As the most powerful tool in discrepancy theory, the partial coloring method has wide applications in many problems including the Beck-Fiala problem [12] and Spencer’s celebrated result [4, 28, 37, 41]. Currently, there are two major algorithmic approaches for the partial coloring method: the first approach uses linear algebraic tools to update the partial coloring for many rounds [4, 6, 7, 25, 28]; and the second one, called Gaussian measure algorithm [36, 37], projects a random Gaussian vector to the feasible region that satisfies all discrepancy constraints in \([-1,1]^n\) . In this work, we explore the advantages of these two approaches and show the following results for them separately. Spencer [42] conjectured that the prefix discrepancy of any \(\textbf{A}\in \{0,1\}^{m \times n}\) is \(O(\sqrt{m})\) , i.e., \(\exists \textbf{x}\in \{\pm 1\}^n\) such that \(\max _{t \le n} \Vert \sum _{i \le t} \textbf{A}(\cdot ,i) \cdot \textbf{x}(i) \Vert _{\infty } = O(\sqrt{m})\) where \(\textbf{A}(\cdot ,i)\) denotes column i of \(\textbf{A}\) . Combining small deviations bounds of Gaussian processes and the Gaussian measure algorithm [37], we show how to find a partial coloring with prefix discrepancy \(O(\sqrt{m})\) and \(\varOmega (n)\) entries in \(\{ \pm 1\}\) efficiently. While this bound was folklore before [34, 43], our argument is simpler and more direct than previous proofs. Moreover, our conceptual contribution here is a connection between Gaussian processes and the Gaussian measure algorithms. Our second result extends the linear algebraic approach to a sampling algorithm in Spencer’s classical setting. On the first hand, besides the six deviation bound [41], Spencer also proved that there are \(1.99^m\) good colorings with discrepancy \(O(\sqrt{m})\) for any \(\textbf{A}\in \{0,1\}^{m \times n}\) . Hence a natural question is to design efficient random sampling algorithms in Spencer’s setting. On the second hand, some applications of discrepancy theory, such as experimental design, prefer a random solution instead of a fixed one [20, 22, 44]. Our second result is an efficient sampling algorithm whose random output has min-entropy \(\varOmega (n)\) and discrepancy \(O(\sqrt{m})\) .