import java.util.Scanner; public class MaxSum { public static void main(String[] args) { Scanner scan = new Scanner(System.in); System.out.println("請輸入整數個數:"); int N = scan.nextInt(); int[] arr = new int[N]; System.out.println("請輸入整數序列(各數據間用空格隔開):"); for(int i=0;i<N;i++){ arr[i] = scan.nextInt(); } System.out.println("各算法所求最大子段和分別爲:"); System.out.println("分治算法最優值:" + maxSumDiv(arr, 0, arr.length - 1)); System.out.println("==========================="); maxSumDp(arr); System.out.println("==========================="); maxSumSimp(arr, 0, 0); } // 簡單算法(需要O(n^2)計算時間) public static void maxSumSimp(int arr[], int bestx, int besty) { int n = arr.length, sum = 0; for (int i = 1; i <= n; i++) { int thissum = 0; //改變子段和的首數字時,須將當前的子段和清零 for (int j = i; j <= n; j++) { thissum += arr[j - 1]; if (thissum > sum) { sum = thissum; bestx = i; besty = j; } } } System.out.println("簡單算法最優值:" + sum); System.out.println("最優解:" + bestx + "-->" + besty); } // 分治算法實現(需要O(nlogn)的計算時間) public static int maxSumDiv(int[] arr, int left, int right) { int sum = 0; if (left == right) { sum = arr[left] > 0 ? arr[left] : 0; } else { int center = (left + right) / 2; int leftSum = maxSumDiv(arr, left, center); int rightSum = maxSumDiv(arr, center + 1, right); int s1 = 0; int lefts = 0; for (int i = center; i >= left; i--) { lefts += arr[i]; if (lefts > s1) { s1 = lefts; } } int s2 = 0; int rights = 0; for (int i = center + 1; i <= right; i++) { rights += arr[i]; if (rights > s2) { s2 = rights; } } sum = s1 + s2; if (sum < leftSum) { sum = leftSum; } if (sum < rightSum) { sum = rightSum; } } return sum; } // 動態規劃算法實現(只需要O(n)計算時間和O(n)空間) public static void maxSumDp(int[] arr) { int sum = 0, b = 0, n = arr.length, bestx = 0, besty = 0; for (int i = 1; i <= n; i++) { //b是累加和 if (b > 0) { b += arr[i - 1]; //和爲正時累加 } else { b = arr[i - 1]; //否則將當前值賦給b bestx = i; } if (b > sum) { sum = b; besty = i; } } System.out.println("動態規劃算法最優值:" + sum); System.out.println("最優解:" + bestx + "-->" + besty); } }
運行結果:
作爲DP的經典案例,仔細體會DP的基本思想:保存已解決的子問題的答案,避免大量的重複計算。