diff --git a/en/docs/chapter_backtracking/subset_sum_problem.md b/en/docs/chapter_backtracking/subset_sum_problem.md index 51638edb93..e72a7ff586 100644 --- a/en/docs/chapter_backtracking/subset_sum_problem.md +++ b/en/docs/chapter_backtracking/subset_sum_problem.md @@ -6,7 +6,7 @@ Given an array of positive integers `nums` and a target positive integer `target`, find all possible combinations such that the sum of the elements in the combination equals `target`. The given array has no duplicate elements, and each element can be chosen multiple times. Please return these combinations as a list, which should not contain duplicate combinations. -For example, for the input set $\{3, 4, 5\}$ and target integer $9$, the solutions are $\{3, 3, 3\}, \{4, 5\}$. Note the following two points. +For example, given the input set $\{3, 4, 5\}$ and target integer $9$, the solutions are $\{3, 3, 3\}, \{4, 5\}$. It is important to note the following two points: - Elements in the input set can be chosen an unlimited number of times. - Subsets do not distinguish the order of elements, for example $\{4, 5\}$ and $\{5, 4\}$ are the same subset. @@ -21,9 +21,9 @@ Unlike the permutation problem, **elements in this problem can be chosen an unli [file]{subset_sum_i_naive}-[class]{}-[func]{subset_sum_i_naive} ``` -Inputting the array $[3, 4, 5]$ and target element $9$ into the above code yields the results $[3, 3, 3], [4, 5], [5, 4]$. **Although it successfully finds all subsets with a sum of $9$, it includes the duplicate subset $[4, 5]$ and $[5, 4]$**. +Inputting the array $[3, 4, 5]$ and target element $9$ into the above code yields the output $[3, 3, 3], [4, 5], [5, 4]$. **Although it successfully finds all subsets with a sum of $9$, it includes the duplicate subset $[4, 5]$ and $[5, 4]$**. -This is because the search process distinguishes the order of choices, however, subsets do not distinguish the choice order. As shown in the figure below, choosing $4$ before $5$ and choosing $5$ before $4$ are different branches, but correspond to the same subset. +This is because the search process distinguishes the order of choices, however, subsets do not. As shown in the figure below, choosing $4$ before $5$ and choosing $5$ before $4$ are different branches, but correspond to the same subset. ![Subset search and pruning out of bounds](subset_sum_problem.assets/subset_sum_i_naive.png) @@ -39,7 +39,7 @@ To eliminate duplicate subsets, **a straightforward idea is to deduplicate the r 1. When choosing $3$ in the first round and $4$ in the second round, all subsets containing these two elements are generated, denoted as $[3, 4, \dots]$. 2. Later, when $4$ is chosen in the first round, **the second round should skip $3$** because the subset $[4, 3, \dots]$ generated by this choice completely duplicates the subset from step `1.`. -In the search process, each layer's choices are tried one by one from left to right, so the more to the right a branch is, the more it is pruned. +During the search process, each layer's choices are attempted one by one from left to right, so the more to the right a branch is, the more it is pruned. 1. First two rounds choose $3$ and $5$, generating subset $[3, 5, \dots]$. 2. First two rounds choose $4$ and $5$, generating subset $[4, 5, \dots]$. @@ -47,7 +47,7 @@ In the search process, each layer's choices are tried one by one from left to ri ![Different choice orders leading to duplicate subsets](subset_sum_problem.assets/subset_sum_i_pruning.png) -In summary, given the input array $[x_1, x_2, \dots, x_n]$, the choice sequence in the search process should be $[x_{i_1}, x_{i_2}, \dots, x_{i_m}]$, which needs to satisfy $i_1 \leq i_2 \leq \dots \leq i_m$. **Any choice sequence that does not meet this condition will cause duplicates and should be pruned**. +In summary, given the input array $[x_1, x_2, \dots, x_n]$, let the choice sequence in the search process be $[x_{i_1}, x_{i_2}, \dots, x_{i_m}]$. This sequence of choice needs to satisfy $i_1 \leq i_2 \leq \dots \leq i_m$. **Any choice sequence that does not meet this condition will cause duplicates and should be pruned**. ### Code implementation @@ -55,7 +55,7 @@ To implement this pruning, we initialize the variable `start`, which indicates t Besides, we have made the following two optimizations to the code. -- Before starting the search, sort the array `nums`. In the traversal of all choices, **end the loop directly when the subset sum exceeds `target`** as subsequent elements are larger and their subset sum will definitely exceed `target`. +- Before starting the search, sort the array `nums`. In the traversal of all choices, **end the loop immediately when the subset sum exceeds `target`**, because the subsequent elements are larger, and their subset sum will definitely exceed `target`. - Eliminate the element sum variable `total`, **by performing subtraction on `target` to count the element sum**. When `target` equals $0$, record the solution. ```src @@ -74,15 +74,15 @@ The figure below shows the overall backtracking process after inputting the arra Compared to the previous question, **this question's input array may contain duplicate elements**, introducing new problems. For example, given the array $[4, \hat{4}, 5]$ and target element $9$, the existing code's output results in $[4, 5], [\hat{4}, 5]$, resulting in duplicate subsets. -**The reason for this duplication is that equal elements are chosen multiple times in a certain round**. In the figure below, the first round has three choices, two of which are $4$, generating two duplicate search branches, thus outputting duplicate subsets; similarly, the two $4$s in the second round also produce duplicate subsets. +**The reason for this duplication is that equal elements are chosen multiple times in a certain round**. In the figure below, the first round has three choices, two of which are $4$, generating two duplicate search branches, thus outputting duplicate subsets; similarly, the two $4$s in the second round will also produce duplicate subsets. ![Duplicate subsets caused by equal elements](subset_sum_problem.assets/subset_sum_ii_repeat.png) ### Equal element pruning -To solve this issue, **we need to limit equal elements to being chosen only once per round**. The implementation is quite clever: since the array is sorted, equal elements are adjacent. This means that in a certain round of choices, if the current element is equal to its left-hand element, it means it has already been chosen, so skip the current element directly. +To solve this issue, **we need to limit equal elements to being chosen only once per round**. The implementation is quite clever: since the array is sorted, equal elements are adjacent to each other. This means that in a certain round of choices, if the current element is equal to its left-hand element, it means it has already been chosen, so skip the current element directly. -At the same time, **this question stipulates that each array element can only be chosen once**. Fortunately, we can also use the variable `start` to meet this constraint: after making the choice $x_{i}$, set the next round to start from index $i + 1$ going forward. This not only eliminates duplicate subsets but also avoids repeated selection of elements. +At the same time, **this question stipulates that each element of the array can only be chosen once**. Fortunately, we can also use the variable `start` to meet this constraint: after making the choice $x_{i}$, set the next round to start traversing from index $i + 1$. This not only eliminates duplicate subsets but also avoids selecting the same elements repeatedly. ### Code implementation