{"id":"traversal-accumulation-universal-across-data-structures","text":"The single-traversal accumulation paradigm — visit each element once while maintaining state via accumulators — is a recurring algorithmic shape across data structures: arrays use left-to-right streaming with scalar variables, and trees use DFS with closure-captured variables. Both achieve efficient single-pass O(n) computation, suggesting that traversal-order discipline is a primary mechanism for correctness in these patterns.","truth_value":"IN","source":"","source_url":"","source_hash":"","justifications":[{"type":"SL","antecedents":["tree-postorder-closure-idiom","single-pass-streaming-dominant-shape"],"outlist":[],"label":"Generalizes the dominant streaming shape beyond linear sequences to recursive structures, with closure variables as the tree analogue of scalar accumulators"}],"dependents":["preprocessing-is-domain-transformation-to-streaming","streaming-universality-through-operation-specialization"],"metadata":{"last_reviewed":"2026-06-07T22:02:22","review_result":"pass"},"created_at":"","updated_at":"","reviewed_at":"","verified_at":"","retracted_at":"","explanation":{"steps":[{"node":"traversal-accumulation-universal-across-data-structures","truth_value":"IN","reason":"SL justification valid","antecedents":["tree-postorder-closure-idiom","single-pass-streaming-dominant-shape"],"label":"Generalizes the dominant streaming shape beyond linear sequences to recursive structures, with closure variables as the tree analogue of scalar accumulators"},{"node":"tree-postorder-closure-idiom","truth_value":"IN","reason":"SL justification valid","antecedents":["closure-dfs-pattern","postorder-accumulator-pattern"],"label":"Closures handle cross-subtree accumulation while postorder handles intra-subtree composition — together they cover all tree computation needs"},{"node":"closure-dfs-pattern","truth_value":"IN","reason":"premise"},{"node":"postorder-accumulator-pattern","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"}]}}