LintCode & LeetCode
  • Introduction
  • Linked List
    • Sort List
    • Merge Two Sorted Lists
    • Merge k Sorted Lists
    • Linked List Cycle
    • Linked List Cycle II
    • Add Two Numbers II
    • Add Two Numbers
    • Odd Even Linked List
    • Intersection of Two Linked Lists
    • Reverse Linked List
    • Reverse Linked List II
    • Remove Linked List Elements
    • Remove Nth Node From End of List
    • Middle of the Linked List
    • Design Linked List
      • Design Singly Linked List
      • Design Doubly Linked List
    • Palindrome Linked List
    • Remove Duplicates from Sorted List
    • Remove Duplicates from Sorted List II
    • Implement Stack Using Singly Linked List
    • Copy List with Random Pointer
  • Binary Search
    • Search in Rotated Sorted Array
    • Search in Rotated Sorted Array II
    • Search in a Sorted Array of Unknown Size
    • First Bad Version
    • Find Minimum in Rotated Sorted Array
    • Find Minimum in Rotated Sorted Array II
    • Find Peak Element
    • Search for a Range
    • Find K Closest Elements
    • Search Insert Position
    • Peak Index in a Mountain Array
    • Heaters
  • Hash Table
    • Jewels and Stones
    • Single Number
    • Subdomain Visit Count
    • Design HashMap
    • Design HashSet
    • Logger Rate Limiter
    • Isomorphic Strings
    • Minimum Index Sum of Two Lists
    • Contains Duplicate II
    • Contains Duplicate III
    • Longest Consecutive Sequence
    • Valid Sudoku
    • Distribute Candies
    • Shortest Word Distance
    • Shortest Word Distance II
  • String
    • Rotate String
    • Add Binary
    • Implement strStr()
    • Longest Common Prefix
    • Reverse Words in a String
    • Reverse Words in a String II
    • Reverse Words in a String III
    • Valid Word Abbreviation
    • Group Anagrams
    • Unique Email Addresses
    • Next Closest Time
    • License Key Formatting
    • String to Integer - atoi
    • Ransom Note
    • Multiply Strings
    • Text Justification
    • Reorder Log Files
    • Most Common Word
    • Valid Parenthesis String
    • K-Substring with K different characters
    • Find All Anagrams in a String
    • Find the Closest Palindrome
    • Simplify Path
  • Array
    • Partition Array
    • Median of Two Sorted Arrays
    • Intersection of Two Arrays
    • Intersection of Two Arrays II
    • Maximum Subarray Sum
    • Minimum Subarray Sum
    • Maximum Subarray II
    • Maximum Subarray III
    • Subarray Sum Closest
    • Subarray Sum
    • Plus One
    • Maximum Subarray Difference
    • Maximum Subarray IV
    • Subarray Sum Equals K
    • Intersection of Two Arrays
    • Intersection of Two Arrays II
    • Find Pivot Index
    • Rotate Array
    • Get Smallest Nonnegative Integer Not In The Array
    • Maximize Distance to Closest Person
    • Sort Colors
    • Next Permutation
    • Rotate Image
    • Pour Water
    • Prison Cells After N Days
    • Majority Element
    • Can Place Flowers
    • Candy
  • Matrix
    • Spiral Matrix
    • Set Matrix Zeroes
    • Diagonal Traverse
  • Queue
    • Design Circular Queue
    • Implement Queue using Stacks
    • Implement Queue by Two Stacks
    • Implement Stack using Queues
    • Moving Average from Data Stream
    • Walls and Gates
    • Open the Lock
    • Sliding Window Maximum
    • Implement Queue Using Fixed Length Array
    • Animal Shelter
  • Stack
    • Valid Parentheses
    • Longest Valid Parentheses
    • Min Stack
    • Max Stack
    • Daily Temperatures
    • Evaluate Reverse Polish Notation
    • Next Greater Element I
    • Next Greater Element II
    • Next Greater Element III
    • Largest Rectangle in Histogram
    • Maximal Rectangle
    • Car Fleet
  • Heap
    • Trapping Rain Water II
    • The Skyline Problem
    • Top K Frequent Words
    • Top K Frequent Words II
    • Top K Frequent Elements
    • Top k Largest Numbers
    • Top k Largest Numbers II
    • Minimum Cost to Hire K Workers
    • Kth Largest Element in an Array
    • Kth Smallest Number in Sorted Matrix
    • Kth Smallest Sum In Two Sorted Arrays
    • K Closest Points to the Origin
    • Merge K Sorted Lists
    • Merge K Sorted Arrays
    • Top K Frequent Words - Map Reduce
  • Data Structure & Design
    • Hash Function
    • Heapify
    • LRU Cache
    • LFU Cache
    • Rehashing
    • Stack Sorting
    • Animal Shelter
    • Sliding Window Maximum
    • Moving Average from Data Stream
    • Find Median from Data Stream
    • Sliding Window Median
    • Design Hit Counter
    • Read N Characters Given Read4 II - Call multiple times
    • Read N Characters Given Read4
    • Flatten 2D Vector
    • Flatten Nested List Iterator
    • Design Search Autocomplete System
    • Time Based Key-Value Store
    • Design Tic-Tac-Toe
    • Insert Delete GetRandom O(1)
  • Union Find
    • Find the Connected Component in the Undirected Graph
    • Find the Weak Connected Component in the Directed Graph
    • Graph Valid Tree
    • Number of Islands
    • Number of Islands II
    • Surrounded Regions
    • Most Stones Removed with Same Row or Column
    • Redundant Connection
  • Trie
    • Implement Trie
    • Add and Search Word
    • Word Search II
    • Longest Word in Dictionary
    • Palindrome Pairs
    • Trie Serialization
    • Trie Service
    • Design Search Autocomplete System
    • Typeahead
  • Trees
    • Binary Tree Inorder Traversal
    • Binary Tree Postorder Traversal
    • Binary Tree Preorder Traversal
    • Binary Tree Level Order Traversal
    • Binary Tree Zigzag Level Order Traversal
    • Binary Tree Vertical Order Traversal
    • N-ary Tree Level Order Traversal
    • N-ary Tree Preorder Traversal
    • N-ary Tree Postorder Traversal
    • Construct Binary Tree from Preorder and Inorder Traversal
    • Populating Next Right Pointers in Each Node
    • Populating Next Right Pointers in Each Node II
    • Maximum Depth of Binary Tree
    • Symmetric Tree
    • Validate Binary Search Tree
    • Convert Sorted Array to Binary Search Tree
    • Path Sum
    • Path Sum II
    • Path Sum III
    • Binary Tree Maximum Path Sum
    • Kth Smallest Element in a BST
    • Same Tree
    • Lowest Common Ancestor of a Binary Tree
    • Lowest Common Ancestor of a Binary Search Tree
    • Nested List Weight Sum II
    • BST Node Distance
    • Minimum Distance (Difference) Between BST Nodes
    • Closet Common Manager
    • N-ary Tree Postorder Traversal
    • Serialize and Deserialize Binary Tree
    • Serialize and Deserialize N-ary Tree
    • Diameter of a Binary Tree
    • Print Binary Trees
  • Segment Tree
    • Segment Tree Build
    • Range Sum Query - Mutable
  • Binary Indexed Tree
  • Graph & Search
    • Clone Graph
    • N Queens
    • Six Degrees
    • Number of Islands
    • Number of Distinct Islands
    • Word Search
    • Course Schedule
    • Course Schedule II
    • Word Ladder
    • Redundant Connection
    • Redundant Connection II
    • Longest Increasing Path in a Matrix
    • Reconstruct Itinerary
    • The Maze
    • The Maze II
    • The Maze III
    • Topological Sorting
    • Island Perimeter
    • Flood Fill
    • Cheapest Flights Within K Stops
    • Evaluate Division
    • Alien Dictionary
    • Cut Off Trees for Golf Event
    • Jump Game II
    • Most Stones Removed with Same Row or Column
  • Backtracking
    • Subsets
    • Subsets II
    • Letter Combinations of a Phone Number
    • Permutations
    • Permutations II
    • Combinations
    • Combination Sum
    • Combination Sum II
    • Combination Sum III
    • Combination Sum IV
    • N-Queens
    • N-Queens II
    • Generate Parentheses
    • Subsets of Size K
  • Two Pointers
    • Two Sum II
    • Triangle Count
    • Trapping Rain Water
    • Container with Most Water
    • Minimum Size Subarray Sum
    • Minimum Window Substring
    • Longest Substring Without Repeating Characters
    • Longest Substring with At Most K Distinct Characters
    • Longest Substring with At Most Two Distinct Characters
    • Fruit Into Baskets
    • Nuts & Bolts Problem
    • Valid Palindrome
    • The Smallest Difference
    • Reverse String
    • Remove Element
    • Max Consecutive Ones
    • Max Consecutive Ones II
    • Remove Duplicates from Sorted Array
    • Remove Duplicates from Sorted Array II
    • Move Zeroes
    • Longest Repeating Character Replacement
    • 3Sum With Multiplicity
    • Merge Sorted Array
    • 3Sum Smaller
    • Backspace String Compare
  • Mathematics
    • Ugly Number
    • Ugly Number II
    • Super Ugly Number
    • Sqrt(x)
    • Random Number 1 to 7 With Equal Probability
    • Pow(x, n)
    • Narcissistic Number
    • Rectangle Overlap
    • Happy Number
    • Add N Days to Given Date
    • Reverse Integer
    • Greatest Common Divisor or Highest Common Factor
  • Bit Operation
    • IP to CIDR
  • Random
    • Random Pick with Weight
    • Random Pick Index
    • Linked List Random Node
  • Dynamic Programming
    • House Robber
    • House Robber II
    • House Robber III
    • Longest Increasing Continuous Subsequence
    • Longest Increasing Continuous Subsequence II
    • Coins in a Line
    • Coins in a Line II
    • Coins in a Line III
    • Maximum Product Subarray
    • Longest Palindromic Substring
    • Stone Game
    • Burst Balloons
    • Perfect Squares
    • Triangle
    • Pascal's Triangle
    • Pascal's Triangle II
    • Min Cost Climbing Stairs
    • Climbing Stairs
    • Unique Paths
    • Unique Paths II
    • Minimum Path Sum
    • Word Break
    • Word Break II
    • Range Sum Query - Immutable
    • Decode Ways
    • Edit Distance
    • Unique Binary Search Trees
    • Unique Binary Search Trees II
    • Maximal Rectangle
    • Maximal Square
    • Regular Expression Matching
    • Wildcard Matching
    • Flip Game II
    • Longest Increasing Subsequence
    • Target Sum
    • Partition Equal Subset Sum
    • Coin Change
    • Jump Game
    • Can I Win
    • Maximum Sum Rectangle in a 2D Matrix
    • Cherry Pick
  • Knapsack
    • Backpack
    • Backpack II
    • Backpack III
    • Backpack IV
    • Backpack V
    • Backpack VI
    • Backpack VII
    • Coin Change
    • Coin Change II
  • High Frequency
    • 2 Sum Closest
    • 3 Sum
    • 3 Sum Closest
    • Sort Colors II
    • Majority Number
    • Majority Number II
    • Majority Number III
    • Best Time to Buy and Sell Stock
    • Best Time to Buy and Sell Stock II
    • Best Time to Buy and Sell Stock III
    • Best Time to Buy and Sell Stock IV
    • Two Sum
    • Two Sum II - Input array is sorted
    • Two Sum III - Data structure design
    • Two Sum IV - Input is a BST
    • 4 Sum
    • 4 Sum II
  • Sorting
  • Greedy
    • Jump Game II
    • Remove K Digits
  • Minimax
    • Nim Game
    • Can I Win
  • Sweep Line & Interval
    • Meeting Rooms
    • Meeting Rooms II
    • Merge Intervals
    • Insert Interval
    • Number of Airplanes in the Sky
    • Exam Room
    • Employee Free Time
    • Closest Pair of Points
    • My Calendar I
    • My Calendar II
    • My Calendar III
    • Add Bold Tag in String
  • Other Algorithms and Data Structure
    • Huffman Coding
    • Reservoir Sampling
    • Bloom Filter
    • External Sorting
    • Construct Quad Tree
  • Company Tag
    • Google
      • Guess the Word
      • Raindrop on Sidewalk
    • Airbnb
      • Display Pages (Pagination)
    • Amazon
  • Problem Solving Summary
    • String or Array Rotation
    • Tips for Avoiding Bugs
    • Substring or Subarray Search
    • Sliding Window
    • K Sums
    • Combination Sum Series
    • Knapsack Problems
    • Depth-first Search
    • Large Number Operation
    • Implementation - Simulation
    • Monotonic Stack & Queue
    • Top K Problems
    • Java Interview Tips
      • OOP in Java
      • Conversion in Java
      • Data Structures in Java
    • Algorithm Optimization Tips
  • Reference
Powered by GitBook
On this page
  • Analysis
  • Solution
  • Reference

Was this helpful?

  1. Dynamic Programming

Triangle

Given a triangle, find the minimum path sum from top to bottom. Each step you may move to adjacent numbers on the row below.

For example, given the following triangle

[
     [2],
    [3,4],
   [6,5,7],
  [4,1,8,3]
]

The minimum path sum from top to bottom is11(i.e.,2+3+5+1= 11).

Note:

Bonus point if you are able to do this using only O(n) extra space, where _n _is the total number of rows in the triangle.

Analysis

DP vs DFS / BFS

此问题乍一看很像Tree, 因此容易想到DFS或者BFS的方案,加上记忆化搜索,可以勉强AC。但是仔细观察,相邻节点总是会有共用的分支,也就是说有重叠的子问题 overlapping subproblems,而且状态转移方程很容易得到:

O(1) 时间复杂度的递推关系 minpath[k][i] = min( minpath[k+1][i], minpath[k+1][i+1]) + triangle[k][i];

其中minpath[k][i]表示第k行,第i列的元素到bottom的最小路径和, 即min path sum

也可以说有一个最优子结构 optimal substructure。这样的情况下DP可能有最佳的时间复杂度。

Top-Down vs Bottom-Up

  • For 'top-down' DP, starting from the node on the very top, we recursively find the minimum path sum of each node. When a path sum is calculated, we store it in an array (memoization); the next time we need to calculate the path sum of the same node, just retrieve it from the array. O(N^2) space.

  • 'Bottom-up' DP, on the other hand, is very straightforward: we start from the nodes on the bottom row; the min pathsums for these nodes are the values of the nodes themselves. From there, the min pathsum at the ith node on the kth row would be the lesser of the pathsums of its two children plus the value of itself.

空间优化小Trick:

Or even better, since the row minpath[k+1] would be useless after minpath[k] is computed, we can simply set minpath as a 1D array, and iteratively update itself:

For the kth level:
minpath[i] = min( minpath[i], minpath[i+1]) + triangle[k][i];

因为每一行的结果在下一行用过之后就不会被用到,可以将计算结果直接存入自身,也就节约了一个维度的空间。O(n).

triangle.get(i).set(j, triangle.get(i).get(j) + Math.min(triangle.get(i + 1).get(j), triangle.get(i + 1).get(j + 1)));

下标优化小Trick:

声明DP数组时,多加一位:int[] minpath = new int[size + 1];

这样就可以直接从Bottom那一行开始运用递推关系,而不用从先复制最后一行赋值给DP数组,再从倒数第二行开始用递推关系。

Solution

DP Buttom Up 1D array (4ms 90.59% AC)

class Solution {
    public int minimumTotal(List<List<Integer>> triangle) {
        int size = triangle.size();

        int[] minpath = new int[size];
        int k = 0;
        for (Integer e : triangle.get(size - 1)) {
            minpath[k++] = e;
        }
        for (int row = size - 2; row >= 0; row--) {
            for (int i = 0; i <= row; i++) {
                minpath[i] = Math.min(minpath[i], minpath[i + 1]) + triangle.get(row).get(i);
            }
        }
        return minpath[0];
    }
}

*DP - Bottom Up - Improved implementation with size = N + 1 array (4ms 90.59% AC)

class Solution {
    public int minimumTotal(List<List<Integer>> triangle) {
        int size = triangle.size();

        int[] minpath = new int[size + 1];
        for (int row = size - 1; row >= 0; row--) {
            for (int i = 0; i <= row; i++) {
                minpath[i] = Math.min(minpath[i], minpath[i + 1]) + triangle.get(row).get(i);
            }
        }
        return minpath[0];
    }
}

DP with O(1) space

public class Solution {
    public int minimumTotal(List<List<Integer>> triangle) {
        for(int i = triangle.size() - 2; i >= 0; i--)
            for(int j = 0; j <= i; j++)
                triangle.get(i).set(j, triangle.get(i).get(j) + Math.min(triangle.get(i + 1).get(j), triangle.get(i + 1).get(j + 1)));
        return triangle.get(0).get(0);
    }
}

Top Down DP with memorization - DFS (102 ms 1.05% AC)

class Solution {
    public int minimumTotal(List<List<Integer>> triangle) {
        if (triangle == null || triangle.size() == 0 || triangle.get(0).size() == 0) return 0;
        return minPathSumHelper(triangle, new HashMap<String, Integer>(), new Coordinate(0, 0));
    }
    public int minPathSumHelper(List<List<Integer>> triangle, HashMap<String, Integer> map, Coordinate cor) {
        String key = cor.row + "," + cor.col;
        if (map.containsKey(key)) {
            return map.get(key);   
        }
        if (cor.row >= triangle.size() || cor.col >= triangle.get(triangle.size() - 1).size()) return 0;
        int val = triangle.get(cor.row).get(cor.col);
        int min = val + 
            Math.min(minPathSumHelper(triangle, map, new Coordinate(cor.row + 1, cor.col)), 
                     minPathSumHelper(triangle, map, new Coordinate(cor.row + 1, cor.col + 1)));
        map.put(key, min);
        return min;
    }

    class Coordinate {
        int row, col;
        Coordinate (int row, int col) {
            this.row = row;
            this.col = col;
        }
    }
}

Reference

PreviousPerfect SquaresNextPascal's Triangle

Last updated 5 years ago

Was this helpful?

@stellari:

此外也可以用triangle本身存储中间结果,那么space就变成O(1),参考::

LeetCode论坛分析
My 8 line DP Java code(4 meaningful lines) with O(1) space
https://leetcode.com/problems/triangle/discuss/38730/DP-Solution-for-Triangle