python-li-chao-tree-floor-division-overflow
Solved and fully verified. The deliverable is ~/lichao_solution.md (self-contained: title, root-cause analysis, exact fix with code, complexity, and verification), with the artifacts:
lichao_fixed.py — the fixed tree (two exact variants: endpoint-comparison and exact-floor-crossing)lichao_buggy.py — faithful buggy reproduction (int((b1-b2)/(m2-m1)))verify_lichao.py, verify_deep.py — verification suitesThe original code located line intersections in floating point and truncated with int(). Two compounding errors:
1. int() truncates, not floors — negative crossings are off by one by construction.
2. Intercepts up to 10¹⁸ exceed 2⁵³, so (b1−b2) already carries float error (~64), and the quotient can round onto the wrong side of the integer routing boundary. A crossing at 10⁹ − 10⁻⁹ (e.g. from m=(10¹⁸, 10¹⁸−10⁹), b=(0, 10¹⁸−1)) computes as exactly 1e9 → int() → 1_000_000_000 ≥ r → the new line is discarded, so the isolated query at x=10⁹ returns 10²⁷ instead of 10²⁷−1.
m1*x+b1 < m2*x+b2, no division at all); a dynamic node-allocating Li Chao tree, O(log N) per insert/query.(b1 - b2) // (m2 - m1) — Python // is exact floor division for every sign, so the floor can never be off by one.got 1000000000000000000000000000, expected 999999999999999999999999999 at x=10⁹).Run again anytime with: python3 verify_lichao.py && python3 verify_deep.py
# Evidence - Problem class: python-li-chao-tree-floor-division-overflow - Model: openrouter/deepseek/deepseek-v4-flash-0731 - Solved: 2026-09-03T22:11:43.884Z - Verification: solution produced by pi in sandbox; see signatures.json
{"description": "Implement a Li Chao segment tree over integer x in [0, 1e9] supporting dynamic insertion of lines y = m*x + b with m and b up to 1e18, plus point minimum queries. The current code computes candidate line intersections in floating point and rounds with int(), which flips the winning line whenever two lines cross at a non-integer x whose floor is off by one, so isolated queries silently return the wrong value near the huge-slope boundary. Replace float math with exact integer comparisons (cross-multiplication plus floor division), keep insertions amortized O(log N), and verify on adversarial crossing sets at the 1e18 slope limit.", "environment": "python3", "language": "python", "model": "openrouter/deepseek/deepseek-v4-flash-0731", "problem_class": "python-li-chao-tree-floor-division-overflow", "provider": "openrouter", "solved_at": "2026-09-03T22:11:43.884Z", "version": "3.11"}Solved and fully verified. The deliverable is ~/lichao_solution.md (self-contained: title, root-cause analysis, exact fix with code, complexity, and verification), with the artifacts:
lichao_fixed.py — the fixed tree (two exact variants: endpoint-comparison and exact-floor-crossing)lichao_buggy.py — faithful buggy reproduction (int((b1-b2)/(m2-m1)))verify_lichao.py, verify_deep.py — verification suitesThe original code located line intersections in floating point and truncated with int(). Two compounding errors:
1. int() truncates, not floors — negative crossings are off by one by construction.
2. Intercepts up to 10¹⁸ exceed 2⁵³, so (b1−b2) already carries float error (~64), and the quotient can round onto the wrong side of the integer routing boundary. A crossing at 10⁹ − 10⁻⁹ (e.g. from m=(10¹⁸, 10¹⁸−10⁹), b=(0, 10¹⁸−1)) computes as exactly 1e9 → int() → 1_000_000_000 ≥ r → the new line is discarded, so the isolated query at x=10⁹ returns 10²⁷ instead of 10²⁷−1.
m1*x+b1 < m2*x+b2, no division at all); a dynamic node-allocating Li Chao tree, O(log N) per insert/query.(b1 - b2) // (m2 - m1) — Python // is exact floor division for every sign, so the floor can never be off by one.got 1000000000000000000000000000, expected 999999999999999999999999999 at x=10⁹).Run again anytime with: python3 verify_lichao.py && python3 verify_deep.py
# Evidence - Problem class: python-li-chao-tree-floor-division-overflow - Model: openrouter/deepseek/deepseek-v4-flash-0731 - Solved: 2026-09-03T22:11:43.884Z - Verification: solution produced by pi in sandbox; see signatures.json
{"description": "Implement a Li Chao segment tree over integer x in [0, 1e9] supporting dynamic insertion of lines y = m*x + b with m and b up to 1e18, plus point minimum queries. The current code computes candidate line intersections in floating point and rounds with int(), which flips the winning line whenever two lines cross at a non-integer x whose floor is off by one, so isolated queries silently return the wrong value near the huge-slope boundary. Replace float math with exact integer comparisons (cross-multiplication plus floor division), keep insertions amortized O(log N), and verify on adversarial crossing sets at the 1e18 slope limit.", "environment": "python3", "language": "python", "model": "openrouter/deepseek/deepseek-v4-flash-0731", "problem_class": "python-li-chao-tree-floor-division-overflow", "provider": "openrouter", "solved_at": "2026-09-03T22:11:43.884Z", "version": "3.11"}