Learning Path Builder Knowledge Graph

A learning path is a scheduling problem over a directed acyclic graph. Topics are nodes, prerequisite relations are edges, and any valid study order is a topological sort of that graph. Two properties then control the timeline. The critical path, meaning the longest chain of dependent topics, sets a floor on elapsed time no matter how many hours per week are available, because those topics must be taken in sequence. Everything off the critical path can in principle be studied in parallel, which is why total effort and total elapsed time are separate quantities. Graph density, the ratio of actual prerequisite edges to the maximum possible, is low in well-designed curricula, typically well under 10 percent, because a subject in which everything depends on everything else cannot be entered at any point except the very beginning. Effort per topic is where estimates usually go wrong. This model assigns 3 hours to beginner topics, 6 to intermediate, 12 to advanced and 20 to expert, then inflates by a review multiplier because retention is not a free by-product of first exposure. The retention target drives the number of spaced review sessions, conventionally scheduled at expanding intervals of roughly 1, 3, 7, 14, 30 and 60 days, with more sessions needed for higher targets. The heuristics have clear limits. The critical path is approximated as 1.5 times the square root of the topic count rather than computed from a real dependency graph, prerequisite counts are assumed rather than mapped, and the parallel-learning figure presumes a learner can genuinely hold several unrelated threads at once, which working memory and context switching costs make doubtful beyond two or three. Treat any output as a lower bound, given the well-documented tendency to underestimate task duration.

The empirical basis for the review schedule is Hermann Ebbinghaus, who between 1879 and 1885 ran memorisation experiments on himself using nonsense syllables and published the results as Uber das Gedachtnis in 1885. He produced the forgetting curve, showing retention decaying steeply within the first day and then flattening, and demonstrated the spacing effect, that repetitions distributed across time produce far more durable retention than the same number massed together. Later work formalised the practical schedule. Paul Pimsleur described a graduated interval recall schedule for language learning in 1967, and Sebastian Leitner popularised a physical implementation in 1972 using a box of flashcard compartments that a card advances through on each successful recall. Computerised scheduling followed. Piotr Wozniak began developing SuperMemo in Poland in the mid 1980s, releasing the first program in 1987 and a succession of SM algorithms that estimate the difficulty of an individual item and predict its optimal next interval, an approach later inherited by open tools such as Anki. The dependency-scheduling half of this model has a separate origin in operations research: the critical path method was developed by James Kelley of Remington Rand and Morgan Walker of DuPont in 1957 for plant maintenance scheduling, alongside PERT at the United States Navy. It is that machinery, applied to prerequisite graphs instead of construction tasks, that produces the distinction between total effort and minimum elapsed time.