Clarify definition of opt(i, j): choose maximum k among optimal candidates

ceciliachan1979 · web-flow · commit 3e7d29f38b20 · 2025-09-13T13:34:18.000-07:00
# Clarify definition of `opt(i, j)` to specify maximum `k` among optimal candidates ## Description This PR updates the documentation to clarify the definition of `opt(i, j)`: - Previously, it was ambiguous when multiple values of `k` achieved the optimum. - The reference explicitly states that we should take the **maximum** value of `k` among all optimal candidates. - This change makes the intended behavior explicit and avoids confusion in implementations. ## Motivation Clarifying this definition ensures consistency with the reference and prevents misinterpretation when different values of `k` yield the same optimum. ## Changes - Clarified wording in `opt(i, j)` definition. - Specified that the chosen `k` must be the maximum among all optimal candidates. > The reference is https://dl.acm.org/doi/pdf/10.1145/800141.804691, specifically, when we define $K_c$ on the left column.
diff --git a/src/dynamic_programming/knuth-optimization.md b/src/dynamic_programming/knuth-optimization.md
@@ -13,7 +13,7 @@ The Speedup is applied for transitions of the form
 
 $$dp(i, j) = \min_{i \leq k < j} [ dp(i, k) + dp(k+1, j) + C(i, j) ].$$
 
-Similar to [divide and conquer DP](./divide-and-conquer-dp.md), let $opt(i, j)$ be the value of $k$ that minimizes the expression in the transition ($opt$ is referred to as the "optimal splitting point" further in this article). The optimization requires that the following holds:
+Similar to [divide and conquer DP](./divide-and-conquer-dp.md), let $opt(i, j)$ be the maximum value of $k$ that minimizes the expression in the transition ($opt$ is referred to as the "optimal splitting point" further in this article). The optimization requires that the following holds:
 
 $$opt(i, j-1) \leq opt(i, j) \leq opt(i+1, j).$$
 
