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Jansen and Wegener (Algorithmica 2002) proved an upper bound of\n \n $$O(\\textrm{poly}(n) + 4^k\/p_c)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n for the (\n \n $$\\mu $$<\/jats:tex-math>\n <\/jats:inline-formula>\n +1)\u00a0Genetic Algorithm ((\n \n $$\\mu $$<\/jats:tex-math>\n <\/jats:inline-formula>\n +1)\u00a0GA), but only for unrealistically small crossover probabilities\u00a0\n \n $$p_c$$<\/jats:tex-math>\n <\/jats:inline-formula>\n . To this date, it remains an open problem to prove similar upper bounds for realistic\u00a0\n \n $$p_c$$<\/jats:tex-math>\n <\/jats:inline-formula>\n ; the best known runtime bound, in terms of function evaluations, for\n \n $$p_c = \\Omega (1)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n is\n \n $$O((n\/\\chi )^{k-1})$$<\/jats:tex-math>\n <\/jats:inline-formula>\n ,\n \n $$\\chi $$<\/jats:tex-math>\n <\/jats:inline-formula>\n a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the (\n \n $$\\mu $$<\/jats:tex-math>\n <\/jats:inline-formula>\n +1)\u00a0GA on\n \n $$\\textsc {Jump} _k$$<\/jats:tex-math>\n <\/jats:inline-formula>\n . The (\n \n $$\\mu $$<\/jats:tex-math>\n <\/jats:inline-formula>\n +1)-\n \n $${\\lambda _c}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n -GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover\n \n $${\\lambda _c}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the (\n \n $$\\mu $$<\/jats:tex-math>\n <\/jats:inline-formula>\n +1)-\n \n $${\\lambda _c}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n -GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of\n \n $$O(\\mu n \\log (\\mu ) + 4^k)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n function evaluations for a range of\u00a0\n k<\/jats:italic>\n under the mild assumptions\n \n $$p_c = O(1\/k)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n and\n \n $$\\mu \\in \\Omega (kn)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n . For all constant\u00a0\n k<\/jats:italic>\n , the restriction is satisfied for some\n \n $$p_c = \\Omega (1)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n and it implies that the expected runtime for all constant\u00a0\n k<\/jats:italic>\n and an appropriate\n \n $$\\mu = \\Theta (kn)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n is bounded by\n \n $$O(n^2 \\log n)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n , irrespective of\u00a0\n k<\/jats:italic>\n . For larger\u00a0\n k<\/jats:italic>\n , the expected time of the (\n \n $$\\mu $$<\/jats:tex-math>\n <\/jats:inline-formula>\n +1)-\n \n $${\\lambda _c}$$<\/jats:tex-math>\n <\/jats:inline-formula>\n -GA is\n \n $$\\Theta (4^k)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n , which is tight for a large class of unbiased black-box algorithms and faster than the original (\n \n $$\\mu $$<\/jats:tex-math>\n <\/jats:inline-formula>\n +1)\u00a0GA by a factor of\n \n $$\\Omega (1\/p_c)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n . We also show that our analysis can be extended to other unitation functions such as\n \n $$\\textsc {Jump} _{k, \\delta }$$<\/jats:tex-math>\n <\/jats:inline-formula>\n and H\n urdle<\/jats:sc>\n .\n <\/jats:p>","DOI":"10.1007\/s00453-025-01323-x","type":"journal-article","created":{"date-parts":[[2025,7,28]],"date-time":"2025-07-28T04:17:02Z","timestamp":1753676222000},"page":"1564-1619","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Achieving Tight $$O(4^k)$$ Runtime Bounds on Jumpk by Proving that Genetic Algorithms Evolve Near-Maximal Population Diversity"],"prefix":"10.1007","volume":"87","author":[{"given":"Andre","family":"Opris","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Johannes","family":"Lengler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dirk","family":"Sudholt","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,7,28]]},"reference":[{"issue":"2","key":"1323_CR1","doi-asserted-by":"publisher","first-page":"274","DOI":"10.1162\/artl_a_00319","volume":"26","author":"J Lehman","year":"2020","unstructured":"Lehman, J., Clune, J., Misevic, D., Adami, C., Altenberg, L., Beaulieu, J., Bentley, P.J., Bernard, S., Beslon, G., Bryson, D.M., Cheney, N., Chrabaszcz, P., Cully, A., Doncieux, S., Dyer, F.C., Ellefsen, K.O., Feldt, R., Fischer, S., Forrest, S., Frenoy, A., Gag\u0144e, C., Le Goff, L., Grabowski, L.M., Hodjat, B., Hutter, F., Keller, L., Knibbe, C., Krcah, P., Lenski, R.E., Lipson, H., MacCurdy, R., Maestre, C., Miikkulainen, R., Mitri, S., Moriarty, D.E., Mouret, J.-B., Nguyen, A., Ofria, C., Parizeau, M., Parsons, D., Pennock, R.T., Punch, W.F., Ray, T.S., Schoenauer, M., Schulte, E., Sims, K., Stanley, K.O., Taddei, F., Tarapore, D., Thibault, S., Watson, R., Weimer, W., Yosinski, J.: The surprising creativity of digital evolution: a collection of anecdotes from the evolutionary computation and artificial life research communities. 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