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

The provided code showcases a recursive algorithm to compute the exponential tree forms of real numbers with integer exponents. This approach is more efficient and accurate than the traditional method of repeated multiplication, particularly for large exponents. The code uses the path function to generate the path of numbers leading to the target exponent, and the treePow function to calculate the exponential tree form. The showPow function prints out the path and the computed result for various inputs.

The path function employs a breadth-first search algorithm to find the path of numbers leading to the desired exponent. It starts with the number 1 and iteratively adds new numbers by summing existing numbers in the path. The function uses a mutable map p to efficiently track the parent of each number and a mutable list lvl to represent the current level of numbers in the path.

The treePow function uses dynamic programming with a mutable map r to compute the exponential tree form based on the path found by the path function. It stores intermediate results in the map to avoid recomputation and ensures accuracy by using the BigDecimal class for decimal operations.

The showPow function provides a user-friendly interface to display the path and the computed result. It formats the output based on whether the exponent is integral or not.

The main function demonstrates the usage of the showPow function for various inputs. It calculates and displays the exponential tree forms of 2.0, 1.1, and 3.0 for different exponents, showcasing the efficiency and accuracy of this approach.

// version 1.1.3

import java.math.BigDecimal

var p = mutableMapOf(1 to 0)
var lvl = mutableListOf(listOf(1))

fun path(n: Int): List<Int> {
    if (n == 0) return emptyList<Int>()
    while (n !in p) {
        val q = mutableListOf<Int>()
        for (x in lvl[0]) {
            for (y in path(x)) { 
                if ((x + y) in p) break
                p[x + y] = x
                q.add(x + y)
            } 
        }
        lvl[0] = q
    }
    return path(p[n]!!) + n
}

fun treePow(x: Double, n: Int): BigDecimal {
    val r = mutableMapOf(0 to BigDecimal.ONE, 1 to BigDecimal(x.toString()))
    var p = 0
    for (i in path(n)) {
        r[i] = r[i - p]!! * r[p]!!
        p = i
    }
    return r[n]!!
}

fun showPow(x: Double, n: Int, isIntegral: Boolean = true) {
    println("$n: ${path(n)}")
    val f = if (isIntegral) "%.0f" else "%f"
    println("${f.format(x)} ^ $n = ${f.format(treePow(x, n))}\n")
} 

fun main(args: Array<String>) {
    for (n in 0..17) showPow(2.0, n)
    showPow(1.1, 81, false)
    showPow(3.0, 191)
}