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https://github.com/morwenn/prime_multiset
A multiset of prime numbers represented by a single integer
https://github.com/morwenn/prime_multiset
data-structures python python-3-5
Last synced: 3 days ago
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A multiset of prime numbers represented by a single integer
- Host: GitHub
- URL: https://github.com/morwenn/prime_multiset
- Owner: Morwenn
- License: mit
- Created: 2017-03-26T19:48:11.000Z (almost 8 years ago)
- Default Branch: master
- Last Pushed: 2017-03-26T19:58:50.000Z (almost 8 years ago)
- Last Synced: 2024-12-22T14:58:13.820Z (3 days ago)
- Topics: data-structures, python, python-3-5
- Language: Python
- Size: 3.91 KB
- Stars: 1
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: license.txt
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README
[](http://doge.mit-license.org)
## A multiset of prime numbers
The `prime_multiset` module contains a `PrimeMultiset` class that represents a multiset that can only store prime
numbers. Under the hood, the prime multiset is represented by a single positive integer. This is possible thanks to the
interesting observation that any natural number can be decomposed in a unique collection of prime factors. In other
words, [any natural number is a unique multiset of prime integers][1]. The underlying integer is accessible through the
`value` member of a `PrimeMultiset`.
This single-integer representation has some fun & unique properties:
* Most set operations can be computed with operations on integers (multiplication, gcd...)
* The complexity of some operations becomes funny (*e.g* `discard` is a mere division)
* Listing the elements is very expensive because it involves prime decomposition (hence not provided)
* The `len` method isn't available for the same reason, but `is_empty` is provided
* The storage for the underlying integer can be higher or lower than for a classic multiset implementation
You can read more about the operations equivalence in the article linked in the first paragraph.
## The `PrimeMultiset` API
### Initialization of the prime multiset
```python
PrimeMultiset(sequence=(), initial_value=1)
```
The constructor initializes the `value` member with `initial_value`, which is the initial integer multiset
representation (`1` represents an empty multiset), to which are added the prime numbers contained in `sequence`.
### Usual multiset operations
```python
multiset & other
multiset &= other
```
Computes the multiset intersection of `multiset` and `other` (retaining the lesser multiplicity per element).
```python
multiset | other
multiset |= other
```
Computes the multiset union of `multiset` and `other` (retaining the greater multiplicity per element).
```python
multiset + other
multiset += other
```
Computes the multiset addition of `multiset` and `other` (retaining the sum of multiplicities per element).
```python
multiset - other
multiset -= other
```
Computes the multiset subtraction of `multiset` and `other` (retaining the differences of multiplicities per element
between one multiset and another). If `other` contains prime numbers that are not in `multiset`, the subtraction is
computed *as if* `other` did not contain those additional prime numbers.
### Multiset equality operations
```python
multiset == other
```
Returns whether `multiset` and `other` contain the same elements.
```python
multiset != other
```
Returns whether `multiset` and `other` contain different elements.
### Multiset inclusion operations
```python
multiset <= other
```
Returns whether every element in `multiset` is also in `other`.
```python
multiset < other
```
Returns whether `multiset` is a proper subset of `other`, that is, whether `multiset <= other and multiset != other`.
```python
multiset >= other
```
Returns whether every element in `other` is also in `multiset`.
```python
multiset > other
```
Returns whether `multiset` is a proper superset of `other`, that is, whether `multiset >= other and multiset != other`.
### Modifying multiset operations
```python
add(element)
```
Adds the prime number `element` to the multiset.
```python
remove(element)
```
Removes the prime number `element` from the multiset. Raises `KeyError` if `element` is not contained in the multiset.
```python
discard(element)
```
Removes the prime number `element` from the multiset if it is present.
```python
clear()
```
Removes all elements from the multiset.
### Non-modifying multiset operations
```python
element in multiset
element not in multiset
```
Returns whether `multiset` contains the prime number `element`.
```python
is_empty
```
Returns whether the multiset does not contain any element.
## Error handling in the API
Checking whether a given integer is a prime number or not is rather expensive, therefore the `PrimeMultiset` API does
not validate its inputs. If integers other than prime numbers are passed to functions expecting prime numbers, the
behaviour of the operations is not guaranteed.
[1]: http://psmay.com/2012/02/17/fun-with-math-a-natural-number-is-a-multiset-of-prime-factors/