# Learn Data Structures by Practicing - Part I

"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.

#### 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.

#### 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
• 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.

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:
• 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

# Learn Data Structures by Practicing - Part I

"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.

#### 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.

#### 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
• 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.

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: