17. Dynamic programming. NumberSolitaire.swift

import Foundation
import Glibc

// Solution @ Sergey Leschev, Belarusian State University

// 17. Dynamic programming. NumberSolitaire.

// A game for one player is played on a board consisting of N consecutive squares, numbered from 0 to N − 1. There is a number written on each square. A non-empty array A of N integers contains the numbers written on the squares. Moreover, some squares can be marked during the game.
// At the beginning of the game, there is a pebble on square number 0 and this is the only square on the board which is marked. The goal of the game is to move the pebble to square number N − 1.
// During each turn we throw a six-sided die, with numbers from 1 to 6 on its faces, and consider the number K, which shows on the upper face after the die comes to rest. Then we move the pebble standing on square number I to square number I + K, providing that square number I + K exists. If square number I + K does not exist, we throw the die again until we obtain a valid move. Finally, we mark square number I + K.
// After the game finishes (when the pebble is standing on square number N − 1), we calculate the result. The result of the game is the sum of the numbers written on all marked squares.

// For example, given the following array:
//     A[0] = 1
//     A[1] = -2
//     A[2] = 0
//     A[3] = 9
//     A[4] = -1
//     A[5] = -2
// one possible game could be as follows:
// the pebble is on square number 0, which is marked;
// we throw 3; the pebble moves from square number 0 to square number 3; we mark square number 3;
// we throw 5; the pebble does not move, since there is no square number 8 on the board;
// we throw 2; the pebble moves to square number 5; we mark this square and the game ends.
// The marked squares are 0, 3 and 5, so the result of the game is 1 + 9 + (−2) = 8. This is the maximal possible result that can be achieved on this board.

// Write a function:
// class Solution { public int solution(int[] A); }
// that, given a non-empty array A of N integers, returns the maximal result that can be achieved on the board represented by array A.

// For example, given the array
//     A[0] = 1
//     A[1] = -2
//     A[2] = 0
//     A[3] = 9
//     A[4] = -1
//     A[5] = -2
// the function should return 8, as explained above.

// Write an efficient algorithm for the following assumptions:
// N is an integer within the range [2..100,000];
// each element of array A is an integer within the range [−10,000..10,000].

public func solution(_ A: inout [Int]) -> Int {
    let count = A.count    
    var sums = [Int: Int](minimumCapacity: count)

    for i in 0 .. <count - 1 {
        let leftSum: Int
        if let s = sums[i] {
            leftSum = s
        } else {
            leftSum = A[i]
        }
    
        for j in (i + 1) .. <min(count, i + 7) {
            let b = A[j]

            if let sum = sums[j] {
                sums[j] = max(sum, leftSum + b)
            } else {
                sums[j] = leftSum + b
            }
        }
    }
    return sums[count - 1]!
}