Step by Step solution about How to resolve the algorithm Knuth's power tree step by step in the Go programming language

This Go code implements an algorithm that computes the powers of a number using a divide-and-conquer approach and a binary tree data structure. Let's break down the code step by step:

1. Global Variables:

   • p is a map that tracks the parent of each node in the binary tree, where the key is the node and the value is its parent.
   • lvl is a slice of slices that represents the levels of the binary tree, with each inner slice representing the nodes at a particular level. Initially, it contains only the root node 1 in the first level.

2. path Function:

   • This function takes an integer n as input and returns the path from the root node to n in the tree.
   • It starts by checking if n is 0, in which case it returns an empty slice.
   • It then iterates until it finds the path to n. If the path is not yet known (i.e., n does not exist in the tree), it calculates the next level of the tree by adding all possible combinations of sums of nodes in the current level.
   • The function adds these new nodes to the tree (map p and slice lvl) and continues the search for n until it finds it.
   • Finally, it returns the path to n in the tree.

3. treePow Function:

   • This function takes a float64 x and an integer n as input and returns the result of x raised to the power of n using the divide-and-conquer approach and the binary tree.
   • It initializes a map r that stores the powers for each node in the tree.
   • It calculates the path from the root node to n in the tree and iteratively multiplies the powers of the nodes along the path to compute x raised to the power of n.
   • Finally, it returns the result as a big.Float object.

4. showPow Function:

   • This function takes a float64 x, an integer n, and a boolean isIntegral as input and prints the result of x raised to the power of n.
   • It prints the path from the root node to n in the tree.
   • It prints x and n in a formatted string and then prints the result returned by the treePow function.

5. main Function:

   • This is the entry point of the program.
   • It iterates through n values from 0 to 17 and calls the showPow function to print the powers of 2 raised to n as integers.
   • It then calls showPow for 1.1 raised to the power of 81 and 3 raised to the power of 191, printing the results as floating-point numbers.

package main

import (
    "fmt"
    "math/big"
)

var (
    p   = map[int]int{1: 0}
    lvl = [][]int{[]int{1}}
)

func path(n int) []int {
    if n == 0 {
        return []int{}
    }
    for {
        if _, ok := p[n]; ok {
            break
        }
        var q []int
        for _, x := range lvl[0] {
            for _, y := range path(x) {
                z := x + y
                if _, ok := p[z]; ok {
                    break
                }
                p[z] = x
                q = append(q, z)
            }
        }
        lvl[0] = q
    }
    r := path(p[n])
    r = append(r, n)
    return r
}

func treePow(x float64, n int) *big.Float {
    r := map[int]*big.Float{0: big.NewFloat(1), 1: big.NewFloat(x)}
    p := 0
    for _, i := range path(n) {
        temp := new(big.Float).SetPrec(320)
        temp.Mul(r[i-p], r[p])
        r[i] = temp
        p = i
    }
    return r[n]
}

func showPow(x float64, n int, isIntegral bool) {
    fmt.Printf("%d: %v\n", n, path(n))
    f := "%f"
    if isIntegral {
        f = "%.0f"
    }
    fmt.Printf(f, x)
    fmt.Printf(" ^ %d = ", n)
    fmt.Printf(f+"\n\n", treePow(x, n))
}

func main() {
    for n := 0; n <= 17; n++ {
        showPow(2, n, true)
    }
    showPow(1.1, 81, false)
    showPow(3, 191, true)
}