您将如何测试给定数字集的所有可能的加法组合，N以便它们加起来为给定的最终数字？
一个简单的例子：
- 要添加的一组数字：
N = {1,5,22,15,0,...} - 期望的结果：
12345
问问题
537845 次
32 回答
299
这个问题可以通过所有可能的和的递归组合来解决，过滤掉那些达到目标的和。这是Python中的算法：
def subset_sum(numbers, target, partial=[]):
s = sum(partial)
# check if the partial sum is equals to target
if s == target:
print "sum(%s)=%s" % (partial, target)
if s >= target:
return # if we reach the number why bother to continue
for i in range(len(numbers)):
n = numbers[i]
remaining = numbers[i+1:]
subset_sum(remaining, target, partial + [n])
if __name__ == "__main__":
subset_sum([3,9,8,4,5,7,10],15)
#Outputs:
#sum([3, 8, 4])=15
#sum([3, 5, 7])=15
#sum([8, 7])=15
#sum([5, 10])=15
这种类型的算法在接下来的斯坦福抽象编程讲座中得到了很好的解释——这个视频非常推荐用于了解递归如何工作以生成解决方案的排列。
编辑
以上作为生成器函数，使其更有用一点。需要 Python 3.3+，因为yield from.
def subset_sum(numbers, target, partial=[], partial_sum=0):
if partial_sum == target:
yield partial
if partial_sum >= target:
return
for i, n in enumerate(numbers):
remaining = numbers[i + 1:]
yield from subset_sum(remaining, target, partial + [n], partial_sum + n)
这是相同算法的 Java 版本：
package tmp;
import java.util.ArrayList;
import java.util.Arrays;
class SumSet {
static void sum_up_recursive(ArrayList<Integer> numbers, int target, ArrayList<Integer> partial) {
int s = 0;
for (int x: partial) s += x;
if (s == target)
System.out.println("sum("+Arrays.toString(partial.toArray())+")="+target);
if (s >= target)
return;
for(int i=0;i<numbers.size();i++) {
ArrayList<Integer> remaining = new ArrayList<Integer>();
int n = numbers.get(i);
for (int j=i+1; j<numbers.size();j++) remaining.add(numbers.get(j));
ArrayList<Integer> partial_rec = new ArrayList<Integer>(partial);
partial_rec.add(n);
sum_up_recursive(remaining,target,partial_rec);
}
}
static void sum_up(ArrayList<Integer> numbers, int target) {
sum_up_recursive(numbers,target,new ArrayList<Integer>());
}
public static void main(String args[]) {
Integer[] numbers = {3,9,8,4,5,7,10};
int target = 15;
sum_up(new ArrayList<Integer>(Arrays.asList(numbers)),target);
}
}
这是完全相同的启发式方法。我的 Java 有点生疏，但我认为很容易理解。
Java 解决方案的 C# 转换： （by @JeremyThompson）
public static void Main(string[] args)
{
List<int> numbers = new List<int>() { 3, 9, 8, 4, 5, 7, 10 };
int target = 15;
sum_up(numbers, target);
}
private static void sum_up(List<int> numbers, int target)
{
sum_up_recursive(numbers, target, new List<int>());
}
private static void sum_up_recursive(List<int> numbers, int target, List<int> partial)
{
int s = 0;
foreach (int x in partial) s += x;
if (s == target)
Console.WriteLine("sum(" + string.Join(",", partial.ToArray()) + ")=" + target);
if (s >= target)
return;
for (int i = 0; i < numbers.Count; i++)
{
List<int> remaining = new List<int>();
int n = numbers[i];
for (int j = i + 1; j < numbers.Count; j++) remaining.Add(numbers[j]);
List<int> partial_rec = new List<int>(partial);
partial_rec.Add(n);
sum_up_recursive(remaining, target, partial_rec);
}
}
红宝石解决方案： （@emaillenin）
def subset_sum(numbers, target, partial=[])
s = partial.inject 0, :+
# check if the partial sum is equals to target
puts "sum(#{partial})=#{target}" if s == target
return if s >= target # if we reach the number why bother to continue
(0..(numbers.length - 1)).each do |i|
n = numbers[i]
remaining = numbers.drop(i+1)
subset_sum(remaining, target, partial + [n])
end
end
subset_sum([3,9,8,4,5,7,10],15)
编辑：复杂性讨论
正如其他人提到的，这是一个NP-hard 问题。它可以在指数时间 O(2^n) 内求解，例如对于 n=10，将有 1024 种可能的解决方案。如果您尝试达到的目标范围较低，则此算法有效。例如：
subset_sum([1,2,3,4,5,6,7,8,9,10],100000)生成 1024 个分支，因为目标永远不会过滤掉可能的解决方案。
另一方面，只subset_sum([1,2,3,4,5,6,7,8,9,10],10)生成 175 个分支，因为要达到的目标10会过滤掉许多组合。
如果N和Target是大数字，则应该进入解决方案的近似版本。
于 2011-01-08T10:50:39.243 回答
40
The solution of this problem has been given a million times on the Internet. The problem is called The coin changing problem. One can find solutions at http://rosettacode.org/wiki/Count_the_coins and mathematical model of it at http://jaqm.ro/issues/volume-5,issue-2/pdfs/patterson_harmel.pdf (or Google coin change problem).
By the way, the Scala solution by Tsagadai, is interesting. This example produces either 1 or 0. As a side effect, it lists on the console all possible solutions. It displays the solution, but fails making it usable in any way.
To be as useful as possible, the code should return a List[List[Int]]in order to allow getting the number of solution (length of the list of lists), the "best" solution (the shortest list), or all the possible solutions.
Here is an example. It is very inefficient, but it is easy to understand.
object Sum extends App {
def sumCombinations(total: Int, numbers: List[Int]): List[List[Int]] = {
def add(x: (Int, List[List[Int]]), y: (Int, List[List[Int]])): (Int, List[List[Int]]) = {
(x._1 + y._1, x._2 ::: y._2)
}
def sumCombinations(resultAcc: List[List[Int]], sumAcc: List[Int], total: Int, numbers: List[Int]): (Int, List[List[Int]]) = {
if (numbers.isEmpty || total < 0) {
(0, resultAcc)
} else if (total == 0) {
(1, sumAcc :: resultAcc)
} else {
add(sumCombinations(resultAcc, sumAcc, total, numbers.tail), sumCombinations(resultAcc, numbers.head :: sumAcc, total - numbers.head, numbers))
}
}
sumCombinations(Nil, Nil, total, numbers.sortWith(_ > _))._2
}
println(sumCombinations(15, List(1, 2, 5, 10)) mkString "\n")
}
When run, it displays:
List(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
List(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2)
List(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2)
List(1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2)
List(1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2)
List(1, 1, 1, 1, 1, 2, 2, 2, 2, 2)
List(1, 1, 1, 2, 2, 2, 2, 2, 2)
List(1, 2, 2, 2, 2, 2, 2, 2)
List(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5)
List(1, 1, 1, 1, 1, 1, 1, 1, 2, 5)
List(1, 1, 1, 1, 1, 1, 2, 2, 5)
List(1, 1, 1, 1, 2, 2, 2, 5)
List(1, 1, 2, 2, 2, 2, 5)
List(2, 2, 2, 2, 2, 5)
List(1, 1, 1, 1, 1, 5, 5)
List(1, 1, 1, 2, 5, 5)
List(1, 2, 2, 5, 5)
List(5, 5, 5)
List(1, 1, 1, 1, 1, 10)
List(1, 1, 1, 2, 10)
List(1, 2, 2, 10)
List(5, 10)
The sumCombinations() function may be used by itself, and the result may be further analyzed to display the "best" solution (the shortest list), or the number of solutions (the number of lists).
Note that even like this, the requirements may not be fully satisfied. It might happen that the order of each list in the solution be significant. In such a case, each list would have to be duplicated as many time as there are combination of its elements. Or we might be interested only in the combinations that are different.
For example, we might consider that List(5, 10) should give two combinations: List(5, 10) and List(10, 5). For List(5, 5, 5) it could give three combinations or one only, depending on the requirements. For integers, the three permutations are equivalent, but if we are dealing with coins, like in the "coin changing problem", they are not.
Also not stated in the requirements is the question of whether each number (or coin) may be used only once or many times. We could (and we should!) generalize the problem to a list of lists of occurrences of each number. This translates in real life into "what are the possible ways to make an certain amount of money with a set of coins (and not a set of coin values)". The original problem is just a particular case of this one, where we have as many occurrences of each coin as needed to make the total amount with each single coin value.
于 2012-11-20T10:31:02.680 回答
38
一个Javascript版本：
function subsetSum(numbers, target, partial) {
var s, n, remaining;
partial = partial || [];
// sum partial
s = partial.reduce(function (a, b) {
return a + b;
}, 0);
// check if the partial sum is equals to target
if (s === target) {
console.log("%s=%s", partial.join("+"), target)
}
if (s >= target) {
return; // if we reach the number why bother to continue
}
for (var i = 0; i < numbers.length; i++) {
n = numbers[i];
remaining = numbers.slice(i + 1);
subsetSum(remaining, target, partial.concat([n]));
}
}
subsetSum([3,9,8,4,5,7,10],15);
// output:
// 3+8+4=15
// 3+5+7=15
// 8+7=15
// 5+10=15于 2015-06-16T14:30:32.520 回答
13
相同算法的 C++ 版本
#include <iostream>
#include <list>
void subset_sum_recursive(std::list<int> numbers, int target, std::list<int> partial)
{
int s = 0;
for (std::list<int>::const_iterator cit = partial.begin(); cit != partial.end(); cit++)
{
s += *cit;
}
if(s == target)
{
std::cout << "sum([";
for (std::list<int>::const_iterator cit = partial.begin(); cit != partial.end(); cit++)
{
std::cout << *cit << ",";
}
std::cout << "])=" << target << std::endl;
}
if(s >= target)
return;
int n;
for (std::list<int>::const_iterator ai = numbers.begin(); ai != numbers.end(); ai++)
{
n = *ai;
std::list<int> remaining;
for(std::list<int>::const_iterator aj = ai; aj != numbers.end(); aj++)
{
if(aj == ai)continue;
remaining.push_back(*aj);
}
std::list<int> partial_rec=partial;
partial_rec.push_back(n);
subset_sum_recursive(remaining,target,partial_rec);
}
}
void subset_sum(std::list<int> numbers,int target)
{
subset_sum_recursive(numbers,target,std::list<int>());
}
int main()
{
std::list<int> a;
a.push_back (3); a.push_back (9); a.push_back (8);
a.push_back (4);
a.push_back (5);
a.push_back (7);
a.push_back (10);
int n = 15;
//std::cin >> n;
subset_sum(a, n);
return 0;
}
于 2013-03-19T10:09:58.580 回答
12
@msalvadores 代码答案的 C# 版本
void Main()
{
int[] numbers = {3,9,8,4,5,7,10};
int target = 15;
sum_up(new List<int>(numbers.ToList()),target);
}
static void sum_up_recursive(List<int> numbers, int target, List<int> part)
{
int s = 0;
foreach (int x in part)
{
s += x;
}
if (s == target)
{
Console.WriteLine("sum(" + string.Join(",", part.Select(n => n.ToString()).ToArray()) + ")=" + target);
}
if (s >= target)
{
return;
}
for (int i = 0;i < numbers.Count;i++)
{
var remaining = new List<int>();
int n = numbers[i];
for (int j = i + 1; j < numbers.Count;j++)
{
remaining.Add(numbers[j]);
}
var part_rec = new List<int>(part);
part_rec.Add(n);
sum_up_recursive(remaining,target,part_rec);
}
}
static void sum_up(List<int> numbers, int target)
{
sum_up_recursive(numbers,target,new List<int>());
}
于 2012-05-08T20:25:36.557 回答
8
Java非递归版本，它只是不断添加元素并在可能的值中重新分配它们。0's 被忽略并且适用于固定列表（你得到的是你可以玩的东西）或可重复数字的列表。
import java.util.*;
public class TestCombinations {
public static void main(String[] args) {
ArrayList<Integer> numbers = new ArrayList<>(Arrays.asList(0, 1, 2, 2, 5, 10, 20));
LinkedHashSet<Integer> targets = new LinkedHashSet<Integer>() {{
add(4);
add(10);
add(25);
}};
System.out.println("## each element can appear as many times as needed");
for (Integer target: targets) {
Combinations combinations = new Combinations(numbers, target, true);
combinations.calculateCombinations();
for (String solution: combinations.getCombinations()) {
System.out.println(solution);
}
}
System.out.println("## each element can appear only once");
for (Integer target: targets) {
Combinations combinations = new Combinations(numbers, target, false);
combinations.calculateCombinations();
for (String solution: combinations.getCombinations()) {
System.out.println(solution);
}
}
}
public static class Combinations {
private boolean allowRepetitions;
private int[] repetitions;
private ArrayList<Integer> numbers;
private Integer target;
private Integer sum;
private boolean hasNext;
private Set<String> combinations;
/**
* Constructor.
*
* @param numbers Numbers that can be used to calculate the sum.
* @param target Target value for sum.
*/
public Combinations(ArrayList<Integer> numbers, Integer target) {
this(numbers, target, true);
}
/**
* Constructor.
*
* @param numbers Numbers that can be used to calculate the sum.
* @param target Target value for sum.
*/
public Combinations(ArrayList<Integer> numbers, Integer target, boolean allowRepetitions) {
this.allowRepetitions = allowRepetitions;
if (this.allowRepetitions) {
Set<Integer> numbersSet = new HashSet<>(numbers);
this.numbers = new ArrayList<>(numbersSet);
} else {
this.numbers = numbers;
}
this.numbers.removeAll(Arrays.asList(0));
Collections.sort(this.numbers);
this.target = target;
this.repetitions = new int[this.numbers.size()];
this.combinations = new LinkedHashSet<>();
this.sum = 0;
if (this.repetitions.length > 0)
this.hasNext = true;
else
this.hasNext = false;
}
/**
* Calculate and return the sum of the current combination.
*
* @return The sum.
*/
private Integer calculateSum() {
this.sum = 0;
for (int i = 0; i < repetitions.length; ++i) {
this.sum += repetitions[i] * numbers.get(i);
}
return this.sum;
}
/**
* Redistribute picks when only one of each number is allowed in the sum.
*/
private void redistribute() {
for (int i = 1; i < this.repetitions.length; ++i) {
if (this.repetitions[i - 1] > 1) {
this.repetitions[i - 1] = 0;
this.repetitions[i] += 1;
}
}
if (this.repetitions[this.repetitions.length - 1] > 1)
this.repetitions[this.repetitions.length - 1] = 0;
}
/**
* Get the sum of the next combination. When 0 is returned, there's no other combinations to check.
*
* @return The sum.
*/
private Integer next() {
if (this.hasNext && this.repetitions.length > 0) {
this.repetitions[0] += 1;
if (!this.allowRepetitions)
this.redistribute();
this.calculateSum();
for (int i = 0; i < this.repetitions.length && this.sum != 0; ++i) {
if (this.sum > this.target) {
this.repetitions[i] = 0;
if (i + 1 < this.repetitions.length) {
this.repetitions[i + 1] += 1;
if (!this.allowRepetitions)
this.redistribute();
}
this.calculateSum();
}
}
if (this.sum.compareTo(0) == 0)
this.hasNext = false;
}
return this.sum;
}
/**
* Calculate all combinations whose sum equals target.
*/
public void calculateCombinations() {
while (this.hasNext) {
if (this.next().compareTo(target) == 0)
this.combinations.add(this.toString());
}
}
/**
* Return all combinations whose sum equals target.
*
* @return Combinations as a set of strings.
*/
public Set<String> getCombinations() {
return this.combinations;
}
@Override
public String toString() {
StringBuilder stringBuilder = new StringBuilder("" + sum + ": ");
for (int i = 0; i < repetitions.length; ++i) {
for (int j = 0; j < repetitions[i]; ++j) {
stringBuilder.append(numbers.get(i) + " ");
}
}
return stringBuilder.toString();
}
}
}
样本输入：
numbers: 0, 1, 2, 2, 5, 10, 20
targets: 4, 10, 25
样本输出：
## each element can appear as many times as needed
4: 1 1 1 1
4: 1 1 2
4: 2 2
10: 1 1 1 1 1 1 1 1 1 1
10: 1 1 1 1 1 1 1 1 2
10: 1 1 1 1 1 1 2 2
10: 1 1 1 1 2 2 2
10: 1 1 2 2 2 2
10: 2 2 2 2 2
10: 1 1 1 1 1 5
10: 1 1 1 2 5
10: 1 2 2 5
10: 5 5
10: 10
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2
25: 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2
25: 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
25: 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
25: 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
25: 1 1 1 2 2 2 2 2 2 2 2 2 2 2
25: 1 2 2 2 2 2 2 2 2 2 2 2 2
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 5
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 5
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 5
25: 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 5
25: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 5
25: 1 1 1 1 1 1 1 1 2 2 2 2 2 2 5
25: 1 1 1 1 1 1 2 2 2 2 2 2 2 5
25: 1 1 1 1 2 2 2 2 2 2 2 2 5
25: 1 1 2 2 2 2 2 2 2 2 2 5
25: 2 2 2 2 2 2 2 2 2 2 5
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 5
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 2 5 5
25: 1 1 1 1 1 1 1 1 1 1 1 2 2 5 5
25: 1 1 1 1 1 1 1 1 1 2 2 2 5 5
25: 1 1 1 1 1 1 1 2 2 2 2 5 5
25: 1 1 1 1 1 2 2 2 2 2 5 5
25: 1 1 1 2 2 2 2 2 2 5 5
25: 1 2 2 2 2 2 2 2 5 5
25: 1 1 1 1 1 1 1 1 1 1 5 5 5
25: 1 1 1 1 1 1 1 1 2 5 5 5
25: 1 1 1 1 1 1 2 2 5 5 5
25: 1 1 1 1 2 2 2 5 5 5
25: 1 1 2 2 2 2 5 5 5
25: 2 2 2 2 2 5 5 5
25: 1 1 1 1 1 5 5 5 5
25: 1 1 1 2 5 5 5 5
25: 1 2 2 5 5 5 5
25: 5 5 5 5 5
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10
25: 1 1 1 1 1 1 1 1 1 1 1 1 1 2 10
25: 1 1 1 1 1 1 1 1 1 1 1 2 2 10
25: 1 1 1 1 1 1 1 1 1 2 2 2 10
25: 1 1 1 1 1 1 1 2 2 2 2 10
25: 1 1 1 1 1 2 2 2 2 2 10
25: 1 1 1 2 2 2 2 2 2 10
25: 1 2 2 2 2 2 2 2 10
25: 1 1 1 1 1 1 1 1 1 1 5 10
25: 1 1 1 1 1 1 1 1 2 5 10
25: 1 1 1 1 1 1 2 2 5 10
25: 1 1 1 1 2 2 2 5 10
25: 1 1 2 2 2 2 5 10
25: 2 2 2 2 2 5 10
25: 1 1 1 1 1 5 5 10
25: 1 1 1 2 5 5 10
25: 1 2 2 5 5 10
25: 5 5 5 10
25: 1 1 1 1 1 10 10
25: 1 1 1 2 10 10
25: 1 2 2 10 10
25: 5 10 10
25: 1 1 1 1 1 20
25: 1 1 1 2 20
25: 1 2 2 20
25: 5 20
## each element can appear only once
4: 2 2
10: 1 2 2 5
10: 10
25: 1 2 2 20
25: 5 20
于 2018-02-22T12:36:20.933 回答
8
到目前为止有很多解决方案，但都是生成然后过滤的形式。这意味着他们可能会花费大量时间在不会导致解决方案的递归路径上工作。
这是一个解决方案，即O(size_of_array * (number_of_sums + number_of_solutions)). 换句话说，它使用动态编程来避免枚举永远不会匹配的可能解决方案。
对于咯咯笑和笑声，我使用正负数来完成这项工作，并使其成为迭代器。它适用于 Python 2.3+。
def subset_sum_iter(array, target):
sign = 1
array = sorted(array)
if target < 0:
array = reversed(array)
sign = -1
# Checkpoint A
last_index = {0: [-1]}
for i in range(len(array)):
for s in list(last_index.keys()):
new_s = s + array[i]
if 0 < (new_s - target) * sign:
pass # Cannot lead to target
elif new_s in last_index:
last_index[new_s].append(i)
else:
last_index[new_s] = [i]
# Checkpoint B
# Now yield up the answers.
def recur(new_target, max_i):
for i in last_index[new_target]:
if i == -1:
yield [] # Empty sum.
elif max_i <= i:
break # Not our solution.
else:
for answer in recur(new_target - array[i], i):
answer.append(array[i])
yield answer
for answer in recur(target, len(array)):
yield answer
这是一个与数组和目标一起使用的示例，其中其他解决方案中使用的过滤方法实际上永远无法完成。
def is_prime(n):
for i in range(2, n):
if 0 == n % i:
return False
elif n < i * i:
return True
if n == 2:
return True
else:
return False
def primes(limit):
n = 2
while True:
if is_prime(n):
yield(n)
n = n + 1
if limit < n:
break
for answer in subset_sum_iter(primes(1000), 76000):
print(answer)
这会在 2 秒内打印出所有 522 个答案。以前的方法很幸运能在宇宙的当前生命周期中找到任何答案。（整个空间有2^168 = 3.74144419156711e+50可能的组合贯穿。那……需要一段时间。）
解释 我被要求解释代码，但解释数据结构通常更具启发性。所以我将解释数据结构。
让我们考虑一下subset_sum_iter([-2, 2, -3, 3, -5, 5, -7, 7, -11, 11], 10)。
在检查点 A，我们已经意识到我们的目标是积极的，所以sign = 1。我们已经对输入进行了排序，以便array = [-11, -7, -5, -3, -2, 2, 3, 5, 7, 11]. 由于我们经常通过索引访问它，这里是从索引到值的映射：
0: -11
1: -7
2: -5
3: -3
4: -2
5: 2
6: 3
7: 5
8: 7
9: 11
通过检查点 B，我们使用动态编程来生成我们的last_index数据结构。它包含什么？
last_index = {
-28: [4],
-26: [3, 5],
-25: [4, 6],
-24: [5],
-23: [2, 4, 5, 6, 7],
-22: [6],
-21: [3, 4, 5, 6, 7, 8],
-20: [4, 6, 7],
-19: [3, 5, 7, 8],
-18: [1, 4, 5, 6, 7, 8],
-17: [4, 5, 6, 7, 8, 9],
-16: [2, 4, 5, 6, 7, 8],
-15: [3, 5, 6, 7, 8, 9],
-14: [3, 4, 5, 6, 7, 8, 9],
-13: [4, 5, 6, 7, 8, 9],
-12: [2, 4, 5, 6, 7, 8, 9],
-11: [0, 5, 6, 7, 8, 9],
-10: [3, 4, 5, 6, 7, 8, 9],
-9: [4, 5, 6, 7, 8, 9],
-8: [3, 5, 6, 7, 8, 9],
-7: [1, 4, 5, 6, 7, 8, 9],
-6: [5, 6, 7, 8, 9],
-5: [2, 4, 5, 6, 7, 8, 9],
-4: [6, 7, 8, 9],
-3: [3, 5, 6, 7, 8, 9],
-2: [4, 6, 7, 8, 9],
-1: [5, 7, 8, 9],
0: [-1, 5, 6, 7, 8, 9],
1: [6, 7, 8, 9],
2: [5, 6, 7, 8, 9],
3: [6, 7, 8, 9],
4: [7, 8, 9],
5: [6, 7, 8, 9],
6: [7, 8, 9],
7: [7, 8, 9],
8: [7, 8, 9],
9: [8, 9],
10: [7, 8, 9]
}
（旁注，它不是对称的，因为条件if 0 < (new_s - target) * sign阻止我们记录过去的任何东西target，在我们的例子中是 10。）
这是什么意思？好吧，进入条目，10: [7, 8, 9]。这意味着我们可以得到最终的总和，10最后选择的数字在索引 7、8 或 9 处。也就是说，最后选择的数字可以是 5、7 或 11。
让我们仔细看看如果我们选择索引 7 会发生什么。这意味着我们以 5 结束。因此，在我们来到索引 7 之前，我们必须达到 10-5 = 5。而 5 的条目读取，5: [6, 7, 8, 9]。所以我们可以选择索引 6，即 3。虽然我们在索引 7、8 和 9 处达到 5，但在索引 7 之前我们没有到达那里。所以我们倒数第二个选择必须是索引 6 处的 3 .
现在我们必须在索引 6 之前达到 5-3 = 2。条目 2 显示为：2: [5, 6, 7, 8, 9]。同样，我们只关心 index 处的答案，5因为其他答案发生得太晚了。所以倒数第三个选择必须是索引 5 处的 2。
最后，我们必须在索引 5 之前到达 2-2 = 0。条目 0 显示为：0: [-1, 5, 6, 7, 8, 9]。同样，我们只关心-1. 但-1它不是一个索引——事实上我用它来表示我们已经完成了选择。
所以我们刚刚找到了解决方案2+3+5 = 10。这是我们打印出来的第一个解决方案。
现在我们进入recur子函数。因为它是在我们的 main 函数内部定义的，所以它可以看到last_index.
首先要注意的是它调用yield，而不是return。这使它成为一个生成器。当你调用它时，你会返回一种特殊的迭代器。当你遍历那个迭代器时，你会得到一个它可以产生的所有东西的列表。但是你会在它生成它们时得到它们。如果它是一个很长的列表，你不要把它放在内存中。（有点重要，因为我们可以得到一个很长的清单。）
将产生的是您recur(new_target, max_i)可以总结为仅使用具有最大索引new_target的元素的所有方式。那就是它的答案：“我们必须在 index 之前到达。” 当然，它是递归的。arraymax_inew_targetmax_i+1
因此recur(target, len(array))，所有解决方案都使用任何索引达到目标。这就是我们想要的。
于 2020-10-15T22:24:58.890 回答
5
另一个 python 解决方案是使用该itertools.combinations模块，如下所示：
#!/usr/local/bin/python
from itertools import combinations
def find_sum_in_list(numbers, target):
results = []
for x in range(len(numbers)):
results.extend(
[
combo for combo in combinations(numbers ,x)
if sum(combo) == target
]
)
print results
if __name__ == "__main__":
find_sum_in_list([3,9,8,4,5,7,10], 15)
输出：[(8, 7), (5, 10), (3, 8, 4), (3, 5, 7)]
于 2012-05-15T21:21:41.403 回答
5
Thank you.. ephemient
我已将上述逻辑从 python 转换为 php ..
<?php
$data = array(array(2,3,5,10,15),array(4,6,23,15,12),array(23,34,12,1,5));
$maxsum = 25;
print_r(bestsum($data,$maxsum)); //function call
function bestsum($data,$maxsum)
{
$res = array_fill(0, $maxsum + 1, '0');
$res[0] = array(); //base case
foreach($data as $group)
{
$new_res = $res; //copy res
foreach($group as $ele)
{
for($i=0;$i<($maxsum-$ele+1);$i++)
{
if($res[$i] != 0)
{
$ele_index = $i+$ele;
$new_res[$ele_index] = $res[$i];
$new_res[$ele_index][] = $ele;
}
}
}
$res = $new_res;
}
for($i=$maxsum;$i>0;$i--)
{
if($res[$i]!=0)
{
return $res[$i];
break;
}
}
return array();
}
?>
于 2012-01-26T19:31:27.020 回答
4
Excel VBA 版本如下。我需要在 VBA 中实现这一点（不是我的偏好，不要评判我！），并使用此页面上的答案作为方法。我正在上传以防其他人也需要 VBA 版本。
Option Explicit
Public Sub SumTarget()
Dim numbers(0 To 6) As Long
Dim target As Long
target = 15
numbers(0) = 3: numbers(1) = 9: numbers(2) = 8: numbers(3) = 4: numbers(4) = 5
numbers(5) = 7: numbers(6) = 10
Call SumUpTarget(numbers, target)
End Sub
Public Sub SumUpTarget(numbers() As Long, target As Long)
Dim part() As Long
Call SumUpRecursive(numbers, target, part)
End Sub
Private Sub SumUpRecursive(numbers() As Long, target As Long, part() As Long)
Dim s As Long, i As Long, j As Long, num As Long
Dim remaining() As Long, partRec() As Long
s = SumArray(part)
If s = target Then Debug.Print "SUM ( " & ArrayToString(part) & " ) = " & target
If s >= target Then Exit Sub
If (Not Not numbers) <> 0 Then
For i = 0 To UBound(numbers)
Erase remaining()
num = numbers(i)
For j = i + 1 To UBound(numbers)
AddToArray remaining, numbers(j)
Next j
Erase partRec()
CopyArray partRec, part
AddToArray partRec, num
SumUpRecursive remaining, target, partRec
Next i
End If
End Sub
Private Function ArrayToString(x() As Long) As String
Dim n As Long, result As String
result = "{" & x(n)
For n = LBound(x) + 1 To UBound(x)
result = result & "," & x(n)
Next n
result = result & "}"
ArrayToString = result
End Function
Private Function SumArray(x() As Long) As Long
Dim n As Long
SumArray = 0
If (Not Not x) <> 0 Then
For n = LBound(x) To UBound(x)
SumArray = SumArray + x(n)
Next n
End If
End Function
Private Sub AddToArray(arr() As Long, x As Long)
If (Not Not arr) <> 0 Then
ReDim Preserve arr(0 To UBound(arr) + 1)
Else
ReDim Preserve arr(0 To 0)
End If
arr(UBound(arr)) = x
End Sub
Private Sub CopyArray(destination() As Long, source() As Long)
Dim n As Long
If (Not Not source) <> 0 Then
For n = 0 To UBound(source)
AddToArray destination, source(n)
Next n
End If
End Sub
输出（写入立即窗口）应该是：
SUM ( {3,8,4} ) = 15
SUM ( {3,5,7} ) = 15
SUM ( {8,7} ) = 15
SUM ( {5,10} ) = 15
于 2014-01-12T14:55:24.467 回答
4
我以为我会使用这个问题的答案，但我不能，所以这是我的答案。它使用的是Structure and Interpretation of Computer Programs中答案的修改版本。我认为这是一个更好的递归解决方案，应该更取悦纯粹主义者。
我的答案是在 Scala 中（如果我的 Scala 很糟糕，我很抱歉，我刚刚开始学习它）。findSumCombinations的疯狂之处在于对递归的原始列表进行排序和唯一性，以防止被欺骗。
def findSumCombinations(target: Int, numbers: List[Int]): Int = {
cc(target, numbers.distinct.sortWith(_ < _), List())
}
def cc(target: Int, numbers: List[Int], solution: List[Int]): Int = {
if (target == 0) {println(solution); 1 }
else if (target < 0 || numbers.length == 0) 0
else
cc(target, numbers.tail, solution)
+ cc(target - numbers.head, numbers, numbers.head :: solution)
}
要使用它：
> findSumCombinations(12345, List(1,5,22,15,0,..))
* Prints a whole heap of lists that will sum to the target *
于 2012-09-21T07:19:18.813 回答
4
这是R中的解决方案
subset_sum = function(numbers,target,partial=0){
if(any(is.na(partial))) return()
s = sum(partial)
if(s == target) print(sprintf("sum(%s)=%s",paste(partial[-1],collapse="+"),target))
if(s > target) return()
for( i in seq_along(numbers)){
n = numbers[i]
remaining = numbers[(i+1):length(numbers)]
subset_sum(remaining,target,c(partial,n))
}
}
于 2017-09-20T18:31:04.287 回答
4
Perl 版本（主要答案）：
use strict;
sub subset_sum {
my ($numbers, $target, $result, $sum) = @_;
print 'sum('.join(',', @$result).") = $target\n" if $sum == $target;
return if $sum >= $target;
subset_sum([@$numbers[$_ + 1 .. $#$numbers]], $target,
[@{$result||[]}, $numbers->[$_]], $sum + $numbers->[$_])
for (0 .. $#$numbers);
}
subset_sum([3,9,8,4,5,7,10,6], 15);
结果：
sum(3,8,4) = 15
sum(3,5,7) = 15
sum(9,6) = 15
sum(8,7) = 15
sum(4,5,6) = 15
sum(5,10) = 15
Javascript版本：
const subsetSum = (numbers, target, partial = [], sum = 0) => {
if (sum < target)
numbers.forEach((num, i) =>
subsetSum(numbers.slice(i + 1), target, partial.concat([num]), sum + num));
else if (sum == target)
console.log('sum(%s) = %s', partial.join(), target);
}
subsetSum([3,9,8,4,5,7,10,6], 15);实际返回结果（而不是打印结果）的 Javascript 单行代码：
const subsetSum=(n,t,p=[],s=0,r=[])=>(s<t?n.forEach((l,i)=>subsetSum(n.slice(i+1),t,[...p,l],s+l,r)):s==t?r.push(p):0,r);
console.log(subsetSum([3,9,8,4,5,7,10,6], 15));还有我最喜欢的带回调的单行代码：
const subsetSum=(n,t,cb,p=[],s=0)=>s<t?n.forEach((l,i)=>subsetSum(n.slice(i+1),t,cb,[...p,l],s+l)):s==t?cb(p):0;
subsetSum([3,9,8,4,5,7,10,6], 15, console.log);于 2020-02-02T09:21:18.623 回答
3
使用我几年前用 C++ 编写的表格的非常有效的算法。
如果您设置 PRINT 1 它将打印所有组合（但不会使用有效的方法）。
它非常高效，可以在不到 10 毫秒的时间内计算超过 10^14 种组合。
#include <stdio.h>
#include <stdlib.h>
//#include "CTime.h"
#define SUM 300
#define MAXNUMsSIZE 30
#define PRINT 0
long long CountAddToSum(int,int[],int,const int[],int);
void printr(const int[], int);
long long table1[SUM][MAXNUMsSIZE];
int main()
{
int Nums[]={3,4,5,6,7,9,13,11,12,13,22,35,17,14,18,23,33,54};
int sum=SUM;
int size=sizeof(Nums)/sizeof(int);
int i,j,a[]={0};
long long N=0;
//CTime timer1;
for(i=0;i<SUM;++i)
for(j=0;j<MAXNUMsSIZE;++j)
table1[i][j]=-1;
N = CountAddToSum(sum,Nums,size,a,0); //algorithm
//timer1.Get_Passd();
//printf("\nN=%lld time=%.1f ms\n", N,timer1.Get_Passd());
printf("\nN=%lld \n", N);
getchar();
return 1;
}
long long CountAddToSum(int s, int arr[],int arrsize, const int r[],int rsize)
{
static int totalmem=0, maxmem=0;
int i,*rnew;
long long result1=0,result2=0;
if(s<0) return 0;
if (table1[s][arrsize]>0 && PRINT==0) return table1[s][arrsize];
if(s==0)
{
if(PRINT) printr(r, rsize);
return 1;
}
if(arrsize==0) return 0;
//else
rnew=(int*)malloc((rsize+1)*sizeof(int));
for(i=0;i<rsize;++i) rnew[i]=r[i];
rnew[rsize]=arr[arrsize-1];
result1 = CountAddToSum(s,arr,arrsize-1,rnew,rsize);
result2 = CountAddToSum(s-arr[arrsize-1],arr,arrsize,rnew,rsize+1);
table1[s][arrsize]=result1+result2;
free(rnew);
return result1+result2;
}
void printr(const int r[], int rsize)
{
int lastr=r[0],count=0,i;
for(i=0; i<rsize;++i)
{
if(r[i]==lastr)
count++;
else
{
printf(" %d*%d ",count,lastr);
lastr=r[i];
count=1;
}
}
if(r[i-1]==lastr) printf(" %d*%d ",count,lastr);
printf("\n");
}
于 2013-10-16T11:45:29.313 回答
3
O(t*N)当复杂度（动态解决方案）大于指数算法时，这是一个非常适合小 N 和非常大目标总和的 Java 版本。我的版本使用了中间攻击，以及一点点移动，以降低从经典幼稚O(n*2^n)到O(2^(n/2)).
如果您想将其用于具有 32 到 64 个元素的集合，您应该将int表示 step 函数中当前子集的 更改为 a，long尽管随着集合大小的增加，性能会明显下降。如果要将其用于具有奇数个元素的集合，则应在集合中添加 0 以使其为偶数。
import java.util.ArrayList;
import java.util.List;
public class SubsetSumMiddleAttack {
static final int target = 100000000;
static final int[] set = new int[]{ ... };
static List<Subset> evens = new ArrayList<>();
static List<Subset> odds = new ArrayList<>();
static int[][] split(int[] superSet) {
int[][] ret = new int[2][superSet.length / 2];
for (int i = 0; i < superSet.length; i++) ret[i % 2][i / 2] = superSet[i];
return ret;
}
static void step(int[] superSet, List<Subset> accumulator, int subset, int sum, int counter) {
accumulator.add(new Subset(subset, sum));
if (counter != superSet.length) {
step(superSet, accumulator, subset + (1 << counter), sum + superSet[counter], counter + 1);
step(superSet, accumulator, subset, sum, counter + 1);
}
}
static void printSubset(Subset e, Subset o) {
String ret = "";
for (int i = 0; i < 32; i++) {
if (i % 2 == 0) {
if ((1 & (e.subset >> (i / 2))) == 1) ret += " + " + set[i];
}
else {
if ((1 & (o.subset >> (i / 2))) == 1) ret += " + " + set[i];
}
}
if (ret.startsWith(" ")) ret = ret.substring(3) + " = " + (e.sum + o.sum);
System.out.println(ret);
}
public static void main(String[] args) {
int[][] superSets = split(set);
step(superSets[0], evens, 0,0,0);
step(superSets[1], odds, 0,0,0);
for (Subset e : evens) {
for (Subset o : odds) {
if (e.sum + o.sum == target) printSubset(e, o);
}
}
}
}
class Subset {
int subset;
int sum;
Subset(int subset, int sum) {
this.subset = subset;
this.sum = sum;
}
}
于 2016-08-08T13:28:52.157 回答
2
这类似于硬币找零问题
public class CoinCount
{
public static void main(String[] args)
{
int[] coins={1,4,6,2,3,5};
int count=0;
for (int i=0;i<coins.length;i++)
{
count=count+Count(9,coins,i,0);
}
System.out.println(count);
}
public static int Count(int Sum,int[] coins,int index,int curSum)
{
int count=0;
if (index>=coins.length)
return 0;
int sumNow=curSum+coins[index];
if (sumNow>Sum)
return 0;
if (sumNow==Sum)
return 1;
for (int i= index+1;i<coins.length;i++)
count+=Count(Sum,coins,i,sumNow);
return count;
}
}
于 2013-08-07T02:55:54.137 回答
2
PHP 版本，灵感来自 Keith Beller 的 C# 版本。
bala 的 PHP 版本对我不起作用，因为我不需要对数字进行分组。我想要一个具有一个目标值和一个数字池的更简单的实现。此功能还将修剪任何重复的条目。
编辑 25/10/2021：添加了精度参数以支持浮点数（现在需要 bcmath 扩展）。
/**
* Calculates a subset sum: finds out which combinations of numbers
* from the numbers array can be added together to come to the target
* number.
*
* Returns an indexed array with arrays of number combinations.
*
* Example:
*
* <pre>
* $matches = subset_sum(array(5,10,7,3,20), 25);
* </pre>
*
* Returns:
*
* <pre>
* Array
* (
* [0] => Array
* (
* [0] => 3
* [1] => 5
* [2] => 7
* [3] => 10
* )
* [1] => Array
* (
* [0] => 5
* [1] => 20
* )
* )
* </pre>
*
* @param number[] $numbers
* @param number $target
* @param array $part
* @param int $precision
* @return array[number[]]
*/
function subset_sum($numbers, $target, $precision=0, $part=null)
{
// we assume that an empty $part variable means this
// is the top level call.
$toplevel = false;
if($part === null) {
$toplevel = true;
$part = array();
}
$s = 0;
foreach($part as $x)
{
$s = $s + $x;
}
// we have found a match!
if(bccomp((string) $s, (string) $target, $precision) === 0)
{
sort($part); // ensure the numbers are always sorted
return array(implode('|', $part));
}
// gone too far, break off
if($s >= $target)
{
return null;
}
$matches = array();
$totalNumbers = count($numbers);
for($i=0; $i < $totalNumbers; $i++)
{
$remaining = array();
$n = $numbers[$i];
for($j = $i+1; $j < $totalNumbers; $j++)
{
$remaining[] = $numbers[$j];
}
$part_rec = $part;
$part_rec[] = $n;
$result = subset_sum($remaining, $target, $precision, $part_rec);
if($result)
{
$matches = array_merge($matches, $result);
}
}
if(!$toplevel)
{
return $matches;
}
// this is the top level function call: we have to
// prepare the final result value by stripping any
// duplicate results.
$matches = array_unique($matches);
$result = array();
foreach($matches as $entry)
{
$result[] = explode('|', $entry);
}
return $result;
}
例子：
$result = subset_sum(array(5, 10, 7, 3, 20), 25);
这将返回一个带有两个数字组合数组的索引数组：
3, 5, 7, 10
5, 20
浮点数示例：
// Specify the precision in the third argument
$result = subset_sum(array(0.40, 0.03, 0.05), 0.45, 2);
这将返回一个匹配：
0.40, 0.05
于 2017-04-11T16:55:44.107 回答
2
首先推导出0。零是加法的一个恒等式，因此在这种特殊情况下，对幺半群定律来说它是无用的。如果您想爬升到正数，也可以推导出负数。否则你还需要减法运算。
所以......你可以在这个特定工作中获得的最快算法如下在 JS 中给出。
function items2T([n,...ns],t){
var c = ~~(t/n);
return ns.length ? Array(c+1).fill()
.reduce((r,_,i) => r.concat(items2T(ns, t-n*i).map(s => Array(i).fill(n).concat(s))),[])
: t % n ? []
: [Array(c).fill(n)];
};
var data = [3, 9, 8, 4, 5, 7, 10],
result;
console.time("combos");
result = items2T(data, 15);
console.timeEnd("combos");
console.log(JSON.stringify(result));这是一个非常快的算法，但如果你对data数组进行降序排序，它会更快。使用.sort()是微不足道的，因为该算法最终将以更少的递归调用结束。
于 2019-03-31T20:12:38.313 回答
2
这是一个更好的版本，具有更好的输出格式和 C++ 11 功能：
void subset_sum_rec(std::vector<int> & nums, const int & target, std::vector<int> & partialNums)
{
int currentSum = std::accumulate(partialNums.begin(), partialNums.end(), 0);
if (currentSum > target)
return;
if (currentSum == target)
{
std::cout << "sum([";
for (auto it = partialNums.begin(); it != std::prev(partialNums.end()); ++it)
cout << *it << ",";
cout << *std::prev(partialNums.end());
std::cout << "])=" << target << std::endl;
}
for (auto it = nums.begin(); it != nums.end(); ++it)
{
std::vector<int> remaining;
for (auto it2 = std::next(it); it2 != nums.end(); ++it2)
remaining.push_back(*it2);
std::vector<int> partial = partialNums;
partial.push_back(*it);
subset_sum_rec(remaining, target, partial);
}
}
于 2017-03-26T01:12:15.933 回答
2
我将 C# 示例移植到 Objective-c 并没有在响应中看到它：
//Usage
NSMutableArray* numberList = [[NSMutableArray alloc] init];
NSMutableArray* partial = [[NSMutableArray alloc] init];
int target = 16;
for( int i = 1; i<target; i++ )
{ [numberList addObject:@(i)]; }
[self findSums:numberList target:target part:partial];
//*******************************************************************
// Finds combinations of numbers that add up to target recursively
//*******************************************************************
-(void)findSums:(NSMutableArray*)numbers target:(int)target part:(NSMutableArray*)partial
{
int s = 0;
for (NSNumber* x in partial)
{ s += [x intValue]; }
if (s == target)
{ NSLog(@"Sum[%@]", partial); }
if (s >= target)
{ return; }
for (int i = 0;i < [numbers count];i++ )
{
int n = [numbers[i] intValue];
NSMutableArray* remaining = [[NSMutableArray alloc] init];
for (int j = i + 1; j < [numbers count];j++)
{ [remaining addObject:@([numbers[j] intValue])]; }
NSMutableArray* partRec = [[NSMutableArray alloc] initWithArray:partial];
[partRec addObject:@(n)];
[self findSums:remaining target:target part:partRec];
}
}
于 2016-01-06T16:18:26.183 回答
1
于 2012-11-30T21:56:18.393 回答
1
@KeithBeller 的回答稍微改变了变量名和一些评论。
public static void Main(string[] args)
{
List<int> input = new List<int>() { 3, 9, 8, 4, 5, 7, 10 };
int targetSum = 15;
SumUp(input, targetSum);
}
public static void SumUp(List<int> input, int targetSum)
{
SumUpRecursive(input, targetSum, new List<int>());
}
private static void SumUpRecursive(List<int> remaining, int targetSum, List<int> listToSum)
{
// Sum up partial
int sum = 0;
foreach (int x in listToSum)
sum += x;
//Check sum matched
if (sum == targetSum)
Console.WriteLine("sum(" + string.Join(",", listToSum.ToArray()) + ")=" + targetSum);
//Check sum passed
if (sum >= targetSum)
return;
//Iterate each input character
for (int i = 0; i < remaining.Count; i++)
{
//Build list of remaining items to iterate
List<int> newRemaining = new List<int>();
for (int j = i + 1; j < remaining.Count; j++)
newRemaining.Add(remaining[j]);
//Update partial list
List<int> newListToSum = new List<int>(listToSum);
int currentItem = remaining[i];
newListToSum.Add(currentItem);
SumUpRecursive(newRemaining, targetSum, newListToSum);
}
}'
于 2015-03-28T19:37:31.107 回答
1
这也可用于打印所有答案
public void recur(int[] a, int n, int sum, int[] ans, int ind) {
if (n < 0 && sum != 0)
return;
if (n < 0 && sum == 0) {
print(ans, ind);
return;
}
if (sum >= a[n]) {
ans[ind] = a[n];
recur(a, n - 1, sum - a[n], ans, ind + 1);
}
recur(a, n - 1, sum, ans, ind);
}
public void print(int[] a, int n) {
for (int i = 0; i < n; i++)
System.out.print(a[i] + " ");
System.out.println();
}
时间复杂度是指数级的。2^n 的顺序
于 2016-05-13T00:45:17.910 回答
1
Java解决方案的Swift 3转换：（@JeremyThompson）
protocol _IntType { }
extension Int: _IntType {}
extension Array where Element: _IntType {
func subsets(to: Int) -> [[Element]]? {
func sum_up_recursive(_ numbers: [Element], _ target: Int, _ partial: [Element], _ solution: inout [[Element]]) {
var sum: Int = 0
for x in partial {
sum += x as! Int
}
if sum == target {
solution.append(partial)
}
guard sum < target else {
return
}
for i in stride(from: 0, to: numbers.count, by: 1) {
var remaining = [Element]()
for j in stride(from: i + 1, to: numbers.count, by: 1) {
remaining.append(numbers[j])
}
var partial_rec = [Element](partial)
partial_rec.append(numbers[i])
sum_up_recursive(remaining, target, partial_rec, &solution)
}
}
var solutions = [[Element]]()
sum_up_recursive(self, to, [Element](), &solutions)
return solutions.count > 0 ? solutions : nil
}
}
用法：
let numbers = [3, 9, 8, 4, 5, 7, 10]
if let solution = numbers.subsets(to: 15) {
print(solution) // output: [[3, 8, 4], [3, 5, 7], [8, 7], [5, 10]]
} else {
print("not possible")
}
于 2016-08-23T09:13:42.617 回答
1
推荐作为答案：
这是使用 es2015生成器的解决方案：
function* subsetSum(numbers, target, partial = [], partialSum = 0) {
if(partialSum === target) yield partial
if(partialSum >= target) return
for(let i = 0; i < numbers.length; i++){
const remaining = numbers.slice(i + 1)
, n = numbers[i]
yield* subsetSum(remaining, target, [...partial, n], partialSum + n)
}
}
使用生成器实际上非常有用，因为它允许您在找到有效子集后立即暂停脚本执行。这与没有生成器（即缺少状态）的解决方案形成对比，后者必须遍历numbers
于 2017-11-17T18:35:09.733 回答
1
我不喜欢上面看到的 Javascript 解决方案。这是我使用部分应用、闭包和递归构建的：
好的，我主要关心的是，如果组合数组可以满足目标要求，希望这个接近你会开始找到其余的组合
这里只需设置目标并传递组合数组。
function main() {
const target = 10
const getPermutationThatSumT = setTarget(target)
const permutation = getPermutationThatSumT([1, 4, 2, 5, 6, 7])
console.log( permutation );
}
我想出的当前实现
function setTarget(target) {
let partial = [];
return function permute(input) {
let i, removed;
for (i = 0; i < input.length; i++) {
removed = input.splice(i, 1)[0];
partial.push(removed);
const sum = partial.reduce((a, b) => a + b)
if (sum === target) return partial.slice()
if (sum < target) permute(input)
input.splice(i, 0, removed);
partial.pop();
}
return null
};
}
于 2020-06-05T22:32:20.490 回答
1
我正在为 scala 分配做类似的事情。想在这里发布我的解决方案：
def countChange(money: Int, coins: List[Int]): Int = {
def getCount(money: Int, remainingCoins: List[Int]): Int = {
if(money == 0 ) 1
else if(money < 0 || remainingCoins.isEmpty) 0
else
getCount(money, remainingCoins.tail) +
getCount(money - remainingCoins.head, remainingCoins)
}
if(money == 0 || coins.isEmpty) 0
else getCount(money, coins)
}
于 2016-09-13T10:32:44.033 回答
0
import java.util.*;
public class Main{
int recursionDepth = 0;
private int[][] memo;
public static void main(String []args){
int[] nums = new int[] {5,2,4,3,1};
int N = nums.length;
Main main = new Main();
main.memo = new int[N+1][N+1];
main._findCombo(0, N-1,nums, 8, 0, new LinkedList() );
System.out.println(main.recursionDepth);
}
private void _findCombo(
int from,
int to,
int[] nums,
int targetSum,
int currentSum,
LinkedList<Integer> list){
if(memo[from][to] != 0) {
currentSum = currentSum + memo[from][to];
}
if(currentSum > targetSum) {
return;
}
if(currentSum == targetSum) {
System.out.println("Found - " +list);
return;
}
recursionDepth++;
for(int i= from ; i <= to; i++){
list.add(nums[i]);
memo[from][i] = currentSum + nums[i];
_findCombo(i+1, to,nums, targetSum, memo[from][i], list);
list.removeLast();
}
}
}
于 2020-04-27T21:00:14.787 回答
0
func sum(array : [Int]) -> Int{
var sum = 0
array.forEach { (item) in
sum = item + sum
}
return sum
}
func susetNumbers(array :[Int], target : Int, subsetArray: [Int],result : inout [[Int]]) -> [[Int]]{
let s = sum(array: subsetArray)
if(s == target){
print("sum\(subsetArray) = \(target)")
result.append(subsetArray)
}
for i in 0..<array.count{
let n = array[i]
let remaning = Array(array[(i+1)..<array.count])
susetNumbers(array: remaning, target: target, subsetArray: subsetArray + [n], result: &result)
}
return result
}
var resultArray = [[Int]]()
let newA = susetNumbers(array: [1,2,3,4,5], target: 5, subsetArray: [],result:&resultArray)
print(resultArray)
于 2020-10-18T14:06:15.373 回答
0
针对此问题的迭代 C++ 堆栈解决方案。与其他一些迭代解决方案不同，它不会制作不必要的中间序列副本。
#include <vector>
#include <iostream>
// Given a positive integer, return all possible combinations of
// positive integers that sum up to it.
std::vector<std::vector<int>> print_all_sum(int target){
std::vector<std::vector<int>> output;
std::vector<int> stack;
int curr_min = 1;
int sum = 0;
while (curr_min < target) {
sum += curr_min;
if (sum >= target) {
if (sum == target) {
output.push_back(stack); // make a copy
output.back().push_back(curr_min);
}
sum -= curr_min + stack.back();
curr_min = stack.back() + 1;
stack.pop_back();
} else {
stack.push_back(curr_min);
}
}
return output;
}
int main()
{
auto vvi = print_all_sum(6);
for (auto const& v: vvi) {
for(auto const& i: v) {
std::cout << i;
}
std::cout << "\n";
}
return 0;
}
输出print_all_sum(6)：
111111
11112
1113
1122
114
123
15
222
24
33
于 2020-12-09T17:24:17.133 回答
0
function solve(n){
let DP = [];
DP[0] = DP[1] = DP[2] = 1;
DP[3] = 2;
for (let i = 4; i <= n; i++) {
DP[i] = DP[i-1] + DP[i-3] + DP[i-4];
}
return DP[n]
}
console.log(solve(5))
这是 JS 的动态解决方案，用于告诉任何人可以通过多少种方式获得一定的金额。如果您考虑时间和空间复杂性，这可能是正确的解决方案。
于 2020-04-05T12:35:03.923 回答