usaco-guide/content/4_Silver/Stacks_Queues.md

id	title	author	description	prerequisites
stacks-queues	Stacks & Queues	Darren Yao		Bronze - Data Structures

Additional Reading

• CPH 4.5
• PAPS 3.2 3.3, 3.4, 6.2, 6.3, 6.5

Stacks

A stack is a Last In First Out (LIFO) data structure that supports three operations, all in O(1) time. Think of it like a real-world stack of papers (or cards).

C++

• push: adds an element to the top of the stack
• pop: removes an element from the top of the stack
• top: retrieves the element at the top without removing it

stack<int> s;
s.push(1); // [1]
s.push(13); // [1, 13]
s.push(7); // [1, 13, 7]
cout << s.top() << endl; // 7
s.pop(); // [1, 13]
cout << s.size() << endl; // 2

Java

• push: adds an element to the top of the stack
• pop: removes an element from the top of the stack
• peek: retrieves the element at the top without removing it

Stack<Integer> s = new Stack<Integer>();
s.push(1); // [1]
s.push(13); // [1, 13]
s.push(7); // [1, 13, 7]
System.out.println(s.peek()); // 7
s.pop(); // [1, 13]
System.out.println(s.size()); // 2

Queues

A queue is a First In First Out (FIFO) data structure that supports three operations, all in O(1) time.

C++

• push: insertion at the back of the queue
• pop, deletion from the front of the queue
• front: which retrieves the element at the front without removing it.

queue<int> q;
q.push(1); // [1]
q.push(3); // [3, 1]
q.push(4); // [4, 3, 1]
q.pop(); // [4, 3]
cout << q.front() << endl; // 3

Java

• add: insertion at the back of the queue
• poll: deletion from the front of the queue
• peek, which retrieves the element at the front without removing it

Java doesn't actually have a Queue class; it's only an interface. The most commonly used implementation is the LinkedList, declared as follows:

Queue<Integer> q = new LinkedList<Integer>();
q.add(1); // [1]
q.add(3); // [3, 1]
q.add(4); // [4, 3, 1]
q.poll(); // [4, 3]
System.out.println(q.peek()); // 3

Deques

A deque (usually pronounced "deck") stands for double ended queue and is a combination of a stack and a queue, in that it supports O(1) insertions and deletions from both the front and the back of the deque. Not very common in Bronze / Silver.

C++

The four methods for adding and removing are push_back, pop_back, push_front, and pop_front.

deque<int> d;
d.push_front(3); // [3]
d.push_front(4); // [4, 3]
d.push_back(7); // [4, 3, 7]
d.pop_front(); // [3, 7]
d.push_front(1); // [1, 3, 7]
d.pop_back(); // [1, 3]

Java

In Java, the deque class is called ArrayDeque. The four methods for adding and removing are addFirst , removeFirst, addLast, and removeLast.

ArrayDeque<Integer> deque = new ArrayDeque<Integer>();
deque.addFirst(3); // [3]
deque.addFirst(4); // [4, 3]
deque.addLast(7); // [4, 3, 7]
deque.removeFirst(); // [3, 7]
deque.addFirst(1); // [1, 3, 7]
deque.removeLast(); // [1, 3]

Priority Queues

A priority queue supports the following operations: insertion of elements, deletion of the element considered highest priority, and retrieval of the highest priority element, all in O(\log n) time according to the number of elements in the priority queue. Priority is based on a comparator function. The priority queue is one of the most important data structures in competitive programming, so make sure you understand how and when to use it.

C++

priority_queue<int> pq;
pq.push(7); // [7]
pq.push(2); // [2, 7]
pq.push(1); // [1, 2, 7]
pq.push(5); // [1, 2, 5, 7]
cout << pq.top() << endl; // 7
pq.pop(); // [1, 2, 5]
pq.pop(); // [1, 2]
pq.push(6); // [1, 2, 6]

Java

In Java, we delete and retrieve the element of lowest priority.

PriorityQueue<Integer> pq = new PriorityQueue<Integer>();
pq.add(7); // [7]
pq.add(2); // [7, 2]
pq.add(1); // [7, 2, 1]
pq.add(5); // [7, 5, 2, 1]
System.out.println(pq.peek()); // 1
pq.poll(); // [7, 5, 2]
pq.poll(); // [7, 5]
pq.add(6); // [7, 6, 5]

General Problems

Problems

(actually go through these and check ...)

Stack

• UVa 00514 - Rails
• UVa 00732 - Anagram by Stack
• UVa 01062 - Containers

Queue / Deque

• UVa 10172 - The Lonesome Cargo
• UVa 10901 - Ferry Loading III
• UVa 11034 - Ferry Loading IV

Monotonic Stack

Consider the following problem:

Given an array, a, of N (1 \le N \le 10^5) integers, for every index i, find the rightmost index j such that j < i and a_i > a_j.

To solve this problem, let's store a stack of pairs of <value, index> and iterate over the array from left to right. For some index i, we will compute ans_i, the rightmost index for i, as follows:

• Keep popping the top element off the stack as long as value \ge a_i. This is because we know that the pair containing value will never be the solution to any index j > i, since a_i is less than or equal to than value and has an index further to the right.
• If value < a_i, set ans[i] to index, because a stack stores the most recently added values first (or in this case, the rightmost ones), index will contain the rightmost value which is less than a_i. Then, pop the top element off the stack, because index can't be the solution for two elements.

The stack we used is called a "monotonic stack" because we keep popping off the top element of the stack which maintains it's monotonicity (the same property needed for algorithms like binary search) because the elements in the stack are increasing.

Further Reading

• CPH 8.2 (Nearest Smaller Element)
• cp-algo - Min Stack)
• Medium - Monotonic Stack

Problems

• Concurrently Balanced Strings (Old Gold)
• Max Histogram Area (Leetcode)

(add more once codeforces comes up)