Very interesting. From your tests the algorithm converges in 1.47.sqrt(N) with dp=0: but 2.sqrt(N) with dp=3: Assuming you used the same jumps, there is only one explanation: the algorithm produces a useful collision in 1.47*sqrt(n) steps but you take a long time to detect it because the walks are too short, they collapse in a loop before to reach a DP. It is not a problem of bad collisions between different tames or between different wilds but "between" the same tame / the same wild (loops). Then a wild and a tame meet frequently (first collision after 1.47.sqrt(n)) but they die immediately after. This issue can be fixed (in my opinion), because these bad collisions (I call them "bad collisions 2" = collision with itself = loop) are distinguishable from the others and you can lower them without increasing the other ones. With DP = 3 means that the lenght of many walks are less than 8!!! It takes +36% (1.47 -> 2) to produce a collision followed by a DP!!! You have to change your jumps. But choose them in a "intelligent" way, I think it is not too difficult. 'Intelligent' means that a man can't do this, let the probability do it. I think that the main problem is that you want to choose the jumps by yourself. Why? You should study separately a sequence of jumps such that the probability to come back before 2^DP steps is low. This study depends only from jumps, DP and m (the average jump), not from points/coordinates/N! For example, with these jumps [+-1,+-2,+-4,+-8,+-16]  you try to generate many sequences (starting from 0). If the number of sequences with lenght < 2^DP (or < k*2^DP) is too high, change randomly the jumps and retry. Lenght of sequence: number of steps before to get back to 0. Deal simply with integer numbers. This setting phase should be done automatically at the start of the program. In this way for each DP you will have configured the jumps in order to get a lower probability of generating loops.
Very interesting. From your tests the algorithm converges in 1.47.sqrt(N) with dp=0: but 2.sqrt(N) with dp=3: Assuming you used the same jumps, there is only one explanation: the algorithm produces a useful collision in 1.47*sqrt(n) steps but you take a long time to detect it because the walks are too short, they collapse in a loop before to reach a DP. It is not a problem of bad collisions between different tames or between different wilds but "between" the same tame / the same wild (loops). Then a wild and a tame meet frequently (first collision after 1.47.sqrt(n)) but they die immediately after. This issue can be fixed (in my opinion), because these bad collisions (I call them "bad collisions 2" = collision with itself = loop) are distinguishable from the others and you can lower them without increasing the other ones. With DP = 3 means that the lenght of many walks are less than 8!!! It takes +36% (1.47 -> 2) to produce a collision followed by a DP!!! You have to change your jumps. But choose them in a "intelligent" way, I think it is not too difficult. 'Intelligent' means that a man can't do this, let the probability do it. I think that the main problem is that you want to choose the jumps by yourself. Why? You should study separately a sequence of jumps such that the probability to come back before 2^DP steps is low. This study depends only from jumps, DP and m (the average jump), not from points/coordinates/N! For example, with these jumps [+-1,+-2,+-4,+-8,+-16]  you try to generate many sequences (starting from 0). If the number of sequences with lenght < 2^DP (or < k*2^DP) is too high, change randomly the jumps and retry. Lenght of sequence: number of steps before to get back to 0. Deal simply with integer numbers. This setting phase should be done automatically at the start of the program. In this way for each DP you will have configured the jumps in order to get a lower probability of generating loops.

Very interesting. From your tests the algorithm converges in 1.47.sqrt(N) with dp=0: but 2.sqrt(N) with dp=3: Assuming you used the same jumps, there is only one explanation: the algorithm produces a useful collision in 1.47*sqrt(n) steps but you take a long time to detect it because the walks are too short, they collapse in a loop before to reach a DP. It is not a problem of bad collisions between different tames or between different wilds but "between" the same tame / the same wild (loops). Then a wild and a tame meet frequently (first collision after 1.47.sqrt(n)) but they die immediately after. This issue can be fixed (in my opinion), because these bad collisions (I call them "bad collisions 2" = collision with itself = loop) are distinguishable from the others and you can lower them without increasing the other ones. With DP = 3 means that the lenght of many walks are less than 8!!! It takes +36% (1.47 -> 2) to produce a collision followed by a DP!!! You have to change your jumps. But choose them in a "intelligent" way, I think it is not too difficult. 'Intelligent' means that a man can't do this, let the probability do it. I think that the main problem is that you want to choose the jumps by yourself. Why? You should study separately a sequence of jumps such that the probability to come back before 2^DP steps is low. This study depends only from jumps, DP and m (the average jump), not from points/coordinates/N! For example, with these jumps [+-1,+-2,+-4,+-8,+-16]  you try to generate many sequences (starting from 0). If the number of sequences with lenght < 2^DP (or < k*2^DP) is too high, change randomly the jumps and retry. Lenght of sequence: number of steps before to get back to 0. Deal simply with integer numbers. This setting phase should be done automatically at the start of the program. In this way for each DP you will have configured the jumps in order to get a lower probability of generating loops.

Very interesting. From your tests the algorithm converges in 1.47.sqrt(N) with dp=0: but 2.sqrt(N) with dp=3: Assuming you used the same jumps, there is only one explanation: the algorithm produces a useful collision in 1.47*sqrt(n) steps but you take a long time to detect it because the walks are too short, they collapse in a loop before to reach a DP. It is not a problem of bad collisions between different tames or between different wilds but "between" the same tame / the same wild (loops). Then a wild and a tame meet frequently (first collision after 1.47.sqrt(n)) but they die immediately after. This issue can be fixed (in my opinion), because these bad collisions (I call them "bad collisions 2" = collision with itself = loop) are distinguishable from the others and you can lower them without increasing the other ones. With DP = 3 means that the lenght of many walks are less than 8!!! It takes +36% (1.47 -> 2) to produce a collision followed by a DP!!! You have to change your jumps. But choose them in a "intelligent" way, I think it is not too difficult. 'Intelligent' means that a man can't do this, let the probability do it. I think that the main problem is that you want to choose the jumps by yourself. Why? You should study separately a sequence of jumps such that the probability to come back before 2^DP steps is low. This study depends only from jumps, DP and m (the average jump), not from points/coordinates/N! For example, with these jumps [+-1,+-2,+-4,+-8,+-16]  you try to generate many sequences (starting from 0). If the number of sequences with lenght < 2^DP (or < k*2^DP) is too high, change randomly the jumps and retry. Lenght of sequence: number of steps before to get back to 0. Deal simply with integer numbers. This setting phase should be done automatically at the start of the program. In this way for each DP you will have configured the jumps in order to get a lower probability of generating loops.