package ec.gp.build; import ec.gp.*; import java.util.*; import java.math.*; import ec.util.*; import ec.*; /* * Uniform.java * * Created Fri Jan 26 14:02:08 EST 2001 * By: Sean Luke */ /** Uniform implements the algorithm described in

Bohm, Walter and Andreas Geyer-Schulz. 1996. "Exact Uniform Initialization for Genetic Programming". In Foundations of Genetic Algorithms IV, Richard Belew and Michael Vose, eds. Morgan Kaufmann. 379-407. (ISBN 1-55860-460-X)

The user-provided requested tree size is either provided directly to the Uniform algorithm, or if the size is NOSIZEGIVEN, then Uniform will pick one at random from the GPNodeBuilder probability distribution system (using either max-depth and min-depth, or using num-sizes).

Further, if the user sets the true-dist parameter, the Uniform will ignore the user's specified probability distribution and instead pick from a distribution between the minimum size and the maximum size the user specified, where the sizes are distributed according to the actual number of trees that can be created with that size. Since many more trees of size 10 than size 3 can be created, for example, size 10 will be picked that much more often.

Uniform also prints out the actual number of trees that exist for a given size, return type, and function set. As if this were useful to you. :-)

The algorithm, which is quite complex, is described in pseudocode below. Basically what the algorithm does is this:

1. For each function set and return type, determine the number of trees of each size which exist for that function set and tree type. Also determine all the permutations of tree sizes among children of a given node. All this can be done with dynamic programming. Do this just once offline, after the function sets are loaded.
2. Using these tables, construct distributions of choices of tree size, child tree size permutations, etc.
3. When you need to create a tree, pick a size, then use the distriutions to recursively create the tree (top-down).

Dealing with Zero Distributions

Some domains have NO tree of a certain size. For example, Artificial Ant's function set can make NO trees of size 2. What happens when we're asked to make a tree of (invalid) size 2 in Artificial Ant then? Uniform presently handles it as follows:

1. If the system specifically requests a given size that's invalid, Uniform will look for the next larger size which is valid. If it can't find any, it will then look for the next smaller size which is valid.
2. If a random choice yields a given size that's invalid, Uniform will pick again.
3. If there is *no* valid size for a given return type, which probably indicates an error, Uniform will halt and complain.

### Pseudocode:

```
*    Func NumTreesOfType(type,size)
*        If NUMTREESOFTYPE[type,size] not defined,       // memoize
*            N[type] = all nodes compatible with type
*            NUMTREESOFTYPE[type,size] = Sum(n in N[type], NumTreesRootedByNode(n,size))
*            return NUMTREESOFTYPE[type,size]
*
*    Func NumTreesRootedByNode(node,size)
*        If NUMTREESROOTEDBYNODE[node,size] not defined,   // memoize
*            count = 0
*            left = size - 1
*            If node.children.length = 0 and left = 0  // a valid terminal
*                count = 1
*            Else if node.children.length <= left  // a valid nonterminal
*                For s is 1 to left inclusive  // yeah, that allows some illegal stuff, it gets set to 0
*                    count += NumChildPermutations(node,s,left,0)
*            NUMTREESROOTEDBYNODE[node,size] = count
*        return NUMTREESROOTEBYNODE[node,size]
*
*
*    Func NumChildPermutations(parent,size,outof,pickchild)
*    // parent is our parent node
*    // size is the size of pickchild's tree that we're considering
*    // pickchild is the child we're considering
*    // outof is the total number of remaining nodes (including size) yet to fill
*        If NUMCHILDPERMUTATIONS[parent,size,outof,pickchild] is not defined,        // memoize
*            count = 0
*            if pickchild = parent.children.length - 1        and outof==size        // our last child, outof must be size
*                count = NumTreesOfType(parent.children[pickchild].type,size)
*            else if pickchild < parent.children.length - 1 and
*                                outof-size >= (parent.children.length - pickchild-1)    // maybe we can fill with terminals
*                cval = NumTreesOfType(parent.children[pickchild].type,size)
*                tot = 0
*                For s is 1 to outof-size // some illegal stuff, it gets set to 0
*                    tot += NumChildPermutations(parent,s,outof-size,pickchild+1)
*                count = cval * tot
*            NUMCHILDPERMUTATIONS [parent,size,outof,pickchild] = count
*        return NUMCHILDPERMUTATIONS[parent,size,outof,pickchild]
*
*
*    For each type type, size size
*        ROOT_D[type,size] = probability distribution of nodes of type and size, derived from
*                            NUMTREESOFTYPE[type,size], our node list, and NUMTREESROOTEDBYNODE[node,size]
*
*    For each parent,outof,pickchild
*        CHILD_D[parent,outof,pickchild] = probability distribution of tree sizes, derived from
*                            NUMCHILDPERMUTATIONS[parent,size,outof,pickchild]
*
*    Func FillNodeWithChildren(parent,pickchild,outof)
*        If pickchild = parent.children.length - 1               // last child
*            Fill parent.children[pickchild] with CreateTreeOfType(parent.children[pickchild].type,outof)
*        Else choose size from CHILD_D[parent,outof,pickchild]
*            Fill parent.pickchildren[pickchild] with CreateTreeOfType(parent.children[pickchild].type,size)
*            FillNodeWithChildren(parent,pickchild+1,outof-size)
*        return
```
Func CreateTreeOfType(type,size) Choose node from ROOT_D[type,size] If size > 1 FillNodeWithChildren(node,0,size-1) return node

Parameters
 base.true-dist bool= true or false (default) (should we use the true numbers of trees for each size as the distribution for picking trees, as opposed to the user-specified distribution?)