Wednesday, October 2, 2013

5.2: How to Convert a Recursive Function to an Iterator

The trick to turn a recursive function is to maintain an agenda of todo items, where each item is the state of each invocation of the recursive version. The agenda replaces the stack management we get when we use recursion. 

The integer partition problem is to break an integer into all possible integer components that sum to the original integer. E.g 6 breaks in to:

6
5 1
4 1 1
3 1 1 1
2 1 1 1 1
1 1 1 1 1 1
2 2 1 1
3 2 1
4 2
2 2 2
3 3

A recursive solution to this problem looks a bit like this:

func IntPartitionRecursive(item []int) {
    fmt.Println(item)
    largest, rest := item[0], item[1:len(item)]
    min, max := 1, largest/2
    if len(rest) > 0 {
        min = rest[0]
    }
    for r := igen.Range(min, max+1); !r.AtEnd(); r.Next() {
        IntPartitionRecursive(append([]int{largest - r.Int(), r.Int()}, rest...))
    }
}

For every slice, starting with, in this case, [6], we print out the slice. We when split it into a first element (6) and rest ([]) and set the min (1) and max (3) values. We then loop through the range [1,4) and call the function again 3 times with the values [5,1], [4,2] and [3,3]. The call with [5,1] will in turn results to calls with [4,1,1] and [3,2,1]. This strategy, of looping only to largest/2, prunes the solution tree so that we don't return solutions such as [2,4] since that's the same as [4,2]. All of our solutions are in decreasing-or-equal order.

To turn this into a iterative solution, we need to identify the essential state of the iteration. In this case, it is the slice of integers that we are generating new solutions from (e.g. the state [5,1] results in new states [4,1,1] and [3,2,1]). On change from the recursive solution is that the iterative solution will return the list in a sorted order such as this:

6
5 1
4 2
4 1 1
3 3
3 2 1
3 1 1 1
2 2 2
2 2 1 1
2 1 1 1 1

To produce the sorted version, we leverage the sort package provided by Go. We need to define three methods on our to-be-sorted collection, Len(), Swap() and Less().

type items [][]int

func (it items) Len() int { 
    return len(it) 
}

func (it items) Swap(i, j int) {
    it[i], it[j] = it[j], it[i]
}

func (it items) Less(i, j int) bool {
    a, b := it[i], it[j]
    for k := 0; k < len(a); k++ {
        if a[k] < b[k] {
            return false
        } else if a[k] > b[k] {
            return true
        }
    }
    // we will never get here since we've alredy pruned the
    // other half of the solution tree
    return true
}

Go's package is geared to sort a container from the smallest element to the largest, we want to sort in reverse order so we switch the expression; return true if item i is larger than j and vice versa.

type intpartition struct {
    agenda items
    item   []int
    err    error
}

agenda is the state of the iteration. item is the current solution that Value() will return.

func IntPartition(n int) i.Forward {
    var p intpartition
    item := make([]int, 1)
    item[0] = n
    p.agenda = append(p.agenda, item)
    p.Next()
    return &p
}

To start the iteration we simply construct a slice from the supplied starting value and call the Next() method.

func (p *intpartition) AtEnd() bool {
    return len(p.agenda) == 0
}
func (p *intpartition) Value() interface{} {
    if p.AtEnd() {
        p.err = fmt.Errorf("Value: Beyond end")
        return nil
    }
    return p.item
}

An empty agenda means the iteration is over.

func (p *intpartition) Next() error {
    if p.AtEnd() {
        p.err = fmt.Errorf("Calling next AtEnd")
        return p.err
    }
    p.item, p.agenda = p.agenda[0], p.agenda[1:len(p.agenda)]
    largest, rest := p.item[0], p.item[1:len(p.item)]
    min, max := 1, largest/2
    if len(rest) > 0 {
        min = rest[0]
    }
    for r := igen.Range(min, max+1); !r.AtEnd(); r.Next() {
        p.agenda = append(p.agenda, (append([]int{largest - r.Int(), r.Int()}, rest...)))
    }
    sort.Sort(p.agenda)
    return nil
}

The Next() method starts py popping the first item from the agenda and assigning it to item. Then we break the item into two parts, largest and rest and proceded exacly like the recursive version, except we push new solutions on the agenda where we would call the function again. Lastly, we sort the agenda.

    itr := IntPartition(n)
    for ; !itr.AtEnd(); itr.Next() {
        fmt.Println(itr.Value())
    }

Looping through the solution space is trivial.

A version that only prints out sums that contain distinct components, e.g. no number repeats itself, is easy. Simply leverage the i.Filter function and write a filter for the serie of integers.

func distinct(itr i.Iterator) bool {
    item, _ := itr.Value().([]int)
    for i := 1; i < len(item); i++ {
        if item[i-1] == item[i] {
            return false
        }
    }
    return true
}

Since the list is in a descending order we simply need to check if the current component is equal to the previous component; if so we return false. If we can run through all components without this happening, we return true.

    itr = hoi.Filter(distinct, IntPartition(n))
    for ; !itr.AtEnd(); itr.Next() {
        fmt.Println(itr.Value())
    }

Get the source at GitHub.