Rare

# (Optional) Unordered Sets & Maps

Authors: Darren Yao, Benjamin Qi

### Prerequisites

Maintaining collections of distinct elements with hashing.

### Warning!

You can (almost always) use the sets introduced in the prerequisite module instead, but it's good to know that these exist.

Resources
IUSACOmodule is based off this

## Hashing

Hashing refers to assigning a unique code to every variable/object which allows insertions, deletions, and searches in $O(1)$ time, albeit with a high constant factor, as hashing requires a large constant number of operations. However, as the name implies, elements are not ordered in any meaningful way, so traversals of an unordered set will return elements in some arbitrary order.

(more in-depth explanation?)

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Introduce unordered_set, unordered_map, but no need to repeat code from previous module.

All of these operations are $O(1)$, but again, due to the hashing, this has a high constant factor.

## Custom Hashing

There is no built in method for hashing pairs or vectors. Namely, unordered_set<vector<int>> does not work. In this case, we can use an ordered map (which supports all of the functions used in the code above) or declare our own hash function.

C++

Resources
Mark NelsonHow to create user-defined hash function for unordered_map.

The link provides an example of hashing pairs of strings. More examples (for pairs of ints)

1#include <bits/stdc++.h>2using namespace std;3
4typedef pair<int,int> pi;5#define f first6#define s second7
8struct hashPi {9    size_t operator()(const pi& p) const { return p.f^p.s; }10};
1#include <bits/stdc++.h>2using namespace std;3
4typedef pair<int,int> pi;5#define f first6#define s second7
8namespace std {9    template<> struct hash<pi> {10        size_t operator()(const pi& p) const { return p.f^p.s; }

Java

Java has its own hash functions for objects. If we wanted to override it, we can define the function hashCode() and set our own custom hash function that will be used in HashMaps.

1import java.io.*; 2import java.util.*; 3
4public class HashTest{5    public static HashMap<Pair, Integer> map;6    public static void main(String[] args)7    {8        map = new HashMap<Pair, Integer>(); //uses custom hash function in class Pair9    }10    static class Pair {

However, this hash function is quite bad; if we insert $(0,0), (1,1), (2,2) \ldots$ then they will all be mapped to the same bucket (so it would easily be hacked).

## Hacking

### Warning!

You don't need to know this for USACO, but you will need this to pass some of the problems in this module.

In USACO contests, unordered sets and maps generally fine, but the built-in hashing algorithm for C++ is vulnerable to pathological data sets causing abnormally slow runtimes. Apparently Java is not vulnerable to this, however.

C++

Resources
CFExplanation of this problem and how to fix it.

Essentially use unordered_map<int, int, custom_hash> defined in the blog above in place of unordered_map<int, int>.

### Another Hash Function

Resources
Benq (from KACTL)
1struct chash { /// use most bits rather than just the lowest ones2    const uint64_t C = ll(2e18*PI)+71; // large odd number3    const int RANDOM = rng(); // random 32-bit number4    ll operator()(ll x) const {5        // https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html6        return __builtin_bswap64((x^RANDOM)*C);7    }8};9template<class K,class V> using um = unordered_map<K,V,chash>;

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(explain assumptions that are required for this to work)