958), motivated by the theory of error-correcting codes. The vertices of a solution to the snake or coil in the box problems can be used as a Gray code that can detect single-bit errors. Such codes have applications in electrical engineering, coding theory, and computer network topologies. In these applications, it is important to devise as long a code as is possible for a given dimension of hypercube. The longer the code, the more effective are its capabilities.

Finding the longest snake or coil becomes notoriously difficult as the dimension number increases and the search space suffers a serious combinatorial explosion. Some techniques for determining the upper and lower bounds for the snake-in-the-box problem include proofs using discrete mathematics and graph theory, exhaustive search of the search space, and heuristic search utilizing evolutionary techniques.

Known lengths and bounds

The maximum length for the snake-in-the-box problem is known for dimensions one through eight; it is

1, 2, 4, 7, 13, 26, 50, 98 (sequence A099155 in the OEIS).

Beyond that length, the exact length of the longest snake is not known; the best lengths found so far for dimensions nine through thirteen are

190, 370, 712, 1373, 2687.

For cycles (the coil-in-the-box problem), a cycle cannot exist in a hypercube of dimension less than two. The maximum lengths of the longest possible cycles are

0, 4, 6, 8, 14, 26, 48, 96 (sequence A000937 in the OEIS).

Beyond that length, the exact length of the longest cycle is not known; the best lengths found so far for dimensions nine through thirteen are

188, 366, 692, 1344, 2594.

Doubled coils are a special case: cycles whose second half repeats the structure of their first half, also known as symmetric coils. For dimensions two through seven the lengths of the longest possible doubled coils are

4, 6, 8, 14, 26, 46.

Beyond that, the best lengths found so far for dimensions eight through thirteen are

94, 186, 362, 662, 1222, 2354.

For both the snake and the coil in the box problems, it is known that the maximum length is proportional to 2n for an n-dimensional box, asymptotically as n grows large, and bounded above by 2n − 1. However the constant of proportionality is not known, but is observed to be in the range 0.3 - 0.4 for current best known values.[1]

Notes
  1. For asymptotic lower bounds, see Evdokimov (1969), Wojciechowski (1989), and Abbot & Katchalski (1991). For upper bounds, see Douglas (1969), Deimer (1985), Solov'eva (1987), Abbot & Katchalski (1988), Snevily (1994), and Zémor (1997).

References
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