在 Leetcode 看到很好的一篇总结,所以我根据看到的再重新写一篇,方便自己理解
首先理解一下什么是回溯,可以用迷宫来做比喻.
通常走出一个迷宫的方法如下:
- 进入迷宫(废话...)
- 选一条道一直走, 一直走到有一个分岔
- 选最左的一条道
- 如果你走到了死胡同, 就回到之前最后一个分岔口, 选从左边数起第二条分岔.
- 一直循环 2 - 5, 直到找到出口位置
伪代码就变成了这个样子:
start
while exist not found:
for each path in the fork:
check whether the path is safe
if yes :
select it and continue walk
if all paths don't resolve the maze, return False
end
接下来看 Leetcode 原题
Subsets
Given a set of distinct integers, nums, return all possible subsets.
Note: The solution set must not contain duplicate subsets.
For example,
If nums = [1,2,3], a solution is:
[
[3],
[1],
[2],
[1,2,3],
[1,3],
[2,3],
[1,2],
[]
]
class DFS: # 也可以把 backtracking 当成 DFS
def subsets(self, S):
self.result = []
self.backtrack(0, sorted(S), [])
return self.result
# 这个是模板啊~
def backtrack(self, start, S, temp):
self.result.append(temp[:])
for i in range(start , len(S)):
temp.append(S[i])
self.backtrack(i + 1, S, temp)
temp.pop()
看 follow up:
Subsets II
Given a collection of integers that might contain duplicates, nums, return all possible subsets.
Note: The solution set must not contain duplicate subsets.
For example,
If nums = [1,2,2], a solution is:
[
[2],
[1],
[1,2,2],
[2,2],
[1,2],
[]
]
这题就是多加了一个判断, 去掉重复的
class DFS:
def subsetsWithDup(self, S):
self.result = []
if len(S) == 0: return self.result
self.helper(0, sorted(S), [])
return self.result
def backtrack(self, start, S, temp):
self.result.append(temp[:])
for i in range(start, len(S)):
if i != start and S[i] == S[i - 1]: # 忽略重复的
continue
temp.append(S[i])
self.helper(i + 1, S, temp)
temp.pop()
另外一类相似的题
Permutations
Given a collection of distinct numbers, return all possible permutations.
For example,
[1,2,3] have the following permutations:
[
[1,2,3],
[1,3,2],
[2,1,3],
[2,3,1],
[3,1,2],
[3,2,1]
]
跟前两道题的区别在于 subset 可以不包括全部.
class Solution:
def permute(self, nums):
self.result = []
self.backtrack(nums, [])
return self.result
def backtrack(self, nums, temp):
if len(temp) == len(nums):
self.result.append(temp[:])
# 由于长度一样, 所以不用知道 index,连 i 都免了
else:
for n in nums:
if n in temp:
continue
temp.append(n)
self.backtrack(nums, temp)
temp.pop()
Permutations II
Given a collection of numbers that might contain duplicates, return all possible unique permutations.
For example,
[1,1,2] have the following unique permutations:
[
[1,1,2],
[1,2,1],
[2,1,1]
]
一样玩法,查重复的
class DFS2:
def permuteUnique(self, nums):
self.result = []
# 查重的方法改变了,通过一个数组来记录有没有走过这个位置的数
self.backtrack(sorted(nums), [], [False for _ in range(len(nums))])
return self.result
def backtrack(self, nums, temp, visited):
if len(nums) == len(temp):
self.result.append(temp[:])
else:
for i in range(len(nums)):
# 保证前一个相同的数也必须包含在结果里
if visited[i] or (i > 0 and nums[i] == nums[i - 1]) and not visited[i - 1]:
continue
visited[i] = True
temp.append(nums[i])
self.backtrack(nums, temp, visited)
visited[i] = False
temp.pop()
另一组题目
Combination Sum
Given a set of candidate numbers (C) (without duplicates) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.
The same repeated number may be chosen from C unlimited number of times.
Note:
All numbers (including target) will be positive integers.
The solution set must not contain duplicate combinations.
For example, given candidate set [2, 3, 6, 7] and target 7,
A solution set is:
[
[7],
[2, 2, 3]
]
class DFS:
# @param candidates, a list of integers
# @param target, integer
# @return a list of lists of integers
def combinationSum(self, candidates, target):
self.result= []
self.backtrack(sorted(candidates), target, 0, 0, [])
return self.result
def backtrack(self, candidates, target, start, total, temp):
if total > target:
return
elif total == target:
self.result.append(temp[:])
else:
for i in range(start, len(candidates)):
temp.append(candidates[i])
# 这里注意用的是 i 而不是 i+ 1, 原因是因为一个数可以重复利用很多次
self.backtrack(candidates, target, i, total + candidates[i], temp)
temp.pop()
题型变化一下, 一样是跟重复有关系
Combination Sum II
Given a collection of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.
Each number in C may only be used once in the combination.
Note:
All numbers (including target) will be positive integers.
The solution set must not contain duplicate combinations.
For example, given candidate set [10, 1, 2, 7, 6, 1, 5] and target 8,
A solution set is:
[
[1, 7],
[1, 2, 5],
[2, 6],
[1, 1, 6]
]
class Solution:
# @param candidates, a list of integers
# @param target, integer
# @return a list of lists of integers
def combinationSum2(self, candidates, target):
self.result = []
self.backtrack(sorted(candidates), 0, 0, target, [])
return self.result
def backtrack(self, candidates, start, total, target, temp):
if total > target:
return
elif total == target:
self.result.append(temp[:])
else:
for i in range(start, len(candidates)):
if i > start and candidates[i] == candidates[i - 1]:
continue
temp.append(candidates[i])
self.backtrack(candidates, i + 1, candidates[i] + total, target, temp)
temp.pop()
接下来的题加了个限制总共可以用多少个数
Combination Sum III
Find all possible combinations of k numbers that add up to a number n, given that only numbers from 1 to 9 can be used and each combination should be a unique set of numbers.
Example 1:
Input: k = 3, n = 7
Output:
[[1,2,4]]
Example 2:
Input: k = 3, n = 9
Output:
[[1,2,6], [1,3,5], [2,3,4]]
要注意题目限制是 1 - 9, 9个数字
class Solution(object):
def combinationSum3(self, k, n):
"""
:type k: int
:type n: int
:rtype: List[List[int]]
"""
self.result = []
if k < 0: return []
self.backtrack(k, n, 1, 0, [])
return self.result
def backtrack(self, k, n, start, total, temp):
if total == n and len(temp) == k:
self.result.append(temp[:])
else:
for i in range(start, 10):
temp.append(i)
self.backtrack(k, n, i + 1, i + total, temp)
temp.pop()
至于第四题 Combination Sum IV, 包含了负数和不限制数量的条件,要用 DP 来解题, 这个以后在别的地方再提吧.
接下来的题其实都大同小异了...
Palindrome Partitioning
Given a string s, partition s such that every substring of the partition is a palindrome.
Return all possible palindrome partitioning of s.
For example, given s = "aab",
Return
[
["aa","b"],
["a","a","b"]
]
变成了 String, 再加个回文的判定
class Solution(object):
# @param s, a string
# @return a list of lists of string
def partition(self, s):
self.result = []
# track each possible partition
self.backtrack(s, 0, [])
return self.result
def backtrack(self, s, start, temp):
if start == len(s):
self.result.append(temp[:])
else:
for i in range(start, len(s)):
string = s[start: i + 1]
if self.isPalindrome(string):
temp.append(string)
self.backtrack(s, i + 1, temp)
temp.pop()
def isPalindrome(self, string):
left, right = 0, len(string) - 1
while left < right:
if string[left] != string[right]:
return False
left+=1
right-=1
return True
Restore IP Addresses
Given a string containing only digits, restore it by returning all possible valid IP address combinations.
For example:
Given "25525511135",
return ["255.255.11.135", "255.255.111.35"]. (Order does not matter)
这道题的用法其实是一样的, 加上有效判断数字的话能快不少
class Solution:
# @param s, a string
# @return a list of strings
def restoreIpAddresses(self, s):
if len(s) < 4 or len(s) > 12:
return []
self.result = []
self.backtrack(s, 0, [])
return self.result
def backtrack(self, s, start, temp):
if len(temp) == 4 and start == len(s):
self.result.append(".".join(temp))
for i in range(start, min(start + 4, len(s))):
string = s[start: i + 1]
if self.isValid(string):
temp.append(string)
self.backtrack(s, i + 1, temp)
temp.pop()
def isValid(self, string):
if not string: return False
if len(string) > 1 and string[0] == '0':
return False
return int(string) <= 255
Factor Combinations
Numbers can be regarded as product of its factors. For example,
8 = 2 x 2 x 2;
= 2 x 4.
Write a function that takes an integer n and return all possible combinations of its factors.
Note:
You may assume that n is always positive.
Factors should be greater than 1 and less than n.
Examples:
input: 1
output:
[]
input: 37
output:
[]
input: 12
output:
[
[2, 6],
[2, 2, 3],
[3, 4]
]
input: 32
output:
[
[2, 16],
[2, 2, 8],
[2, 2, 2, 4],
[2, 2, 2, 2, 2],
[2, 4, 4],
[4, 8]
]
相同的算法但是 Python 超时了...这个题意很简单,就是要找到除了1和自己以外有因数的解, n <= 1 && temp.size() > 1这个条件可以把1 和自己本身的数去掉.
import java.util.List;
import java.util.ArrayList;
public class Solution {
public static void main(String[] args) {
System.out.println(new Solution().getFactors(8));
}
private List> result;
public List> getFactors(int n) {
result = new ArrayList<>();
List temp = new ArrayList<>();
backtrack(n, temp, 2);
return result;
}
private void backtrack(int n, List temp, int start){
if (n <= 1 && temp.size() > 1) {
result.add(new ArrayList(temp));
}else {
for (int i = start; i < n + 1; i++) {
if (n % i == 0) {
temp.add(i);
backtrack(n/i, temp, i);
temp.remove(temp.size() - 1);
}
}
}
}
}