Faster Bi-Objective Non-Dominated Sorting with O(N log N) Complexity

[richardcsuwandi]

This PR introduces a faster non-dominated sorting algorithm that leverages geometric properties to reduce complexity from O(N^2) to O(N log N), resulting in up to 60.92x faster performance for bi-objective problems.

Before (O(N^2) for all cases):

# Original algorithm for all objective counts
for i in range(n):
    for j in range(i + 1, n):
        rel = M[i, j]  # Full dominance matrix calculation
        if rel == 1:
            is_dominating[i].append(j)
            n_dominated[j] += 1
        elif rel == -1:
            is_dominating[j].append(i)
            n_dominated[i] += 1

After (O(N log N) for bi-objective, O(N^2) for others):

# Specialized algorithm for bi-objective problems
if n_objectives == 2:
    return _fast_biobjective_nondominated_sort(F)  # O(N log N) algorithm
else:
    # Fall back to optimized original approach for multi-objective

📊 Performance Results

Size	Original (s)	Evolved (s)	Speedup
50	0.000328	0.000036	9.00x
100	0.001024	0.000078	13.18x
500	0.034089	0.000842	40.49x
1000	0.104040	0.002246	46.33x
2000	0.278000	0.004547	60.92x

Average speedup: 33.98x

Added comprehensive performance test (test_comparison.py) that:

• Uses the same interface as the original pymoo implementation
• Tests with actual bi-objective optimization scenarios
• Verifies identical results between original and optimized methods
• Demonstrates performance improvements across multiple problem sizes

[blankjul]

Amazing thanks for your contribution!

I was under the impression that this already exists somewhere in the codebase, but doesn't seem like it!

[MLopez-Ibanez]

@richardcsuwandi @blankjul

How does this new implementation compare with https://multi-objective.github.io/moocore/python/reference/generated/moocore.pareto_rank.html#moocore.pareto_rank

which already uses the O(n log n) algorithm by Jensen?

Of course, you do whatever you want with pymoo but it does seem wasteful to re-implement functionality that is already provided by moocore which you already have as a dependency. That effort would be better spent on improving pymoo functionality not present in moocore or improving moocore itself.

For benchmarking, you could use the benchmarking code of moocore, see: https://github.com/multi-objective/moocore/blob/main/python/benchmarks/bench_ndom.py

[MLopez-Ibanez]

Also, why would you want the file test_comparison.py in the root of the source directory and not under tests/? Is this a file that should be available to users of the package?

[blankjul]

all fair points. benchmark against the current implementation as well:

Bi-Objective Non-dominated Sorting Performance Benchmark
======================================================================
Size       Default (ms)    Native BiObj (ms)  Speedup    Correct
----------------------------------------------------------------------
50         0.107           0.036              2.98      x Yes
100        0.103           0.100              1.03      x Yes
500        1.038           1.065              0.98      x Yes
1000       3.205           3.094              1.04      x Yes
2000       8.615           8.642              1.00      x Yes
----------------------------------------------------------------------
All results correct: Yes

Decided not to make it default.

and thanks for the callout. decided to make a benchmark folder within tests.

[MLopez-Ibanez]

all fair points. benchmark against the current implementation as well:

I recently added moocore benchmarks for nondominated sorting: https://multi-objective.github.io/moocore/python/#nondominated-sorting-pareto-ranking

As you can see the current implementation of pymoo is the slowest of the ones tested.