[Leetcode][39. Combination Sum] The Key of the Backtrack is Deduplicating and Pruning

By Long Luo

This article is the solution of Problem 39. Combination Sum.

Backtracking

It’s a typical backtracking algorithm.

We can easily draw the tree below.

Tree Path

Take target=7target = 7 as the root node and subtract the value of the edge while creating a child node.

Stop it when the value of the node is 00 or a negative number.

All paths from the root node to leaf node 00 are the results of the question.

Let’s coding.

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class Solution {
public List<List<Integer>> combinationSum(int[] candidates, int target) {
List<List<Integer>> ans = new ArrayList<>();
if (candidates == null || target == 0) {
return ans;
}

backtrack(ans, new ArrayList<>(), candidates, target);
return ans;
}

public static void backtrack(List<List<Integer>> res, List<Integer> list, int[] candidates, int remain) {
if (remain < 0) {
return;
}

if (remain == 0) {
res.add(new ArrayList<>(list));
return;
}

int len = candidates.length;
for (int i = 0; i < len; i++) {
int num = candidates[i];
list.add(num);
backtrack(res, list, candidates, remain - num);
list.remove(list.size() - 1);
}
}
}

Take target=7target = 7 for example, [[2,2,3],[2,3,2],[3,2,2],[7]][[2, 2, 3], [2, 3, 2], [3, 2, 2], [7]] are the answer.

However, the correct answer is [[7],[2,2,3]][[7], [2, 2, 3]].

Since it requires that each solution is not in order, so the [2,2,3][2, 2, 3] and [2,3,2][2, 3, 2] are the same answer.

What’s problem?

When and how did we duplicate the path?

Why Duplicated?

At each node of the tree, all the candidates were searched and used, so there is a duplicate list.

Is it possible to deduplicate while searching?

Of course we can.

We have to search the elements in order while searching, since the output is not by order. We search from the certain position in the next round.

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public static void backtrack(List<List<Integer>> res, List<Integer> list, int[] candidates, int start, int remain) {
if (remain < 0) {
return;
}

if (remain == 0) {
res.add(new ArrayList<>(list));
return;
}

int len = candidates.length;
for (int i = start; i < len; i++) {
int num = candidates[i];
list.add(num);
backtrack(res, list, candidates, i, remain - num);
list.remove(list.size() - 1);
}
}

By now we deduplicating it and complete the problem.

Pruning

We can prune the tree to make it work more efficiency.

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class Solution {
public List<List<Integer>> combinationSum(int[] candidates, int target) {
List<List<Integer>> ans = new ArrayList<>();
if (candidates == null || target == 0) {
return ans;
}

Arrays.sort(candidates);
backtrack(ans, new ArrayList<>(), candidates, 0, target);
return ans;
}

public static void backtrack(List<List<Integer>> res, List<Integer> list, int[] candidates, int begin, int remain) {
if (remain < 0) {
return;
}

if (remain == 0) {
res.add(new ArrayList<>(list));
return;
}

int len = candidates.length;
for (int i = begin; i < len; i++) {
int num = candidates[i];
if (remain - num < 0) {
break;
}
list.add(num);
backtrack(res, list, candidates, i, remain - num);
list.remove(list.size() - 1);
}
}
}

Analysis

  • Time Complexity: O(Sum)O(Sum)SumSum is all the path.
  • Space Complexity: O(target)O(\textit{target}).

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