Nov 2, 2024 · Oct 31, 2024
diff --git a/src/dynamic_programming/knapsack.md b/src/dynamic_programming/knapsack.md

 The grouping is made more efficient by using binary grouping.

 Specifically, $A_{i, j}$ holds $2^j$ individual items ($j\in[0,\lfloor \log_2(k_i+1)\rfloor-1]$).If $k_i + 1$ is not an integer power of $2$, another bundle of size $k_i-2^{\lfloor \log_2(k_i+1)\rfloor-1}$ is used to make up for it.
 Specifically, $A_{i, j}$ holds $2^j$ individual items ($j\in[0,\lfloor \log_2(k_i+1)\rfloor-1]$).If $k_i + 1$ is not an integer power of $2$, another bundle of size $k_i-(2^{\lfloor \log_2(k_i+1)\rfloor}-1)$ is used to make up for it.

 Through the above splitting method, it is possible to obtain any sum of $\leq k_i$ items by selecting a few $A_{i, j}$'s. After splitting each item in the described way, it is sufficient to use 0-1 knapsack method to solve the new formulation of the problem.
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Expand Up		@@ -116,7 +116,7 @@ Let $A_{i, j}$ denote the $j^{th}$ item split from the $i^{th}$ item. In the tri

		The grouping is made more efficient by using binary grouping.

		Specifically, $A_{i, j}$ holds $2^j$ individual items ($j\in[0,\lfloor \log_2(k_i+1)\rfloor-1]$).If $k_i + 1$ is not an integer power of $2$, another bundle of size $k_i-2^{\lfloor \log_2(k_i+1)\rfloor-1}$ is used to make up for it.
		Specifically, $A_{i, j}$ holds $2^j$ individual items ($j\in[0,\lfloor \log_2(k_i+1)\rfloor-1]$).If $k_i + 1$ is not an integer power of $2$, another bundle of size $k_i-(2^{\lfloor \log_2(k_i+1)\rfloor}-1)$ is used to make up for it.

		Through the above splitting method, it is possible to obtain any sum of $\leq k_i$ items by selecting a few $A_{i, j}$'s. After splitting each item in the described way, it is sufficient to use 0-1 knapsack method to solve the new formulation of the problem.

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