交替数组元素的顺序

Question

给定数组[1, 2, 3, a, b, c]，编写一个方法来重新排列数组，以便每个其他元素都改变它的位置。输出应为[1, a, 2, b, 3, c]。

1.写一个更快的algorithm并使用O(1)空格。 2.如果允许一个extra-space，则可以实现time-complexity的改进。

这是一个面试问题。 SO中有几个帖子，Interview test - rearrange the array，Reordering of array elements，Array movement from a1,..,an,b1,..,bn to a1,b1,..,an,bn和elsewhere，其中提供了各种解释，但对我来说并不清楚和具体。 对于上面的第一个问题，我有一个O(n^2)解决方案。但似乎有可能改善它。但我真的不会理解这些帖子的解释，尤其是一些提到理论论文的解释。如果为第一个问题提高时间复杂度（优于O(n^2)）的简单伪代码将是很好的

对于第二个问题，通过使用一个存储阵列，可以将时间复杂度降低到O(n)。 但我认为没有办法进一步改进它。如果有可能，我会很乐意听到它。

我第一次尝试的代码（不使用辅助存储器）在这里：

import java.util.Arrays;

public class SwapHalfArrayElements {

    public static void main(String[] args){
        Object[] arr = {1,2,3,'a','b','c'};
        System.out.println("before sorting "+Arrays.toString(arr));
        rearrageArray(arr);
        System.out.println("After sorting "+ Arrays.toString(arr));
    }

    private static void swap(Object[] arr, int index1, int index2){
        Object temp=arr[index1];
        arr[index1]=arr[index2];
        arr[index2]=temp;
    }

    private static void rearrageArray(Object[] arr){
        int n=arr.length/2;
        for (int i=1;i<n;i++){
            for (int j=-i;j<i;j+=2){
                swap(arr,n+j,n+j+1);

            }
        }
    }
}

Answer 1

我认为我的东西可能比O（n²）更快，而不使用超过O（1）的内存。这个想法如下：从元素2开始。计算它需要去的地方。将它存储在那里并计算被替换元素需要去的位置等，直到我们再次达到2。

然后查找我们尚未交换的元素。我们可以通过检查元素的替换序列是否接触具有较低索引的元素来找到它。

问题：

• 如何证明替换序列实际上会回到start元素？
• 找到下一个元素看起来是O（n²）-ish但我认为它应该更好。

代码（它应该计算交换多少元素以获得更好的终止条件）：

public class Scratch {

  public static int newPos(int i, int len) {
    return i < len/2 ? i * 2 : (i - len/2) * 2 + 1;
  }

  public static int swapChain(int index0, String[] arr) {
    int i = index0;
    int nextStart = i + 1;
    int newPos = newPos(i, arr.length);
    String s = arr[newPos];
    arr[newPos] = arr[i];
    i = newPos;
    do {
      if (i == nextStart) {
        nextStart++;
      }
      newPos = newPos(i, arr.length);
      String t = arr[newPos];
      arr[newPos] = s;
      s = t;
      i = newPos;
    } while (i != index0);

    boolean valid;
    do {
      valid = true;
      i = nextStart;
      do {
        i = newPos(i, arr.length);
        if (i < nextStart) {
          nextStart++;
          if (nextStart >= arr.length / 2) {
            return nextStart;
          }
          valid = false;
          break;
        }
      } while (i != nextStart);
    } while (!valid);
    return nextStart;
  }

  public static void shuffle(String[] arr) {
    int start = 1;
    do {
      start = swapChain(start, arr);
    } while (start < arr.length / 2);
    System.out.println(Arrays.toString(arr));
  }

  public static void main(String[] args) {
    shuffle(new String[]{"1", "2", "3", "a", "b", "c"});
  }
}

更新：

我认为newPos（）在有限集上是双射的这一事实意味着该算法是正确的。

一些最坏情况数据（len是数组长度，步骤调用newPos（），因子是步/ len，平均因子是sum（步）/ sum（len））：

len: 8 steps: 9 factor 1.125 average factor: 0.8333333333333334
len: 16 steps: 29 factor 1.8125 average factor: 1.0857142857142856
len: 22 steps: 43 factor 1.9545454545454546 average factor: 1.2384615384615385
len: 28 steps: 72 factor 2.5714285714285716 average factor: 1.4663461538461537
len: 34 steps: 89 factor 2.6176470588235294 average factor: 1.644736842105263
len: 36 steps: 113 factor 3.138888888888889 average factor: 1.8029411764705883
len: 50 steps: 189 factor 3.78 average factor: 2.0462962962962963
len: 76 steps: 315 factor 4.144736842105263 average factor: 2.2432432432432434
len: 78 steps: 349 factor 4.4743589743589745 average factor: 2.3549422336328627
len: 112 steps: 504 factor 4.5 average factor: 2.5752351097178683
len: 136 steps: 664 factor 4.882352941176471 average factor: 2.8520255863539448
len: 142 steps: 712 factor 5.014084507042254 average factor: 2.823874755381605
len: 148 steps: 757 factor 5.114864864864865 average factor: 2.915825522710887
len: 162 steps: 861 factor 5.314814814814815 average factor: 3.050753012048193
len: 176 steps: 1041 factor 5.9147727272727275 average factor: 3.058876117496807
len: 202 steps: 1195 factor 5.915841584158416 average factor: 3.1066019417475728
len: 204 steps: 1329 factor 6.514705882352941 average factor: 3.1727913175932976
len: 244 steps: 1628 factor 6.672131147540983 average factor: 3.391762196747534
len: 304 steps: 2075 factor 6.8256578947368425 average factor: 3.726240646770448
len: 344 steps: 2589 factor 7.526162790697675 average factor: 3.908449284129865
len: 372 steps: 2983 factor 8.018817204301076 average factor: 3.934933870040253
len: 414 steps: 3380 factor 8.16425120772947 average factor: 4.053607098062898
len: 582 steps: 4902 factor 8.422680412371134 average factor: 4.413604801694715
len: 634 steps: 5615 factor 8.85646687697161 average factor: 4.502837189000436
len: 730 steps: 6701 factor 9.17945205479452 average factor: 4.637317723148786
len: 750 steps: 7097 factor 9.462666666666667 average factor: 4.679881984141619
len: 918 steps: 8880 factor 9.673202614379084 average factor: 4.8680720666104635
len: 1044 steps: 10267 factor 9.834291187739463 average factor: 5.031746787592856
len: 1212 steps: 12529 factor 10.337458745874587 average factor: 5.264601457155285
len: 1380 steps: 14839 factor 10.752898550724638 average factor: 5.3897728130741545
len: 1884 steps: 20660 factor 10.966029723991507 average factor: 5.784626659341847
len: 2052 steps: 23829 factor 11.612573099415204 average factor: 5.873873018885831
len: 2430 steps: 28415 factor 11.693415637860083 average factor: 6.030116322986142
len: 2598 steps: 30610 factor 11.782140107775211 average factor: 6.180337159160489
len: 2724 steps: 32969 factor 12.103157121879589 average factor: 6.215429400065934
len: 3270 steps: 39688 factor 12.137003058103975 average factor: 6.467267421298626
len: 3438 steps: 42335 factor 12.313845258871437 average factor: 6.534241807866802
len: 3564 steps: 44661 factor 12.531144781144782 average factor: 6.560849072043468
len: 3900 steps: 50485 factor 12.944871794871794 average factor: 6.655069539654636
len: 5224 steps: 68497 factor 13.111983154670751 average factor: 7.054594665556264
len: 5244 steps: 69375 factor 13.229405034324943 average factor: 7.058061034933604
len: 5412 steps: 72399 factor 13.377494456762749 average factor: 7.109842815290902
len: 5580 steps: 75029 factor 13.446057347670251 average factor: 7.135766175139542
len: 6588 steps: 90712 factor 13.769277474195507 average factor: 7.350829503005787
len: 6630 steps: 91307 factor 13.771794871794873 average factor: 7.36897875631633
len: 7134 steps: 100175 factor 14.041911970843847 average factor: 7.445842926414864
len: 7428 steps: 105366 factor 14.184975767366721 average factor: 7.5209458113740535
len: 8310 steps: 120431 factor 14.492298435619736 average factor: 7.654482597990361
len: 8814 steps: 128257 factor 14.551508963013388 average factor: 7.747129962677958
len: 9612 steps: 142392 factor 14.81398252184769 average factor: 7.856045162329174
len: 11460 steps: 171110 factor 14.931064572425829 average factor: 8.085284774991209
len: 12784 steps: 196548 factor 15.374530663329162 average factor: 8.24557688280267
len: 13812 steps: 213424 factor 15.452070663191428 average factor: 8.34207247251243
len: 14358 steps: 225811 factor 15.727190416492547 average factor: 8.403979957946827
len: 14694 steps: 233376 factor 15.882400979991834 average factor: 8.431773037753628
len: 17844 steps: 285690 factor 16.010423671822462 average factor: 8.698396681443686
len: 19860 steps: 323874 factor 16.30785498489426 average factor: 8.856597387159667
len: 19902 steps: 326059 factor 16.38322781629987 average factor: 8.860489779349878
len: 20028 steps: 328974 factor 16.425704014379868 average factor: 8.862442462977043
len: 23094 steps: 382677 factor 16.570407898155366 average factor: 9.046465277516655
len: 24900 steps: 415401 factor 16.68277108433735 average factor: 9.154661149194464
len: 28638 steps: 478313 factor 16.702039248550875 average factor: 9.359179435956479
len: 28764 steps: 489089 factor 17.00351133361146 average factor: 9.363446080908416
len: 29772 steps: 509199 factor 17.103284965739622 average factor: 9.412757619449271
len: 29982 steps: 513747 factor 17.1351811086652 average factor: 9.428669816872958
len: 32460 steps: 557422 factor 17.172581638940233 average factor: 9.528070456961768
len: 34644 steps: 612173 factor 17.67039025516684 average factor: 9.614116225080016
len: 38340 steps: 687700 factor 17.936880542514345 average factor: 9.75928795391779
len: 43380 steps: 782907 factor 18.047648686030428 average factor: 9.931165436868616
len: 44094 steps: 818495 factor 18.562502834852815 average factor: 9.954259890345835
len: 55182 steps: 1030147 factor 18.668170780326918 average factor: 10.283520352739814
len: 55350 steps: 1048684 factor 18.946413730803975 average factor: 10.287452618361032
len: 69462 steps: 1333151 factor 19.192522530304338 average factor: 10.618773030829923
len: 83532 steps: 1607630 factor 19.245678302925825 average factor: 10.87596852886678
len: 87774 steps: 1700299 factor 19.371328639460433 average factor: 10.951728617842308
len: 90088 steps: 1762592 factor 19.565225113222628 average factor: 10.988007959921369
len: 95964 steps: 1888910 factor 19.68352715601684 average factor: 11.074210030942767
len: 96804 steps: 2002173 factor 20.682750712780464 average factor: 11.086712711809684

Answer 2

<div class="container">
  <div class="block-element">lorem ipsum dolor sit ametlorem ipsum dolor sit ametlorem ipsum dolor sit amet</div>
  <div class="block-element">lorem ipsum dolor sit ametlorem ipsum dolor sit ametlorem ipsum dolor sit amet</div>
  <div class="block-element">lorem ipsum dolor sit ametlorem ipsum dolor sit ametlorem ipsum dolor sit amet</div>
  <div class="block-element">lorem ipsum dolor sit ametlorem ipsum dolor sit ametlorem ipsum dolor sit amet</div>
  <div class="block-element">lorem ipsum dolor sit ametlorem ipsum dolor sit ametlorem ipsum dolor sit amet</div>
  <div class="block-element">lorem ipsum dolor sit ametlorem ipsum dolor sit ametlorem ipsum dolor sit amet</div>
  <div class="block-element">lorem ipsum dolor sit ametlorem ipsum dolor sit ametlorem ipsum dolor sit amet</div>
  <div class="block-element">lorem ipsum dolor sit ametlorem ipsum dolor sit ametlorem ipsum dolor sit amet</div>
</div>