Dynamic graph — where it appears
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Also named here as invalidation — the same set of essays touches all of them, so they are one junction rather than several.
How long a reweighting stays true
Johnson's one Bellman–Ford run costs under one per cent of an all-pairs computation because it is divided over every source. Asked one query at a time it is divided over nothing, and it still repays itself after 2.7 queries — because the preparation is one Bellman–Ford and every query saves a third of another. What decides the trade is not the query count but whether the graph holds still: at half a per cent of arcs redrawn between queries the stored potential is worth exactly nothing, and its life is geometric at a per-arc failure rate of 6.6%.
A potential mended where it broke
A stored reweighting on a 256-vertex graph with negative arcs costs 10,045 relaxations to rebuild, and rebuilding it every time an update breaks it stops paying once half a per cent of arcs change between queries. Mending it from the arcs that broke costs 16 to 442 relaxations instead, and the stored potential stays at two thirds of the per-query cost at every rate of change. When the change is a vertex whose costs all move together, a repair reaches nearly every vertex. It still costs a third of a rebuild.
Named alongside it
The objects these essays reach for when they reach for this one.
AmortisationBreak-evenInvalidationNegative weightPotential functionReweightingShortest pathBellman–FordDijkstra's algorithmHonest limitPreprocessingRegime