I am working on solving a recent recreational mathematical problem on Project Euler <http://projecteuler.net> . I have a solution, which works fine for small N up to 10^5 but it takes too long to compute for the actual problem, where N is of the order 2*10^7. The problem is nested loops, and I am hoping to avoid one level in the loop by using clever numpy magic (as I have done often before). However, there is one step, which I cannot figure out how to do using numpy operations alone, and I am hoping for some help
The subproblem is that I have in principle k = 1, ..., N sets of boolean arrays f_k and g_k each of length N. For each k I need to find the number of elements i where both f_k[i] and g_k[i] are True and sum that up over all N values of k. A problem of the order 4*10^14 if I just do it brute force. This takes way too long (there is a one minute rule). However, after a lot of thinking and by using some properties of the f_k and g_k I have managed to construct using only pure numpy function and only a single loop over k, arrays f_k_changes_at g_k_changes_at which contain the indices i at which the functions change it boolean value from True to False or False to True. It so happens that the number of changes is only a small fraction of N, the fraction decreases with larger N, so the size of these changes_at arrays contains perhaps only 1000 elements instead of 10000000 for each k, a significant reduction of complexity. Now, my problem is to figure out for how many values of i both f_k and g_k are True given the changes_at arrays. As this may be a little hard to understand here is a specific example of how these arrays can look like for k = 2 and N = 150 f_2_changes_at = [ 2 3 39 41 58 59 65 66 93 102 145] g_2_changes_at = [ 2 94 101 146 149] with the boundary condition that f_2[0] = g_2[0] = False Which expands to i f_2 g_2 f_2 and g_2 0 F F F 1 F F F <- 2 T T T <- 3 F T F 4 F T F ... 38 F T F <- 39 T T T 40 T T T <- 41 F T F 42 F T F ... 57 F T F <- 58 T T T <- 59 F T F 60 F T F ... 64 F T F <- 65 T T T <- 66 F T F ... 92 F T F <- 93 T T T <- 94 T F F ... 100 T F F <- 101 T T T <- 102 F T F ... 144 F T F <- 145 T T T <- 146 T F F 147 T F F 148 T F F <- 149 T T T <- With the sum of elements fulfilling the condition being (see arrows) (2 - 1) + (40 - 38) + (58 - 57) + (65 - 64) + (93 - 92) + (101 - 100) + (145 - 144) + (149 - 148) = 1 + 2 + 1 + 1 + 1 + 1 + 1 + 1 = 9 So, is there a numpy recipe for doing the equivalent process without expanding it into the full arrays? I have tried looping over each element in the changes_at arrays and build up the sums, but that is too inefficient as I then have an inner for loop containing conditional branching code Thanks in advance, Slaunger -- View this message in context: http://numpy-discussion.10968.n7.nabble.com/Is-there-a-pure-numpy-recipe-for-this-tp37077.html Sent from the Numpy-discussion mailing list archive at Nabble.com. _______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
