consecutivecondition

Consecutivecondition is a term used to describe a constraint that a certain property must hold over a consecutive run of elements in a sequence. It is used across mathematics, computer science, and data analysis to express the idea of a streak or sustained condition rather than a single isolated event. The term is not anchored to a single formal definition, but is often used to name problems or criteria involving runs of length k where a predicate P is satisfied for k successive items.

Formally, let (a_i) be a sequence and P a predicate on elements of the sequence. A consecutive condition of length k holds if there exists an index i such that P(a_i), P(a_{i+1}), ..., P(a_{i+k-1}) are all true. Equivalently, one can define a run of k consecutive true values for the derived sequence b_i = 1 if P(a_i) else 0; then the condition holds if some b_i, b_{i+1}, ..., b_{i+k-1} are all 1.

Example: In the binary sequence 0, 1, 1, 1, 0, there is a run of three consecutive items satisfying P(x) = (x = 1) at positions 2–4.

Applications and methods: The concept appears in quality control, sports statistics, and data streams to detect streaks of events or outcomes. It can be identified efficiently with a sliding window that counts satisfied predicates, or via finite automata and regular expressions. The complexity is typically linear in the sequence length.

See also: runs, streaks, consecutive ones problem, pattern detection.