December 1999 Programmer's Challenge

Costas Arrays

Mail solutions to: progchallenge@mactech.com
Due Date: 11:59pm ET, Thursday, 9 December 1999

A Costas array of order N is an NxN array of 1s and 0s satisfying two constraints. First, the array must have exactly N 1s and N*(N-1) 0s, with exactly one 1 in each row and column. Second, no two lines between pairs of 1s may have exactly the same length and the same slope. So, for example, there are exactly 12 Costas arrays of order 4:

		1000  0001  0010  0010  1000  0001
		0010  1000  1000  0100  0001  0100 
		0001  0010  0100  0001  0100  1000 
		0100  0100  0001  1000  0010  0010 
 
		0100  0100  1000  0001  0100  0010
		0010  0001  0100  0010  1000  0001
		1000  0010  0001  1000  0010  0100
		0001  1000  0010  0100  0001  1000

So why would one care about Costas arrays? Because of the asymmetries imposed by the two constraints, Costas arrays make ideal waveforms for certain sensor applications, reducing ambiguity in interpreting radar and sonar returns. The mathematics are interesting in other ways. For example, according to the CRC Standard Mathematical Tables, the number C(n) of Costas arrays of order n increases as a function of n until n==16, after which it decreases, at least until n==23:

n	C(n)	n	C(n)
1	1	2	2
3	4	4	12
5	40	6	116
7	200	8	444
9	760	10	2160
11	4368	12	7852
13	12828	14	17252
15	19612	16	21104
17	18276	18	15096
19	10240	20	6464
21	3536	22	2052
23	872	24	>=1

The number of Costas arrays of order 24 and greater is the subject of continuing research. Your Challenge is to aid these researchers by writing efficient code to enumerate Costas arrays.

The prototype for the code you should write is:

#if defined(__cplusplus)
extern "C" {
#endif

long /* number of arrays */ EnumerateCostas(
  int n,                /* enumerate Costas arrays of order n */
  long *costasArrays    /* preallocated storage for returning your results */
        /* row r of array k is in costasArrays[k*n + r], r,k are origin 0 */
);


#if defined(__cplusplus)
}
#endif

Your EnumerateCostas routine will be asked to enumerate all Costas arrays of order n and return the results in the preallocated costasArrays. Each cell of an array is represented by one bit. The bits for row r of the k-th Costas array are to be returned in costasArrays[k*n + r], r=0..n-1, k>=0. The cell representing column c, c=0..n-1, is the bit in 1<<c. EnumerateCostas must produce all valid Costas arrays, with no duplicates, and return the number of arrays produced.

The winner will be the solution that correctly enumerates the Costas arrays in the minimum time.

This will be a native PowerPC Challenge, using the latest CodeWarrior environment. Solutions may be coded in C, C++, Pascal, or Java.

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Test code for this Challenge is available.

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