二進代数学アルゴリズム

The objective of the problem was to count the number of (X,Y) pairs such that the value X XOR Y was less than or equal to C, while X<=A and Y<=B, where A, B, and C are givens.
The author attempted to address this using dynamic programming on the bit index of the outcome of X^Y (where ^ means XOR), moving from left to right. However, two problems were noted: firstly correctness (for cases of four bits or more, the number of pairs found was different from the expected outcome) and secondly complexity (for cases where A, B, and C were 10^9, the algorithm ran indefinitely). It is unclear whether these two issues can be remedied. For 32 bits, any branch factor of 2 or greater is untenable (2^32 is 4096*10^6).
An oversight in the case of correctness was the failure to search some X and Y bits such that, while bit1^bit2>cBit, the bits left of this bit in X^Y caused the overall outcome to be less than C regardless. An extra boolean, cEq needs to be passed to recursive calls to determine whether or not this is the case. This actually increases the branch factor of the depth first search tree and thus increases complexity. The solutions in the TopCoder match editorial might also be consulted for a different (likely to be more efficient) algorithm.
By passing boolean values for aEq, bEq, and cEq referring to the fact that the left bits in X were equal to A, the left bits in Y were equal to B, and the left bits of X^Y were equal to C, I solved test cases of up to 10 bits efficiently and correctly (with about 2500 recursive method calls). However, large cases with A on the order of 10^9 (about 30 bits) caused the code to run for several minutes, although the answer was correct. The problem seems to be in constraining the branching. One modification is, given that the A, B, and C constraints have been released, we know the number of answers is (2^(nBits to be considered))^2. However, if the C constraint is not released, it is unclear how to compute the number of X,Y pairs, even if the A and B constraints are released.
The search can be truncated in multiple fashions. The first, as mentioed above, is the number-theoretical approach when no constraints on x, y, or x^y exist. However, this is effective only in branches of the search tree where !aEq&&!bEq&&!cEq hold. The more effective restriction on search is via memoization, where a table mapping the (currBit,aEq,bEq,cEq) tuple to a 64-bit long integer representing the number of pairs found in that case is maintained. If a recursive call is made with parameters matching the table entry, no further tree traversal is needed and the table's entry returned. The number of recursive calls is thus bounded at O(30*2^3)=O(232), since there are at most 30 bits and the other values are boolean.