Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the (μ+1) GA using uniform crossover on the fitness function Jump_k. All previous results in this context only hold for unrealistically low crossover probability p_c=O(k/n), while we give analyses for the setting of constant p_c < 1 in all but one case. Our bounds show a dependence on the problem size~, the jump length k, the population size μ, and the crossover probability p_c. For the typical case of constant k > 2 and constant p_c, we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of μ:

• O}(n^k-1) for duplicate elimination/minimisation,
• O}(n² log n) for maximising the convex hull,
• O(n log n) for det. crowding (assuming p_c = k/n),
• O(n log n) for maximising the Hamming distance,
• O(n log n) for fitness sharing,
• O(n log n) for the single-receiver island model.

This proves a sizeable advantage of all variants of the (μ+1) GA compared to the (1+1) EA, which requires θ(n^k). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally.