題目出處

125 · Backpack II

難度

medium

個人範例程式碼

from typing import (
    List,
)

class Solution:
    """
    @param m: An integer m denotes the size of a backpack
    @param a: Given n items with size A[i]
    @param v: Given n items with value V[i]
    @return: The maximum value
    """
    def back_pack_i_i(self, m: int, a: List[int], v: List[int]) -> int:
        # write your code here
        # dp[i][j] = max price (i this round take, j volume)
        n = len(a)
        dp = [[0] * (m+1) for _ in range(n+1)]

        for i in range(1, n+1):
            for j in range(1, m+1):
                dp[i][j] = dp[i-1][j]
                last_volume = j - a[i-1]
                if last_volume >= 0:
                    dp[i][j] = max(dp[i][j], dp[i-1][last_volume] + v[i-1])

        print(dp[-1])
        return max(dp[-1])

算法說明

本題是背包系列問題的第 2 題，建議初學者可以從第一題開始去學：
第 1 題：最基本的背包問題，不重複 size，物品只有一個，計算組合可能性
第 2 題：最基本的背包問題，不重複 size，物品只有一個，計算最大價值
第 3 題：完全背包問題，有重複 size，物品無限數量，計算最大價值
第 4 題：有重複 size，物品無限數量，計算可能組合 ，「(1,2,3) 與 (2,3,1) 視為相同答案」
第 5 題：有重複 size，物品有限數量，計算可能組合 ，「(1,2,3) 與 (2,3,1) 視為相同答案」
第 6 題：有重複 size，物品無限數量，計算可能「排列」 ，也就是說「(1,2,3) 與 (2,3,1) 視為不同答案」
第 7 題：有重複物品，有限制物品數數量上限，問可能組合
第 8 題：有重複物品，有限制物品數數量上限，問最大價值
第 9 題：變化問題，有重複 size，計算機率
第 10 題：變化問題，只是把 size 面額提高

這題比起第一題，我們多考慮物品的價值，第一題只需要考慮 size 即可。
我們用 dp 保存當下背包的 max value

input handling

一同在 dp 內處理

Boundary conditions

用兩層 for 來控制搜尋範圍，
外層 for size (表示我們選擇新的項目)
內層 for volume (表示紀錄目前已有的選擇下，可否達到裝滿的目標，並記錄最大價值)

Reference

⭐ Leetcode 解題紀錄 ⭐		題型	資料結構	Python Solution	C++ Solution	Note
⭐BFS 相關題型 ⭐
104	Maximum Depth of Binary Tree	BFS (分層)		Python
94	Binary Tree Inorder Traversal	BFS (分層)	Tree	Python		內含 處理 Tree 樹問題的重點
102	Binary Tree Level Order Traversal	BFS (分層)	Tree	Python
103	Binary Tree Zigzag Level Order Traversal	BFS (分層)	Tree	Python
107	Binary Tree Level Order Traversal II	BFS (分層)	Tree	Python
133	Clone Graph	BFS (分層)	Graph	Python		Graph 的基本操作 #重要題型
127	Word Ladder	BFS (分層), DFS	Graph	Python
[Lint] 127	Topological Sorting	BFS (拓撲)		Python		內有 indegree, outdegree 介紹 #重要題型
207	Course Schedule	BFS (拓樸)	Graph	Python
210	Course Schedule II	BFS (拓樸)	Graph	Python
[Lint] 892	Alien Dictionary	BFS (拓樸)	Graph	Python
[Lint] 431	Connected Component in Undirected Graph	BFS (連通塊)	Graph	Python		內含 BFS 模板 #重要題型
1091	Shortest Path in Binary Matrix	BFS (最短路徑)	Matrix	Python
⭐ Binary Serach 相關題型 ⭐
33	Search in Rotated Sorted Array	Binary Serach	Array	Python		#重要題型
34	Find First and Last Position of Element in Sorted Array	Binary Serach		Python
50	Pow(x, n)	Binary Serach		Python
153	Find Minimum in Rotated Sorted Array	Binary Serach	Array	Python
162	Find Peak Element	Binary Serach		Python
278	First Bad Version	Binary Serach		Python
658	Find K Closest Elements	Binary Serach		Python
704	Binary Search	Binary Serach		Python
852	Peak Index in a Mountain Array	Binary Serach		Python
[Lint] 14	First Position of Target	Binary Serach		Python
[Lint] 140	Fast Power	Binary Serach		Python
[Lint] 447	Search in a Big Sorted Array	Binary Serach	Array	Python
[Lint] 458	Last Position of Target	Binary Serach		Python
191	Number of 1 Bits	Bit Manipulation		Python
⭐ Data Stream 相關題型 ⭐
[Lint] 642	Moving Average from Data Stream	Data Stream	Queue	Python
⭐ Design 相關題型 ⭐
155	Min Stack	Design		Python
[Lint] 659	Encode and Decode Strings	Design		Python
232	Implement Queue using Stacks	Design	Queue, Stack	Python
⭐ DFS 相關題型 ⭐
98	Validate Binary Search Tree	DFS		Python
2265	Count Nodes Equal to Average of Subtree	DFS		Python		292nd Weekly Contest
2261	K Divisible Elements Subarrays	DFS		Python		內含 python substring 常見作法 / 291st LeetCode Weekly Contest
22	Generate Parentheses	DFS		Python
79	Word Search	DFS	Matrix	Python
126	Word Ladder II	DFS		Python
212	Word Search II	DFS	Tree	Python
290	Word Pattern	DFS		Python
[Lint] 829	Word Pattern II	DFS		Python
31	Next Permutation	DFS (排列)		Python
46	Permutations	DFS (排列)		Python		#重要題型
47	Permutations II	DFS (排列)		Python		#重要題型
51	N-Queens	DFS (排列)		Python
52	N-Queens II	DFS (排列)		Python
[Lint] 862	Next Closest Time	DFS (排列)		Python		內有 set 判斷是否 subset 的用法
10	Regular Expression Matching	DFS (組合)		Python
77	Combinations	DFS (組合)		Python		#重要題型
39	Combination Sum	DFS (組合)		Python
40	Combination Sum II	DFS (組合)		Python
216	Combination Sum III	DFS (組合)		Python
377	Combination Sum IV	DFS (組合)		Python
44	Wildcard Matching	DFS (組合)		Python
78	Subsets	DFS (組合)		Python		#重要題型
90	Subsets II	DFS (組合)		Python		#重要題型
131	Palindrome Partitioning	DFS (組合)		Python
139	Word Break	DFS (組合)		Python
140	Word Break II	DFS (組合)		Python
[Lint] 90	k Sum II	DFS (組合)		Python
[Lint] 680	Split String	DFS (組合)		Python
173	Binary Search Tree Iterator	DFS (BST)	BST	Python		#重要題型
230	Kth Smallest Element in a BST	DFS (BST)	BST	Python
[Lint] 448	Inorder Successor in BST	DFS (BST)	BST	Python
[Lint] 900	Closest Binary Search Tree Value	DFS (BST)	BST	Python
[Lint] 901	Closest Binary Search Tree Value II	DFS (BST)	BST	Python		#綜合難題
208	Implement Trie (Prefix Tree)	DFS (Graph)	Tree	Python
17	Letter Combinations of a Phone Number	DFS (Graph)	Graph	Python
332	Reconstruct Itinerary	DFS (Graph)	Graph	Python		#重要題型
110	Balanced Binary Tree	DFS (Tree)	Tree	Python
226	Invert Binary Tree	DFS (Tree)	Tree	Python
572	Subtree of Another Tree	DFS (Tree)	Tree	Python
105	Construct Binary Tree from Preorder and Inorder Traversal	DFS (Tree)	Tree	Python
112	Path Sum	DFS (Tree)		Python
113	Path Sum II	DFS (Tree)	Tree	Python
235	Lowest Common Ancestor of a Binary Search Tree	DFS (Tree)	Tree	Python
236	Lowest Common Ancestor of a Binary Tree	DFS (Tree)	Tree	Python		#重要題型
257	Binary Tree Paths	DFS (Tree)	Tree	Python
[Lint] 474	Lowest Common Ancestor II	DFS (Tree)	Tree	Python
[Lint] 578	Lowest Common Ancestor III	DFS (Tree)	Tree	Python
[Lint] 596	Minimum Subtree	DFS (Tree)	Tree	Python
543	Diameter of Binary Tree	DFS (Tree)		Python
144	Binary Tree Preorder Traversal	DFS (Tree)	Tree	Python		內含 處理 Tree 樹問題的重點
145	Binary Tree Postorder Traversal	DFS (Tree)	Tree	Python
114	Flatten Binary Tree to Linked List	DFS (Tree)	Tree	Python
⭐ Dynamic Programming 相關題型 ⭐
338	Counting Bits	DP		Python
309	Best Time to Buy and Sell Stock with Cooldown	DP		Python
2266	Count Number of Texts	DP		Python		292nd Weekly Contest
2262	Total Appeal of A String	DP		Python		291st Weekly Contest
91	Decode Ways	DP		Python	C++
70	Climbing Stairs	DP		Python	C++
221	Maximal Square	DP		Python
53	Maximum Subarray	DP (一維)	Array	Python	C++	內含 C++ vector max 用法整理
91	Decode Ways	DP (一維)		Python
55	Jump Game	DP (一維)		Python		#重要題型
45	Jump Game II	DP (一維)		Python		#重要題型
198	House Robber	DP (一維)		Python
213	House Robber II	DP (一維)		Python
509	Fibonacci Number	DP (一維)		Python
122	Best Time to Buy and Sell Stock II	DP (一維)		Python
300	Longest Increasing Subsequence	DP (一維接龍, LIS)		Python		#重要題型
62	Unique Paths	DP (二維)		Python	C++
63	Unique Paths II	DP (二維)		Python	C++
152	Maximum Product Subarray	DP (二維)		Python		#重要題型
118	Pascal’s Triangle	DP (二維)		Python		內含 python sum of two list (list add 相加方法整理)
322	Coin Change	DP (背包問題)		Python		內含 DP 背包問題模板 #重要題型
518	Coin Change 2	DP (背包問題)		Python		#重要題型
[Lint] 92	Backpack	DP (背包問題)		Python
[Lint] 125	Backpack II	DP (背包問題)		Python
[Lint] 440	Backpack III	DP (背包問題)		Python
[Lint] 562	Backpack IV	DP (背包問題)		Python
[Lint] 563	Backpack V	DP (背包問題)		Python
[Lint] 798	Backpack VII	DP (背包問題)		Python
[Lint] 799	Backpack VIII	DP (背包問題)		Python
[Lint] 800	Backpack IX	DP (背包問題)		Python
[Lint] 801	Backpack X	DP (背包問題)		Python
1143	Longest Common Subsequence	DP (LCS)		Python
494	Target Sum	DP (Memoization)		Python
2267	Check if There Is a Valid Parentheses String Path	DP (Memoization)		Python		內含：Memoization 記憶化搜索筆記 / 292nd Weekly Contest
⭐ Hash 相關題型 ⭐
13	Roman to Integer	Hash		Python
73	Set Matrix Zeroes	Hash		Python	C++	內含 python while-else 用法說明
692	Top K Frequent Words	Hash		Python		內含 python 自定義排序 key function 的使用方法
846	Hand of Straights	Hash		Python
347	Top K Frequent Elements	Hash		Python
205	Isomorphic Strings	Hash		Python
268	Missing Number	Hash		Python
242	Valid Anagram	Hash		Python
763	Partition Labels	Hash	Interval	Python
383	Ransom Note	Hash		Python
387	First Unique Character in a String	Hash		Python
2186	Minimum Number of Steps to Make Two Strings Anagram II	Hash		Python		282nd Weekly Contest
[Lint] 793	Intersection of Arrays	Hash		Python
146	LRU Cache	Hash		Python
387	First Unique Character in a String	Hash		Python
[Lint] 920	Meeting Rooms	Hash	Interval	Python		#重要題型
[Lint] 920	Meeting Rooms	Hash	Interval	Python		#重要題型
[Lint] 919	Meeting Rooms II	Hash	Interval	Python		#重要題型
[Lint] 919	Meeting Rooms II	Hash	Interval	Python		#重要題型
349	Intersection of Two Arrays	Hash		Python
217	Contains Duplicate	Hash	Array	Python	C++	內含 C++ set, unordered_set 用法整理
137	Single Number II	Hash		Python
350	Intersection of Two Arrays II	Hash		Python
2248	Intersection of Multiple Arrays	Hash		Python		290th Weekly Contest
49	Group Anagrams	Hash (Anagrams)		Python
217	Contains Duplicate	Hash (Duplicate)		Python
128	Longest Consecutive Sequence	Hash (LCS)		Python
136	Single Number	Hash, Bit Manipulation		Python
⭐ Heap 相關題型 ⭐
264	Ugly Number II	Heap		Python
⭐ Stack 相關題型 ⭐
739	Daily Temperatures	Stack		Python
496	Next Greater Element I	Stack		Python
503	Next Greater Element II	Stack		Python
20	Valid Parentheses	Stack (Parentheses)		Python		內含用 python List 組出 Stack, Queue 的方法整理
⭐ Two pointers 相關題型 ⭐
11	Container With Most Water	Two pointers		Python
42	Trapping Rain Water	Two pointers		Python
74	Search a 2D Matrix	Two pointers	Matrix	Python
141	Linked List Cycle	Two pointers	Linked List	Python
142	Linked List Cycle II	Two pointers	Linked List	Python		內含 python while-else 用法介紹
283	Move Zeroes	Two pointers		Python
876	Middle of the Linked List	Two pointers		Python
2260	Minimum Consecutive Cards to Pick Up	Two pointers (快慢)		Python		291st Weekly Contest
21	Merge Two Sorted Lists	Two pointers (Merge)	Linked List	Python		內含 python Linked List 基本操作 (for 新手教學)
88	Merge Sorted Array	Two pointers (Merge)		Python	C++	內含 python while-else 用法說明
1	Two Sum	Two pointers (NSum)	Array	Python	C++	內有 Python list comprehesion / dict comprehesion 整理 / C++ map find 方法補充 (dict find)
15	3Sum	Two pointers (NSum)	Array	Python
16	3Sum Closest	Two pointers (NSum)		Python
167	Two Sum II – Input Array Is Sorted	Two pointers (NSum)		Python
[Lint] 382	Triangle Count	Two pointers (NSum)		Python
[Lint] 533	Two Sum – Closest to Target	Two pointers (NSum)		Python
[Lint] 587	Two Sum – Unique Pairs	Two pointers (NSum)		Python
5	Longest Palindromic Substring	Two pointers (Palindrome)		Python	C++	內含 C++ string.substr() 用法筆記
9	Palindrome Number	Two pointers (Palindrome)		Python
125	Valid Palindrome	Two pointers (Palindrome)	String	Python		內含 python isalpha(), isalnum() 的整理
680	Valid Palindrome II	Two pointers (Palindrome)		Python
409	Longest Palindrome	Two pointers (Palindrome)		Python
647	Palindromic Substrings	Two pointers (Palindrome)		Python
234	Palindrome Linked List	Two pointers (Palindrome)	Linked List	Python		內含 reverse LinkedList 方法
75	Sort Colors	Two pointers (partition)		Python		#重要題型
[Lint] 5	Kth Largest Element	Two pointers (partition)		Python		#重要題型
[Lint] 31	Partition Array	Two pointers (partition)		Python		#重要題型
[Lint] 143	Sort Colors II	Two pointers (partition)		Python		#重要題型
[Lint] 461	Kth Smallest Numbers in Unsorted Array	Two pointers (partition)		Python		#重要題型
3	Longest Substring Without Repeating Characters	Two pointers (Sliding Window)		Python
76	Minimum Window Substring	Two pointers (Sliding Window)		Python
239	Sliding Window Maximum	Two pointers (Sliding Window)		Python
⭐ 其他題型 / 待分類 ⭐
2	Add Two Numbers		Linked List	Python
7	Reverse Integer			Python
19	Remove Nth Node From End of List		Linked List	Python
36	Valid Sudoku			Python
48	Rotate Image		Matrix	Python
621	Task Scheduler			Python
202	Happy Number			Python
238	Product of Array Except Self			Python
222	Count Complete Tree Nodes		Tree	Python
674	Longest Continuous Increasing Subsequence			Python
435	Non-overlapping Intervals		Interval	Python
88	Merge Sorted Array			Python		內含 python while-else 用法說明
2264	Largest 3-Same-Digit Number in String			Python		292nd Weekly Contest
56	Merge Intervals		Interval	Python		內含：python sorted key 搭配 lambda 的用法範例
57	Insert Interval		Interval	Python
61	Rotate List		Linked List	Python
2259	Remove Digit From Number to Maximize Result			Python		291st Weekly Contest
53	Maximum Subarray			Python
54	Spiral Matrix			Python
228	Summary Ranges			Python
263	Ugly Number			Python
203	Remove Linked List Elements		Linked List	Python		內含 Linked List remove 操作 part 2 (for 新手教學)
206	Reverse Linked List		Linked List	Python		內含 Linked List reverse 反轉操作 part 3 (for 新手教學)
189	Rotate Array		Array	Python
2185	Counting Words With a Given Prefix		String	Python		282nd Weekly Contest
134	Gas Station			Python
121	Best Time to Buy and Sell Stock			Python
83	Remove Duplicates from Sorted List		Linked List	Python
566	Reshape the Matrix		Matrix	Python		內含 python array 初始化, index 操作與控制範例
2243	Calculate Digit Sum of a String			Python		289th Weekly Contest
2244	Minimum Rounds to Complete All Tasks			Python		289th Weekly Contest
2249	Count Lattice Points Inside a Circle			Python		290th Weekly Contest
⭐【喜歡我的文章嗎? 歡迎幫我按讚~ 讓基金會請創作者喝一杯咖啡! 】 如果喜歡我的文章，請幫我在下方【按五下Like】 (Google, Facebook 免註冊)，會由 「LikeCoin」 贊助作者鼓勵繼續創作，讀者們「只需幫忙按讚，完全不用出錢」哦!