{"id":"dp-bounded-lookback-reduces-to-streaming","text":"Dynamic programming with bounded lookback reduces to the streaming paradigm's dominant shape: replacing an O(n) DP table with O(1) rolling variables transforms a DP recurrence into a single-pass scan with scalar accumulators — proving that streaming's coverage extends beyond pure accumulation problems to subsume a class of dynamic programming problems.","truth_value":"IN","source":"","source_url":"","source_hash":"","justifications":[{"type":"SL","antecedents":["dp-to-streaming-via-rolling-variable-reduction","single-pass-streaming-dominant-shape"],"outlist":[],"label":"Connects the orphaned DP-to-streaming chain (depth-1) into the streaming hierarchy (depth-2)"}],"dependents":[],"metadata":{"source_type":"derived","last_reviewed":"2026-06-07T22:02:22","review_result":"pass"},"created_at":"","updated_at":"","reviewed_at":"","verified_at":"","retracted_at":"","explanation":{"steps":[{"node":"dp-bounded-lookback-reduces-to-streaming","truth_value":"IN","reason":"SL justification valid","antecedents":["dp-to-streaming-via-rolling-variable-reduction","single-pass-streaming-dominant-shape"],"label":"Connects the orphaned DP-to-streaming chain (depth-1) into the streaming hierarchy (depth-2)"},{"node":"dp-to-streaming-via-rolling-variable-reduction","truth_value":"IN","reason":"SL justification valid","antecedents":["min-cost-dp-uses-constant-space","min-cost-loop-invariant","min-cost-final-answer-is-min-of-last-two"],"label":"Three structural properties of min-cost-climbing-stairs jointly demonstrate the bounded-lookback DP-to-streaming reduction pattern"},{"node":"min-cost-dp-uses-constant-space","truth_value":"IN","reason":"premise"},{"node":"min-cost-loop-invariant","truth_value":"IN","reason":"premise"},{"node":"min-cost-final-answer-is-min-of-last-two","truth_value":"IN","reason":"premise"},{"node":"single-pass-streaming-dominant-shape","truth_value":"IN","reason":"SL justification valid","antecedents":["early-exit-optimizations-pervasive","o1-space-via-running-accumulators","extend-or-reset-canonical-consecutive-pattern"],"label":"these three patterns compose into a unified streaming shape — accumulators provide O(1) state, extend-or-reset handles consecutive-element logic, and early-exit bounds work to the minimum needed"},{"node":"early-exit-optimizations-pervasive","truth_value":"IN","reason":"SL justification valid","antecedents":["three-consecutive-odds-early-exit","path-crossing-early-exit","isomorphic-early-return","first-violation-sufficiency"],"label":"Early exit is the default control flow strategy, not an optimization afterthought"},{"node":"three-consecutive-odds-early-exit","truth_value":"IN","reason":"premise"},{"node":"path-crossing-early-exit","truth_value":"IN","reason":"premise"},{"node":"isomorphic-early-return","truth_value":"IN","reason":"premise"},{"node":"first-violation-sufficiency","truth_value":"IN","reason":"premise"},{"node":"o1-space-via-running-accumulators","truth_value":"IN","reason":"SL justification valid","antecedents":["highest-altitude-single-pass","longest-task-single-pass-o1-space","iterative-reversal-O1-space","min-tracking-pattern-shared"],"label":"Running accumulators trade re-traversal impossibility for constant memory across streaming-style problems"},{"node":"highest-altitude-single-pass","truth_value":"IN","reason":"premise"},{"node":"longest-task-single-pass-o1-space","truth_value":"IN","reason":"premise"},{"node":"iterative-reversal-O1-space","truth_value":"IN","reason":"premise"},{"node":"min-tracking-pattern-shared","truth_value":"IN","reason":"premise"},{"node":"extend-or-reset-canonical-consecutive-pattern","truth_value":"IN","reason":"SL justification valid","antecedents":["extend-or-reset-pattern","maxpower-eager-max-update","no-post-loop-fixup-needed"],"label":"Run-tracking with eager max avoids off-by-one errors that end-of-array special cases introduce"},{"node":"extend-or-reset-pattern","truth_value":"IN","reason":"premise"},{"node":"maxpower-eager-max-update","truth_value":"IN","reason":"premise"},{"node":"no-post-loop-fixup-needed","truth_value":"IN","reason":"premise"}]}}