Combinatorial Search

code
algorithms

March 31, 2020

This article explains a recursive algorithm that performs a combinatorial search. The concept will be used in a future post to find all possible piano voicings for a given chord.

The algorithm

The following algorithm can be used to generate permutations of any kind:

class Permutation {
  static search<T>(
    finder: (path: T[], solutions: T[][]) => T[],
    validator: (path: T[], solutions: T[][]) => boolean,
    concatFn = (_path: T[], _candidate: T): T[] => [..._path, _candidate],
    path: T[] = [],
    solutions: T[][] = []
  ): T[][] {
    // get candidates for current path
    const candidates = finder(path, solutions);
    // runs current path through validator to either get a new solution or nothing
    if (validator(path, solutions)) {
      solutions.push(path);
    }
    // if no candidates found, we cannot go deeper => either solution or dead end
    if (!candidates.length) {
      return solutions;
    }
    // go deeper
    return candidates.reduce(
      (_, candidate) => Permutation.search(finder, validator, concatFn, concatFn(path, candidate), solutions),
      []
    );
  }
}

The basic three steps are:

  1. call the finder to get possible candidates based on the current path and solutions.
  2. validate current path, and if valid, add to solutions
  3. if no more candidates => done, if more candidates => repeat

Classic Combinatorics: Urn model

Let's implement the classic combinatoric urn model with it.

Basic Urn Implementation

function urn(items) {
  return Permutation.search(
    // all balls that are not yet collected, are available => unique
    (collected) => items.filter((ball) => !collected.includes(ball)),
    // all collections that have the length of the items are valid => pull till empty
    (collected) => collected.length === items.length
  );
}

Here, we pass two functions:

  • The first function returns a set of items we can pull at a given state. In this case, we return all items that have not been collected yet
  • The second function returns a validation function for a given collection to be accepted as "solution". In this case, we only accept collections of the items length

The functions in this case lead to:

  • Order is important
  • Balls can be pulled once
  • We pull till it's "empty"
Balls in Urn:
-+

Extension 1: pull sample

We can extend the implementation by passing a number of balls that should be pulled:

function urn(items, number = items.length) {
  return Permutation.search(
    (collected) => items.filter((ball) => !collected.includes(ball)),
    (collected) => collected.length === number
  );
}

This is like pulling Lotto numbers:

  • Order is important
  • Each ball is unique (only one can be pulled)
  • pulling only a subset (e.g. 6 of 49)
Balls in Urn:
-+

The above implementation works, but has a performance flaw: The first function does not include the sample so it will always run till the end, despite the fact that it won't find new valid combinations. Fix:

function urn(items, number = items.length) {
  return Permutation.search(
    (collected) => (collected.length >= number ? [] : items.filter((ball) => !collected.includes(ball))),
    (collected) => collected.length === number
  );
}

Returning an empty array means, we have no candidates that could be added => recursion stops.

Extension 2: ignore order

We could also ignore the order of items:

function isEqual(collectionA, collectionB) {
  return collectionA.sort().join('-') === collectionB.sort().join('-');
}

function urn(items, number = items.length, strictOrder = true) {
  return Permutation.search(
    (collected) => (collected.length >= number ? [] : items.filter((ball) => !collected.includes(ball))),
    (collected, solutions) =>
      collected.length === number &&
      (strictOrder || !solutions.find((solution) => Permutation.isEqual(collected, solution)))
  );
}

Note: This implementation of isEqual only works with strings.

A real world usage would be pulling a hand in a card game:

  • Order is not important
  • Each card is unique
  • Pulling only a subset of the deck
Balls in Urn:
-+


Extension 3: Balls can be pulled multiple times:

Lets add a flag called unique. If we switch it to false, a ball can be picked multiple times.

A good mental model for this is a combination lock:

  • Order is important
  • Each number can be used multiple times
  • The amount of picks is not related to the amount of available numbers

Another mental model for this without ordering is throwing multiple dice:

  • Order is not important
  • Each number can be used multiple times
  • We throw how many dices we want
Balls in Urn:
+



function urn(
  items,
  number = items.length,
  strictOrder = true,
  unique = true
) {
  return Permutation.search(
    collected =>
      collected.length >= number
        ? []
        : unique
        ? items.filter(ball => !collected.includes(ball)),
        : items
    (collected, solutions) =>
      collected.length === number &&
      (strictOrder ||
        !solutions.find(solution => Permutation.isEqual(collected, solution)))
  );
}

Final Thoughts

Now we should have a clear understanding of how to perform combinatorial search with a functional approach. In the next post(s), we will use that knowledge to generate voicings, melodies and rhythms!

Felix Roos 2022