1. Introduction

Python implementation of basic algorithms in Introduction to Algorithms class. Here is the video 1 and video 2.

2. Data structure

2.1 Array
- Applications
  - 53. Maximum Subarray
  - 152. Maximum Product Subarray
2.2 Linked list
- Applications
2.3 Stack
- FILO: elements are added from the front and removed from the front
- Applications
  - 20. Valid Parentheses
  - 394. Decode String
  - Recursive
  - DFS in graph
2.4 Queue
- FIFO: elements are added from the back and removed from the front
- Applications
  - BFS in graph
  - PriorityQueue
    - 23. Merge k Sorted Lists
2.5 Hash
- It is set: unique and non-order
- (key, value), the key point of Hash is index of key
- Applications
  - Unique n numbers with range [0, m)
    - Solution 1: Quicksort with time complexity O(nlogn) and space complexity O(1)
    - Solution 2: Countsort with time complexity O(n + m) and space complexity O(m)
    - Solution 3: hash
      - Mod operation
  - 128. Longest Consecutive Sequence
  - 1. Two Sum
2.6 Tree
- Useful for dynamic set(i.e., insert or delete element) query
- Applications
2.7 Heap
- It is always complete binary tree: root i, left 2*i, right 2*i+1
- Applications
  - 347. Top K Frequent Elements
  - 23. Merge k Sorted Lists
2.8 Graph
- G = (V, E)
- Storage
  - Adjacency matrix, Adjacency table, edge lists
- Traversal
  - DFS by stack
  - BFS by queue
  - Topological sorting in DAG
    - Time complexity: O(V+E)
    - It is used to schedule jobs from the given dependencies among jobs
    - There may exist several topological sorting results for one graph
    - Applications
      - 207. Course Schedule
- Shortest path
  - Dijkstra
    - From source node s to any other node
    - Key
      - Relax operation to maintain shortest distance from s to any other node
      - Weight must be positive
    - Time complexity: O(n^2)
    - Applications
      - 787. Cheapest Flights Within K Stops
  - Floyd
    - All pairs shortest path

3. Algorithm

3.1 Algorithm attributes

Finite (有穷性)
Determinate (确定性)
Feasible (可行性)
Input & output

3.2 Algorithm complexity

Time
- Basic operation times
- O(nlgn)
  - Expectation time of quicksort
  - Lower-bound of comparison-based sorting algorithms
Space
- Memory usage
Difference
- Space can be reused
- Time and space can be interchanged (e.g. Hash)
O(1) < O(lgn) < O(n^(1/2)) < O(n) < O(nlgn) < O(n^2) < O(n^3) < O(2^n) < O(n!)

3.3 Algorithm design technique

3.3.1 Exhaustive (穷举法)
- N-queen
- N permutation
3.3.2 Divide and Conquer
- Binary search
- MergeSort
3.3.3 Dynamic Programming
- Principle
  - Cache: 1d array, 2d matrix, 3d tensor, map
  - bottom-up
- Scenario
  - [1] Most: big (最大), small (最小), long (最长), optimum (最优), counting (计数)
  - [2] Recursive formula: f(n) = F(n-1)
- Applications
  - 509. Fibonacci Number
  - 198. House Robber 0/1 knapsack without conditions
  - 416. Partition Equal Subset Sum, 0/1 knapsack
  - 72. Edit Distance
  - 70. Climbing Stairs, counting
  - 322. Coin Change
  - Dynamic Programming Practice Problems
3.3.4 Greedy
- Topological sorting
- Dijkstra
  - Local optimal is global optimal
- Prim
- Kruskal

4. Lectures

Lecture 1: introduction, analysis of algorithms, insertion sort, merge sort.
Lecutre 2: asymptotic notation, solving recurrences: substitution, recurrsion tree, master.
Lecutre 3: divide and conquer, binary search, merge sort, x to the power of n, matrix multiplication of Fibonacci numbers, Strassen matrix multiplication.
Lecutre 4: quick sort, randomized quick sort.
Lecutre 5: comparison sorting, decision tree model, sort in linear time: couting sort, radix sort, stable sorting algorithm
Lecutre 6: find the k-th element: quick select, worst-case linear-time order statistics
Lecutre 7: Hash, hash function, load factor, collision: chaining, open addressing
Lecutre 8: universal hashing
Lecutre 9: binary search tree, bst sort, Jensen's equality, convex, expected random build bst height is O(lgn)
Lecutre 10: red-black tree, rb-insert, rotation
Lecutre 11: dynamic order statistics, data structure augmentation, interval tree
Lecutre 12: dynamic programming two hallmarks, longest common subsequence
Lecutre 13
Lecutre 14
Lecutre 15
Lecutre 16
Lecutre 17
Lecutre 18
Lecutre 19
Lecutre 20
Lecutre 21
Lecutre 22

5. Coding interview Book

Codes
Chapter 1
- 1.1 How to test a algorithm?
  - First: it works
  - Second: robust to boundary value or invalid value
  - Third: optimize it by considering time and space complexity
- 1.2 How to solve a hard problem?
  - First: think it at a whole
  - Second: from n=1, n=2; make an example; visualize it

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