Learn Data Structures by Practicing - Part I

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Learn Data Structures by Practicing - Part I

The structure of this article is derived from here. However the organization of this wikipedia article is a mess. It need to be updated urgently in order not to mislead the newbies.

“Algorithm is to construct a proper structure, and insert data. “ —kulasama

Recomendation on hints: Use as few hints as possible.

Data types

Primitive types

Objectives: Knowing how the datum is stored, exploiting intrinsic features of it and avoid making mistakes.

  • You should be able to declare, assign, read or print variables of these types.
  • You should be able to apply all possible operators to variables of these types and predict the results.
  • You should be able to predict the results of conversions between these types.
  • You should know the limits of these types and should be able to predict the results of exceeding them.

Boolean

Character

Floating point

  • Including single precision floats, double precision (IEEE 754) floats, etc.

    Fixed-point numbers

  • Integer, including signed and unsigned integer
  • Reference, pointer, or handle

    Enumerated types

Literatures

  • Hacker’s Delight, Henry S. Warren, Jr.

Excercises

  • Given two 32-bit signed integers $a$ and $b$, print how many bits changes when turning $a$ to $b$.
    • Hint: Hamming weight of $a\underline\vee b$
  • Given a number of height in inches and a number of height in centimeters, tell whether they equal each other.
  • Explain ASCII code $0$, $9$, $10$, $13$, and declare variables of them in charactor literals.
  • How to process emojis?
  • Given $n$ integers, each of them appears twice except for one, which appears exactly once. Find that single one.
    • Advanced: Given $n$ integers, each of them appears three times except for one, which appears exactly once. Find that single one.

**Only premitive types are allowed in these excersices. **

Composite types or non-primitive type

Objectives: Getting familiar with how multiple data are organized basically.

Array

Record, tuple, or structure

String

Union

Tagged union, variant, variant record, discriminated union, or disjoint union

Excercises

  • Given a string of a heximal number (might not be an integer), print it in decimal form.
  • Write a programm of encryption and decryption of Caesar ciphering.
  • Name algorithms of string searching and compare their advantages and disadvantages.
  • Implement a expression evaluator supporting decimal numbers (with or without seperator), $+$ and $-$.
  • Store sparse matrices with various methods and compare where they should be applied. (Note that some of them depends pointers or references)
    • Dictionary of keys
    • List of lists
    • Coordinate list
    • Compressed sparse row
    • Compressed sparese column
    • Diagnal
    • Orthogonal linked list
    • ELLPACK
    • ELLPACK + Coordinates
  • Implement a hash table.
    • How do you hash the keys and how do you handle the conflictions?
    • Hint: Consider there are $n$ key-value pairs and the keys are respectfully $k\ldots k+n$, where $k$ is an constant integer, try to design a structure storing and retrieving values by keys in $O\left(1\right)$.
      • What if the keys are $3*k$, where $k$ is in $1\ldots n$?
      • What if the keys are distinct integers?
      • What if the keys are mostly distinct integers?
      • What if the keys are strings?

Basic data structures

Objective: Understanding the principles of basic data structures, and knowing when to use them.

Linked list

  • Singly linked list
  • Doubly linked list
  • XOR linked list

Excercises

  • Use arrays to implement linked lists.
    • Append a node into a given list.
    • Insert a node after a given node.
    • Remove a node from a given list.
    • Empty a list.
  • Use pointers or references to implement linked lists.
  • Revert a given linked list(unless otherwise specified, linked lists refer to sigly linked list of number)
  • Find $n$’th node from the end of a given linked list.
  • Find the middle node of a given linked list.
  • Sort a given linked list.
    • If you get stucked on this problem, you may also try the following problems first.
    • Find and delete a specified node in a given linked list.
    • Swap two nodes on a given linked list.
    • Implement bubble sort on linked lists.
    • Given an ordered linked list, insert a new number without destroying its order.
    • Implement insertion sort on linked lists.
    • Given a linked list, divide them into two even halves.
    • Given two ordered linked lists, merge them into one ordered linked list.
    • Implement merge sort on linked lists.
    • Given a linked list, divide them into two halves(might not be even) and meanwhile let each number in the first half be greater than all numbers in the second half.
    • Implement quick sort on linked lists.
  • Given a linked list(assume it is), tell whether there is a loop and find the entry of it.
  • Given two linked list, tell whether and where they intersect each other. What if there can be loops?

Stack

Excercises

  • Use array to implement stacks.
    • Push a node into a given stack.
    • Pop a node from a given stack.
    • Peak the top node of a given stack.
    • Empty a given stack.
  • Use pointers or references to implement stacks. Including the operations above.
  • Implement undo/redo functionality(or back/forward navigation in explorer).
  • Given a sequence of push operations and a sequence of pop operations, tell whether it can be valid.
  • Implement a queue supporting push(). pop() and getMin().
  • Without recursion, use backtracking to solve n queens problem.
  • Based on the expression evaluator above, add $\times$ and $\div$ support.
  • Based on the expression evaluator above, add brackets support.

Queue

Excercises

  • Use arrays to implement queue.
    • Enqueue a node into a given queue.
    • Dequeue a node from a given queue.
    • Empty a queue.
  • Use pointers or references to implement linked lists.
  • Given a sequence of enqueue operations and a sequence of dequeue operations, tell whether it can be valid.
  • Implement a queue supporting enqueue(). dequeue() and getMin().
    • Hint: You may first think of implementing a queue with stacks.
  • Implement a circular buffer. // TODO: Better problem needed
  • Implement a message queue. // TODO: Better problem needed

Tree

Excercises

  • Explain binary tree, full binary tree, complete binary tree.
  • Given the root node of a tree, print its pre-order traversal, in-order traversal, post-order traversal and level-order traversal.
    • Same problem, without recursion.
  • Given the post-order traversal and in-order traversal of a tree, print its pre-order traversal.
  • Given a tree with a in-order traversal of which the data are in increasing order, i.e. BST, insert a new node while keeping this property.
    • Implement a sorting algorithm with it (tree sort).
  • Given $n$, how many structurally unique BST’s (binary search trees) that store values $1\ldots n$?
    • Hint: Catalan Number.
  • Analyze the complexity of BST, tell in which situation it behaves bad.
  • Implement a binary heap.
    • Consider a complete binary tree. Can it be properly stored in an array? How to get parent / child node of a given node?
    • If every node of this tree either has no parent (it is the root! ) or the datum of its parent is larger than its, it is called a heap. Can you insert a new node, and keep its properties (complete binary tree, parent datum larger than children datum)?
    • If the root node is removed, can you transform the rest nodes into a heap?
    • Implement a sorting algorithm with it (heap sort).
  • Given a tree (no root node specified), print its diameter.
    • The diameter of a tree ($T=\left(V, E\right)$) is defined as $max_{u,v\in V}\delta\left(u, v\right)$, which means, the length of longest path among all shortest paths between all vertices.
  • Given a set of strings, find them in a text.
    • Hint: Aho-Corasick algorithm
  • Construct Huffman tree with a given set of nodes and their weights.

Graph

  • Store a graph with:
    • Adjacency matrix
    • Adjacency list
  • Explain the possible meaning of powers of adjacency matrices.
  • Generate minimum spanning tree of a given graph.
    • Prim, Krustal, etc.
  • Calculate shortest paths from a source node $s$ to a target node $t$ in a given graph.
    • Hint: DFS, BFS, Bidirectional BFS, Dijkstra, Bellman-ford, etc.
    • Compare their complexity and tell what kind of graphs fit them best.
  • Calculate shortest paths from a source node $s$ to every other node in a given graph.
  • Calculate shortest paths from every node to every other node in a given graph.
    • Floyd-Warshall
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