Patterns for Subarray

Pattern 1

Given a sequence with $N$ elements and an integer $K$, find (total number of) pairs $(i, j)$ such that $f(i, j) = K$, where $i, j$ are the indices of elements in the sequence and $0 \leq i < j \leq N-1$.

Two Sum

$f(i, j) = Seq[i] + Seq[j] = K$

, where $Seq[x]$ is the element at index $x$ in the sequence.

Subarray Sum Equals K

$\begin{aligned} & f(i, j) = \Sigma_{x = i} ^ {j} Seq[x] = K \\ \Rightarrow & \Sigma_{x = 0} ^ {j} Seq[x] - \Sigma_{x = 0} ^ {i-1} Seq[x] = K \end{aligned}$