Computing small temporal modules in time logarithmic in history length

A temporal graph $${\mathcal {G}}$$ is a sequence of static graphs indexed by set integers representing time instants. Given $$\varDelta$$ an integer, -module vertices having the same neighbourhood outside for consecutive We address specific cases enumeration, when $$|A| = 2$$ or $$\varDelta \infty$$ . Our main parameter complexity analysis history length $$\tau \max \{t:G_t\in {\mathcal {G}}\text { not empty }\}-\min }\}$$ Using red-black tree data structure, we give solutions to above enumeration problems in logarithmic $$\tau$$ For general problem, preprocessing using overlapping properties minimal -modules. Numerical our implementation on collected from real-world scales up $$10^8$$