Cuckoo Cycle Speed Challenge; $2500 in bounties

[tromp]

For each level of the breadth, use one O(N) pass through the entire edge set by generating the edges and matching them against the leading edge nodes in the BFS tree.

This description alone is effective for edge trimming, but building on this for directly detecting 42-cycles requires more careful specification of how to handle the BFS graph representation and what to do about cycles in it.

Hmm, seems we were overlooking the obvious.

You can just feed all the edges generated in each pass (having one endpoint already present in subset P or as a key in cuckoo_hash) to my cycle finder (lines 357-390 of cuckoo_miner.h, tweaked to ignore duplicates/2-cycles). It doesn't care where the edges come from or how they are located between successive D_i layers. It will even happily report cycles that bounce back and forth multiple times between layers.

This should be relatively easy to code...

Having a rough version done, it looks like the slowdown for saving a factor k in memory is roughly k times L, where L is cycle length. So, to use only half the memory of the edge-trimming implementation causes two orders of magnitude slowdown. If this is the best one can do, then Cuckoo Cycle is still memory-hard in a practical sense, with cycle-length becoming a memory-hardness parameter.

I'll have more detailed numbers after fine-tuning the implementation...

[aminorex]

.

[Hashbender]

I remember seeing this algo suggested @ Nyancoin's reddit several moths ago. Thought nobody would give it a try, and now there is even a nice bounty to make it work..!

[tromp]

This miner will be available as tomato_miner.h / tomato_miner.cpp
I will experiment with this some more before updating the TMTO bounty...

The tomato miner is now up, and bounty miner is restored as a copy of cuckoo miner.

The TMTO bounty in the OP has been updated, allowing an extra factor 10 of slowdown.

Below is the slightly improved output of the tomato miner, shown running almost twice as fast by using 2 threads.

Happy continued bounty hunting!

-John

Code:

> time ./tomato25 -t 2 -h 157
Looking for 42-cycle on cuckoo25("157") with 50% edges, 1/64 memory, 24/512 parts, 2 threads
Using 4MB node memory.
vpart 0 depth 21 load 78%
   4-cycle found at 1:5
   4-cycle found at 1:6
   4-cycle found at 1:7
   4-cycle found at 1:8
   4-cycle found at 1:9
   4-cycle found at 1:10
   4-cycle found at 1:11
   4-cycle found at 1:12
   4-cycle found at 1:13
   4-cycle found at 1:14
   4-cycle found at 1:15
   4-cycle found at 1:16
   4-cycle found at 1:17
   4-cycle found at 1:18
   4-cycle found at 1:19
   4-cycle found at 1:20
vpart 1 depth 21 load 78%
vpart 2 depth 21 load 78%
vpart 3 depth 21 load 75%
vpart 4 depth 21 load 77%
vpart 5 depth 21 load 76%
vpart 6 depth 21 load 78%
vpart 7 depth 21 load 76%
vpart 8 depth 21 load 77%
vpart 9 depth 21 load 79%
vpart 10 depth 21 load 78%
vpart 11 depth 21 load 76%
vpart 12 depth 21 load 76%
vpart 13 depth 21 load 77%
vpart 14 depth 21 load 76%
vpart 15 depth 21 load 78%
vpart 16 depth 21 load 78%
vpart 17 depth 21 load 78%
vpart 18 depth 21 load 78%
vpart 19 depth 21 load 75%
vpart 20 depth 21 load 77%
vpart 21 depth 21 load 75%
vpart 22 depth 21 load 76%
  42-cycle found at 1:20
vpart 23 depth 21 load 79%
Solution 223e7 56b1b 86b2f cb0a2 106629 233b7e 253ee1 290e4e 2e15cc 2efe4b 32bc8e 356dfd 35ef96 3dd854 534d37 57ee85 5abaa1 5d7d49 60ca66 662933 718651 78554b 7d8e01 7f7360 886c55 8a6448 8e4fd2 9674a2 98e431 b00d71 b16050 b1b561 b362d4 b9e539 ca3ab0 cb4f28 d2a53b d53bc3 e213cc e40bf1 ed5b2a faef36

real    7m5.287s
user    12m27.616s
sys     0m2.345s

[tromp]

This miner will be available as tomato_miner.h / tomato_miner.cpp
I will experiment with this some more before updating the TMTO bounty...

Weird, the post I was quoting has disappeared entirely.

In that post I showed a comparison between the reference and tmto implementation at memory parity, which showed a factor 70 slowdown and a factor 1-1/e reduced solution finding rate (due to sampling a fraction 1/42 of all vertices as starting points for the depth 21 breadth first search).
Discounting for the lower solution finding rate, the tmto implementation was 111 times slower.

For completeness, I include below the output of the reference implementation running on 2 threads,
showing a factor 61 difference (rather than the 70 single threaded).

-John

Code:

> time ./cuckoo25.1 -t 2  -h 157
Looking for 42-cycle on cuckoo25("157") with 50% edges, 11 trims, 2 threads
Using 2MB edge and 2MB node memory.
initial load 6400%
round 1 part U0 load 5222%
round 1 part U1 load 4045%
round 1 part V0 load 2970%
round 1 part V1 load 1895%
round 2 part U0 load 1508%
round 2 part U1 load 1121%
round 2 part V0 load 934%
round 2 part V1 load 746%
round 3 part U0 load 641%
round 3 part U1 load 535%
round 3 part V0 load 469%
round 3 part V1 load 403%
round 4 part U0 load 359%
round 4 part U1 load 315%
round 4 part V0 load 284%
round 4 part V1 load 253%
round 5 part U0 load 230%
round 5 part U1 load 208%
round 5 part V0 load 191%
round 5 part V1 load 174%
round 6 part U0 load 161%
round 6 part U1 load 147%
round 6 part V0 load 137%
round 6 part V1 load 127%
round 7 part U0 load 118%
round 7 part U1 load 110%
round 7 part V0 load 103%
round 7 part V1 load 96%
round 8 part U0 load 91%
round 8 part U1 load 85%
round 8 part V0 load 80%
round 8 part V1 load 76%
round 9 part U0 load 72%
round 9 part U1 load 68%
round 9 part V0 load 64%
round 9 part V1 load 61%
round 10 part U0 load 58%
round 10 part U1 load 55%
round 10 part V0 load 53%
round 10 part V1 load 50%
round 11 part U0 load 48%
round 11 part U1 load 46%
round 11 part V0 load 44%
round 11 part V1 load 42%
   4-cycle found at 1:90%
 166-cycle found at 0:98%
  42-cycle found at 1:98%
  76-cycle found at 1:99%
Solution 223e7 56b1b 86b2f cb0a2 106629 233b7e 253ee1 290e4e 2e15cc 2efe4b 32bc8e 356dfd 35ef96 3dd854 534d37 57ee85 5abaa1 5d7d49 60ca66 662933 718651 78554b 7d8e01 7f7360 886c55 8a6448 8e4fd2 9674a2 98e431 b00d71 b16050 b1b561 b362d4 b9e539 ca3ab0 cb4f28 d2a53b d53bc3 e213cc e40bf1 ed5b2a faef36

real    0m6.957s
user    0m10.709s
sys     0m0.077s

[tromp]

In that post I showed a comparison between the reference and tmto implementation at memory parity, which showed a factor 70 slowdown and a factor 1-1/e reduced solution finding rate (due to sampling a fraction 1/42 of all vertices as starting points for the depth 21 breadth first search).
Discounting for the lower solution finding rate, the tmto implementation was 111 times slower.

I tried another approach, which turns out to be superior. First of all, assume that the graph we're working on has a unique 42 cycle, and we hope to find it by sampling a small fraction of nodes, and exploring their local neighbourhood with a depth-limited breadth-first-search (BFS).

Instead of doing a depth 21 BFS , which (roughly) requires starting from a node on the 42-cycle, we can do a much deeper search, say depth 42. Then we'll find the 42-cycle as long as we start from any node with distance at most 21 to the 42-cycle, which seems much more likely.

Apart from doubling the BFS depth, there is another difference. In the former approach, we only care about nodes on the cycle, and can eliminate any that are known to be off the cycle, such as those with only one incident edge. In the latter approach, we cannot eliminate these. So the BFS tree will be much larger in size relative to the starting set, not only due to doubled depth, but to less elimination as well.

To keep space usage down, we thus need to shrink the starting sets even more.
I ran some experiments which indicate that the increased likelihood of finding the cycle more than makes up for having smaller starting sets.

The updated tomato_miner.h can now use k times less memory with a (discounted) slowdown of roughly 25*k. Further details will appear in the whitepaper, in a week or two.

With this progress on the time-memory trade-off, the partial bounty payout, and the relaxation of the slowdown, I'm reducing the TMTO bounty from $1000 to $500.