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Bellman applies dynamic-programming cost-to-go equations to positive-time finite routing and gives policy-initialized monotone successive approximations that reach the optimal travel cost after a bounded number of iterations. The quickest route is not necessarily the shortest in distance or the road that looks best at the first junction. Bellman assigns each city the best remaining travel time to the destination. Updating a city means comparing its outgoing road time plus the currently known remaining time from the next city. Repeating these local comparisons improves the whole route estimate. For the positive travel times of the original finite model, an optimal route does not need a cycle, giving a bound on the necessary iterations. The paper explains initialization and monotone convergence and acknowledges an independently obtained version of the same iteration. Modern negative-edge variants and distributed routing systems are later, separate developments.
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Scientific scope
The original model uses finitely many cities with strictly positive pairwise travel times and a fixed destination. Its initialization, recursion and iteration bound belong to that model. Nonnegative/negative weights, sparse-graph implementations and modern Bellman–Ford variants need their own conditions. The result does not provide a universal routing guarantee under arbitrary changing network conditions.
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People and systems
Richard Bellman
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English catalogue narrative
Bellman: improve a route by updating what remains
Bellman applies dynamic-programming cost-to-go equations to positive-time finite routing and gives policy-initialized monotone successive approximations that reach the optimal travel cost after a bounded number of iterations.
The quickest route is not necessarily the shortest in distance or the road that looks best at the first junction. Bellman assigns each city the best remaining travel time to the destination. Updating a city means comparing its outgoing road time plus the currently known remaining time from the next city. Repeating these local comparisons improves the whole route estimate. For the positive travel times of the original finite model, an optimal route does not need a cycle, giving a bound on the necessary iterations. The paper explains initialization and monotone convergence and acknowledges an independently obtained version of the same iteration. Modern negative-edge variants and distributed routing systems are later, separate developments.
Scientific scope
The original model uses finitely many cities with strictly positive pairwise travel times and a fixed destination. Its initialization, recursion and iteration bound belong to that model. Nonnegative/negative weights, sparse-graph implementations and modern Bellman–Ford variants need their own conditions. The result does not provide a universal routing guarantee under arbitrary changing network conditions.
People and systems
Richard Bellman