Optimize max flow with Dinic's blocking flow algorithm

[shakayami]

Summary

Refactor the dfs function in the max flow implementation to use Dinic's blocking flow algorithm, which processes an entire level graph in one phase rather than finding augmenting paths one at a time. This improves the time complexity from O(V²E) to O(E√V) for unit-capacity graphs.

Key Changes

maxflow.py:

• Rewrote dfs to compute a complete blocking flow for the current level graph in a single call, rather than returning after finding one augmenting path
• Changed from a stack-based iterative approach to a path-tracing algorithm that walks backwards from sink to source
• Implemented flow pushing in one sweep after finding the bottleneck capacity, then resuming from the first saturated edge instead of restarting
• Optimized edge iteration by advancing past dead and saturated edges, with level updates to prevent revisiting unreachable nodes
• Updated the main loop to call bfs() once per phase (not once per augmenting path), with Iter array reset per phase

benchmarks/workloads.py & benchmarks/README.md:

• Updated bipartitematching cost factor from 0.15 to 5.0, reflecting the improved algorithm performance
• Updated documentation to remove outdated notes about the O(V·E·flow) complexity and the low cost factor being a CI budget workaround
• Clarified that only range_affine_range_sum and exp/log_of_formal_power_series run at reduced scale due to CPython limitations

Implementation Details

The new algorithm maintains a level graph via BFS and uses DFS to find augmenting paths within that level. Key optimizations include:

• Reusing saturated edges' positions to resume path-finding efficiently
• Computing bottleneck capacity across the entire path before updating residual capacities
• Preventing redundant work by marking unreachable nodes with level = n

https://claude.ai/code/session_019maXFnaUfCveCBZFxmRmo7

[github-actions]

📊 Performance Benchmark Results

Workloads ran at ACL_BENCH_SCALE=0.2.

⏱️ Library Checker workloads (best of N, lower is better)

Benchmark	Base	PR	Delta	Noise
assignment	2.772s	2.788s	+0.5% ≈	±15.9%
bipartitematching	3.914s	3.750s	-4.2% ≈	±4.3%
convolution_mod	1.823s	1.810s	-0.7% ≈	±4.2%
exp_of_formal_power_series	2.691s	2.721s	+1.1% ≈	±1.7%
factorize	119.38ms	110.56ms	-7.4% ≈	±13.4%
inv_of_formal_power_series	3.070s	3.106s	+1.2% ≈	±3.4%
log_of_formal_power_series	2.663s	2.671s	+0.3% ≈	±1.9%
number_of_substrings	1.575s	1.531s	-2.7% ≈	±17.3%
point_add_range_sum	1.015s	1.003s	-1.3% ≈	±7.0%
point_set_range_composite	1.808s	1.756s	-2.9% ≈	±8.9%
primality_test	345.09ms	358.03ms	+3.8% ≈	±4.1%
range_affine_range_sum	2.255s	2.187s	-3.0% ≈	±6.2%
scc	1.333s	1.305s	-2.1% ≈	±2.0%
staticrmq	1.461s	1.475s	+0.9% ≈	±11.2%
suffixarray	1.167s	1.170s	+0.2% ≈	±29.1%
sum_of_floor_of_linear	813.56ms	790.40ms	-2.8% ≈	±2.8%
two_sat	1.375s	1.422s	+3.3% ⚠️	±3.2%
unionfind	97.78ms	101.34ms	+3.6% ≈	±8.6%
zalgorithm	162.38ms	165.37ms	+1.8% ≈	±2.2%

🧪 API coverage workloads (not judge-calibrated)

Benchmark	Base	PR	Delta	Noise
extra_acl_math_crt	827.14ms	835.98ms	+1.1% ≈	±1.6%
extra_dsu_groups	316.92ms	298.53ms	-5.8% ✅	±4.4%
extra_maxflow_edges	183.62ms	168.25ms	-8.4% ≈	±9.1%
extra_mincostflow_slope	1.751s	1.746s	-0.3% ≈	±4.0%
extra_prime_fact_divisors	1.384s	1.366s	-1.4% ≈	±5.1%
extra_segtree_binary_search	174.50ms	171.75ms	-1.6% ≈	±1.7%

📏 Workload sizes

Benchmark	Library Checker max	Ran at	Note
assignment	N = 500	50%	O(N^3)-ish in CPython; ACL_BENCH_SCALE=1 is far past the judge's TL
bipartitematching	L = R = 100,000, M = 200,000	100%
convolution_mod	N = M = 524,288	20%
exp_of_formal_power_series	N = 500,000	10%
factorize	Q = 100, A <= 10^18	100%
inv_of_formal_power_series	N = 500,000	20%
log_of_formal_power_series	N = 500,000	10%
number_of_substrings	N = 500,000	100%
point_add_range_sum	N = Q = 500,000	60%
point_set_range_composite	N = Q = 500,000	40%	segtree with a Python-level op; measures op-dominated cost
primality_test	Q = 100,000, N <= 10^18	100%
range_affine_range_sum	N = Q = 500,000	10%
scc	N = M = 500,000	100%
staticrmq	N = Q = 500,000	60%	segtree with a C-level op (min); measures tree overhead
suffixarray	N = 500,000	100%
sum_of_floor_of_linear	T = 100,000	100%
two_sat	N = M = 500,000	60%
unionfind	N = Q = 200,000	100%
zalgorithm	N = 500,000	100%
extra_acl_math_crt	Q = 100,000	100%	inv_mod / inv_gcd / crt
extra_dsu_groups	N = 200,000	100%	size / leader / groups
extra_maxflow_edges	N = 20,000, M = 60,000	100%	get_edge / change_edge / edges / min_cut
extra_mincostflow_slope	N = 2,000, M = 8,000	100%	slope / get_edge / edges
extra_prime_fact_divisors	Q = 10,000	20%	divisors / totient / lcm
extra_segtree_binary_search	N = Q = 200,000	20%	max_right / min_left

💾 Peak memory (lower is better)

Benchmark	Base	PR	Delta
assignment	32.8MiB	32.8MiB	0.0%
bipartitematching	4.3MiB	142.7MiB	+3195.8% ⚠️
convolution_mod	30.2MiB	30.2MiB	0.0%
exp_of_formal_power_series	21.7MiB	21.7MiB	0.0%
extra_acl_math_crt	484.0B	484.0B	0.0%
extra_dsu_groups	30.5MiB	30.5MiB	0.0%
extra_maxflow_edges	29.4MiB	29.4MiB	0.0%
extra_mincostflow_slope	5.5MiB	5.5MiB	0.0%
extra_prime_fact_divisors	8.6KiB	8.6KiB	0.0%
extra_segtree_binary_search	2.3MiB	2.3MiB	0.0%
factorize	836.0B	836.0B	0.0%
inv_of_formal_power_series	21.4MiB	21.4MiB	0.0%
log_of_formal_power_series	24.6MiB	24.6MiB	0.0%
number_of_substrings	82.5MiB	82.5MiB	0.0%
point_add_range_sum	12.1MiB	12.1MiB	0.0%
point_set_range_composite	34.9MiB	34.9MiB	0.0%
primality_test	804.0B	804.0B	0.0%
range_affine_range_sum	14.0MiB	14.0MiB	0.0%
scc	139.5MiB	139.5MiB	0.0%
staticrmq	8.1MiB	8.1MiB	0.0%
suffixarray	82.5MiB	82.5MiB	0.0%
sum_of_floor_of_linear	516.0B	516.0B	0.0%
two_sat	169.8MiB	169.8MiB	0.0%
unionfind	1.7MiB	1.7MiB	0.0%
zalgorithm	3.8MiB	3.8MiB	0.0%

_{Only the workloads whose module the change touches are measured; a scheduled sweep covers the rest. Each workload mirrors the linked Library Checker problem: same query mix, same operators, same value ranges as that problem's max_random generator. Times are the best observation across interleaved passes; Noise is the spread between passes. ✅ improvement / ⚠️ regression are only shown when the delta exceeds both 3% and the measured noise; ≈ means the difference is not distinguishable from noise. API-coverage workloads exist to catch regressions in methods no judge problem exercises — do not quote their timings as speedups. "n/a" rows mean one side has no such benchmark.}