现在我总是听说从随机选择的数据构建二叉搜索树比有序数据更快，这仅仅是因为有序数据需要显式重新平衡以保持树的高度最小。
最近我实现了一个不可变的 treap ，一种特殊的二叉搜索树，它使用随机化来保持自身相对平衡。与我的预期相反，我发现与无序数据相比，我可以始终如一地以快 2 倍的速度构建 treap，并且通常更好地平衡有序数据——我不知道为什么。
这是我的 treap 实现:
这是一个测试程序:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Diagnostics;
namespace ConsoleApplication1
{
class Program
{
static Random rnd = new Random();
const int ITERATION_COUNT = 20;
static void Main(string[] args)
{
List<double> rndTimes = new List<double>();
List<double> orderedTimes = new List<double>();
rndTimes.Add(TimeIt(50, RandomInsert));
rndTimes.Add(TimeIt(100, RandomInsert));
rndTimes.Add(TimeIt(200, RandomInsert));
rndTimes.Add(TimeIt(400, RandomInsert));
rndTimes.Add(TimeIt(800, RandomInsert));
rndTimes.Add(TimeIt(1000, RandomInsert));
rndTimes.Add(TimeIt(2000, RandomInsert));
rndTimes.Add(TimeIt(4000, RandomInsert));
rndTimes.Add(TimeIt(8000, RandomInsert));
rndTimes.Add(TimeIt(16000, RandomInsert));
rndTimes.Add(TimeIt(32000, RandomInsert));
rndTimes.Add(TimeIt(64000, RandomInsert));
rndTimes.Add(TimeIt(128000, RandomInsert));
string rndTimesAsString = string.Join("\n", rndTimes.Select(x => x.ToString()).ToArray());
orderedTimes.Add(TimeIt(50, OrderedInsert));
orderedTimes.Add(TimeIt(100, OrderedInsert));
orderedTimes.Add(TimeIt(200, OrderedInsert));
orderedTimes.Add(TimeIt(400, OrderedInsert));
orderedTimes.Add(TimeIt(800, OrderedInsert));
orderedTimes.Add(TimeIt(1000, OrderedInsert));
orderedTimes.Add(TimeIt(2000, OrderedInsert));
orderedTimes.Add(TimeIt(4000, OrderedInsert));
orderedTimes.Add(TimeIt(8000, OrderedInsert));
orderedTimes.Add(TimeIt(16000, OrderedInsert));
orderedTimes.Add(TimeIt(32000, OrderedInsert));
orderedTimes.Add(TimeIt(64000, OrderedInsert));
orderedTimes.Add(TimeIt(128000, OrderedInsert));
string orderedTimesAsString = string.Join("\n", orderedTimes.Select(x => x.ToString()).ToArray());
Console.WriteLine("Done");
}
static double TimeIt(int insertCount, Action<int> f)
{
Console.WriteLine("TimeIt({0}, {1})", insertCount, f.Method.Name);
List<double> times = new List<double>();
for (int i = 0; i < ITERATION_COUNT; i++)
{
Stopwatch sw = Stopwatch.StartNew();
f(insertCount);
sw.Stop();
times.Add(sw.Elapsed.TotalMilliseconds);
}
return times.Average();
}
static void RandomInsert(int insertCount)
{
Treap<double> tree = new Treap<double>((x, y) => x.CompareTo(y));
for (int i = 0; i < insertCount; i++)
{
tree = tree.Insert(rnd.NextDouble());
}
}
static void OrderedInsert(int insertCount)
{
Treap<double> tree = new Treap<double>((x, y) => x.CompareTo(y));
for(int i = 0; i < insertCount; i++)
{
tree = tree.Insert(i + rnd.NextDouble());
}
}
}
}
这是一张以毫秒为单位比较随机和有序插入时间的图表:
Insertions Random Ordered RandomTime / OrderedTime
50 1.031665 0.261585 3.94
100 0.544345 1.377155 0.4
200 1.268320 0.734570 1.73
400 2.765555 1.639150 1.69
800 6.089700 3.558350 1.71
1000 7.855150 4.704190 1.67
2000 17.852000 12.554065 1.42
4000 40.157340 22.474445 1.79
8000 88.375430 48.364265 1.83
16000 197.524000 109.082200 1.81
32000 459.277050 238.154405 1.93
64000 1055.508875 512.020310 2.06
128000 2481.694230 1107.980425 2.24
我在代码中没有看到任何使有序输入比无序输入渐进更快的东西，所以我无法解释其中的差异。
为什么从有序输入构建 treap 比随机输入快得多？
最佳答案
我运行了你的代码，我认为它与旋转次数有关。在有序输入时，旋转次数是最优的，树永远不会旋转回来。
在随机输入过程中，树将不得不执行更多的旋转，因为它可能不得不来回旋转。
要真正找出答案，我必须为每次运行的左右旋转次数添加计数器。你可能可以自己做。
更新:
我在 rotateleft 和 rotateright 上设置了断点。在有序输入期间，从不使用 rotateright。随机输入的时候，两个都命中，我觉得用的比较频繁。
更新 2:
我向 50 项有序运行添加了一些输出(为清楚起见用整数代替)，以了解更多信息:
TimeIt(50, OrderedInsert)
LastValue = 0, Top.Value = 0, Right.Count = 0, Left.Count = 0
RotateLeft @value=0
LastValue = 1, Top.Value = 1, Right.Count = 0, Left.Count = 1
LastValue = 2, Top.Value = 1, Right.Count = 1, Left.Count = 1
LastValue = 3, Top.Value = 1, Right.Count = 2, Left.Count = 1
RotateLeft @value=3
RotateLeft @value=2
RotateLeft @value=1
LastValue = 4, Top.Value = 4, Right.Count = 0, Left.Count = 4
LastValue = 5, Top.Value = 4, Right.Count = 1, Left.Count = 4
LastValue = 6, Top.Value = 4, Right.Count = 2, Left.Count = 4
RotateLeft @value=6
LastValue = 7, Top.Value = 4, Right.Count = 3, Left.Count = 4
LastValue = 8, Top.Value = 4, Right.Count = 4, Left.Count = 4
RotateLeft @value=8
RotateLeft @value=7
LastValue = 9, Top.Value = 4, Right.Count = 5, Left.Count = 4
LastValue = 10, Top.Value = 4, Right.Count = 6, Left.Count = 4
RotateLeft @value=10
RotateLeft @value=9
RotateLeft @value=5
RotateLeft @value=4
LastValue = 11, Top.Value = 11, Right.Count = 0, Left.Count = 11
LastValue = 12, Top.Value = 11, Right.Count = 1, Left.Count = 11
RotateLeft @value=12
LastValue = 13, Top.Value = 11, Right.Count = 2, Left.Count = 11
RotateLeft @value=13
LastValue = 14, Top.Value = 11, Right.Count = 3, Left.Count = 11
LastValue = 15, Top.Value = 11, Right.Count = 4, Left.Count = 11
RotateLeft @value=15
RotateLeft @value=14
LastValue = 16, Top.Value = 11, Right.Count = 5, Left.Count = 11
LastValue = 17, Top.Value = 11, Right.Count = 6, Left.Count = 11
RotateLeft @value=17
LastValue = 18, Top.Value = 11, Right.Count = 7, Left.Count = 11
LastValue = 19, Top.Value = 11, Right.Count = 8, Left.Count = 11
RotateLeft @value=19
LastValue = 20, Top.Value = 11, Right.Count = 9, Left.Count = 11
LastValue = 21, Top.Value = 11, Right.Count = 10, Left.Count = 11
RotateLeft @value=21
LastValue = 22, Top.Value = 11, Right.Count = 11, Left.Count = 11
RotateLeft @value=22
RotateLeft @value=20
RotateLeft @value=18
LastValue = 23, Top.Value = 11, Right.Count = 12, Left.Count = 11
LastValue = 24, Top.Value = 11, Right.Count = 13, Left.Count = 11
LastValue = 25, Top.Value = 11, Right.Count = 14, Left.Count = 11
RotateLeft @value=25
RotateLeft @value=24
LastValue = 26, Top.Value = 11, Right.Count = 15, Left.Count = 11
LastValue = 27, Top.Value = 11, Right.Count = 16, Left.Count = 11
RotateLeft @value=27
LastValue = 28, Top.Value = 11, Right.Count = 17, Left.Count = 11
RotateLeft @value=28
RotateLeft @value=26
RotateLeft @value=23
RotateLeft @value=16
RotateLeft @value=11
LastValue = 29, Top.Value = 29, Right.Count = 0, Left.Count = 29
LastValue = 30, Top.Value = 29, Right.Count = 1, Left.Count = 29
LastValue = 31, Top.Value = 29, Right.Count = 2, Left.Count = 29
LastValue = 32, Top.Value = 29, Right.Count = 3, Left.Count = 29
RotateLeft @value=32
RotateLeft @value=31
LastValue = 33, Top.Value = 29, Right.Count = 4, Left.Count = 29
RotateLeft @value=33
RotateLeft @value=30
LastValue = 34, Top.Value = 29, Right.Count = 5, Left.Count = 29
RotateLeft @value=34
LastValue = 35, Top.Value = 29, Right.Count = 6, Left.Count = 29
LastValue = 36, Top.Value = 29, Right.Count = 7, Left.Count = 29
LastValue = 37, Top.Value = 29, Right.Count = 8, Left.Count = 29
RotateLeft @value=37
LastValue = 38, Top.Value = 29, Right.Count = 9, Left.Count = 29
LastValue = 39, Top.Value = 29, Right.Count = 10, Left.Count = 29
RotateLeft @value=39
LastValue = 40, Top.Value = 29, Right.Count = 11, Left.Count = 29
RotateLeft @value=40
RotateLeft @value=38
RotateLeft @value=36
LastValue = 41, Top.Value = 29, Right.Count = 12, Left.Count = 29
LastValue = 42, Top.Value = 29, Right.Count = 13, Left.Count = 29
RotateLeft @value=42
LastValue = 43, Top.Value = 29, Right.Count = 14, Left.Count = 29
LastValue = 44, Top.Value = 29, Right.Count = 15, Left.Count = 29
RotateLeft @value=44
LastValue = 45, Top.Value = 29, Right.Count = 16, Left.Count = 29
LastValue = 46, Top.Value = 29, Right.Count = 17, Left.Count = 29
RotateLeft @value=46
RotateLeft @value=45
LastValue = 47, Top.Value = 29, Right.Count = 18, Left.Count = 29
LastValue = 48, Top.Value = 29, Right.Count = 19, Left.Count = 29
LastValue = 49, Top.Value = 29, Right.Count = 20, Left.Count = 29
自然地，有序的项目总是被添加到树的右侧。当右侧大于左侧时，就会发生 rotateleft。向右旋转永远不会发生。每次树翻倍时，都会大致选择一个新的顶部节点。优先级值的随机性使它有点抖动，因此在这次运行中它变为 0、1、4、11、29。
随机运行揭示了一些有趣的东西:
TimeIt(50, RandomInsert)
LastValue = 0,748661640914465, Top.Value = 0,748661640914465, Right.Count = 0, Left.Count = 0
LastValue = 0,669427539533669, Top.Value = 0,748661640914465, Right.Count = 0, Left.Count = 1
RotateRight @value=0,669427539533669
LastValue = 0,318363281115127, Top.Value = 0,748661640914465, Right.Count = 0, Left.Count = 2
RotateRight @value=0,669427539533669
LastValue = 0,33133987678743, Top.Value = 0,748661640914465, Right.Count = 0, Left.Count = 3
RotateLeft @value=0,748661640914465
LastValue = 0,955126694382693, Top.Value = 0,955126694382693, Right.Count = 0, Left.Count = 4
RotateRight @value=0,669427539533669
RotateLeft @value=0,33133987678743
RotateLeft @value=0,318363281115127
RotateRight @value=0,748661640914465
RotateRight @value=0,955126694382693
LastValue = 0,641024029180884, Top.Value = 0,641024029180884, Right.Count = 3, Left.Count = 2
LastValue = 0,20709771951991, Top.Value = 0,641024029180884, Right.Count = 3, Left.Count = 3
LastValue = 0,830862050331599, Top.Value = 0,641024029180884, Right.Count = 4, Left.Count = 3
RotateRight @value=0,20709771951991
RotateRight @value=0,318363281115127
LastValue = 0,203250563798123, Top.Value = 0,641024029180884, Right.Count = 4, Left.Count = 4
RotateLeft @value=0,669427539533669
RotateRight @value=0,748661640914465
RotateRight @value=0,955126694382693
LastValue = 0,701743399585478, Top.Value = 0,641024029180884, Right.Count = 5, Left.Count = 4
RotateLeft @value=0,669427539533669
RotateRight @value=0,701743399585478
RotateLeft @value=0,641024029180884
LastValue = 0,675667521858433, Top.Value = 0,675667521858433, Right.Count = 4, Left.Count = 6
RotateLeft @value=0,33133987678743
RotateLeft @value=0,318363281115127
RotateLeft @value=0,203250563798123
LastValue = 0,531275219531392, Top.Value = 0,675667521858433, Right.Count = 4, Left.Count = 7
RotateRight @value=0,748661640914465
RotateRight @value=0,955126694382693
RotateLeft @value=0,701743399585478
LastValue = 0,704049674190604, Top.Value = 0,675667521858433, Right.Count = 5, Left.Count = 7
RotateRight @value=0,203250563798123
RotateRight @value=0,531275219531392
RotateRight @value=0,641024029180884
RotateRight @value=0,675667521858433
LastValue = 0,161392807104342, Top.Value = 0,161392807104342, Right.Count = 13, Left.Count = 0
RotateRight @value=0,203250563798123
RotateRight @value=0,531275219531392
RotateRight @value=0,641024029180884
RotateRight @value=0,675667521858433
RotateLeft @value=0,161392807104342
LastValue = 0,167598206162266, Top.Value = 0,167598206162266, Right.Count = 13, Left.Count = 1
LastValue = 0,154996359793002, Top.Value = 0,167598206162266, Right.Count = 13, Left.Count = 2
RotateLeft @value=0,33133987678743
LastValue = 0,431767346538495, Top.Value = 0,167598206162266, Right.Count = 14, Left.Count = 2
RotateRight @value=0,203250563798123
RotateRight @value=0,531275219531392
RotateRight @value=0,641024029180884
RotateRight @value=0,675667521858433
RotateLeft @value=0,167598206162266
LastValue = 0,173774613614089, Top.Value = 0,173774613614089, Right.Count = 14, Left.Count = 3
RotateRight @value=0,830862050331599
LastValue = 0,76559642412029, Top.Value = 0,173774613614089, Right.Count = 15, Left.Count = 3
RotateRight @value=0,76559642412029
RotateLeft @value=0,748661640914465
RotateRight @value=0,955126694382693
RotateLeft @value=0,704049674190604
RotateLeft @value=0,675667521858433
LastValue = 0,75742144871383, Top.Value = 0,173774613614089, Right.Count = 16, Left.Count = 3
LastValue = 0,346844367844446, Top.Value = 0,173774613614089, Right.Count = 17, Left.Count = 3
RotateRight @value=0,830862050331599
LastValue = 0,787565814232251, Top.Value = 0,173774613614089, Right.Count = 18, Left.Count = 3
LastValue = 0,734950566540915, Top.Value = 0,173774613614089, Right.Count = 19, Left.Count = 3
RotateLeft @value=0,20709771951991
RotateRight @value=0,318363281115127
RotateLeft @value=0,203250563798123
RotateRight @value=0,531275219531392
RotateRight @value=0,641024029180884
RotateRight @value=0,675667521858433
RotateRight @value=0,75742144871383
RotateLeft @value=0,173774613614089
LastValue = 0,236504829598826, Top.Value = 0,236504829598826, Right.Count = 17, Left.Count = 6
RotateLeft @value=0,830862050331599
RotateLeft @value=0,787565814232251
RotateLeft @value=0,76559642412029
RotateRight @value=0,955126694382693
LastValue = 0,895606500048007, Top.Value = 0,236504829598826, Right.Count = 18, Left.Count = 6
LastValue = 0,599106418713511, Top.Value = 0,236504829598826, Right.Count = 19, Left.Count = 6
LastValue = 0,8182332901369, Top.Value = 0,236504829598826, Right.Count = 20, Left.Count = 6
RotateRight @value=0,734950566540915
LastValue = 0,704216948572647, Top.Value = 0,236504829598826, Right.Count = 21, Left.Count = 6
RotateLeft @value=0,346844367844446
RotateLeft @value=0,33133987678743
RotateRight @value=0,431767346538495
RotateLeft @value=0,318363281115127
RotateRight @value=0,531275219531392
RotateRight @value=0,641024029180884
RotateRight @value=0,675667521858433
RotateRight @value=0,75742144871383
LastValue = 0,379157059536854, Top.Value = 0,236504829598826, Right.Count = 22, Left.Count = 6
RotateLeft @value=0,431767346538495
LastValue = 0,46832062046431, Top.Value = 0,236504829598826, Right.Count = 23, Left.Count = 6
RotateRight @value=0,154996359793002
LastValue = 0,0999000217299443, Top.Value = 0,236504829598826, Right.Count = 23, Left.Count = 7
RotateLeft @value=0,20709771951991
LastValue = 0,229543754006524, Top.Value = 0,236504829598826, Right.Count = 23, Left.Count = 8
RotateRight @value=0,8182332901369
LastValue = 0,80358425984326, Top.Value = 0,236504829598826, Right.Count = 24, Left.Count = 8
RotateRight @value=0,318363281115127
LastValue = 0,259324726769386, Top.Value = 0,236504829598826, Right.Count = 25, Left.Count = 8
RotateRight @value=0,318363281115127
LastValue = 0,307835293145774, Top.Value = 0,236504829598826, Right.Count = 26, Left.Count = 8
RotateLeft @value=0,431767346538495
LastValue = 0,453910283024381, Top.Value = 0,236504829598826, Right.Count = 27, Left.Count = 8
RotateLeft @value=0,830862050331599
LastValue = 0,868997387527021, Top.Value = 0,236504829598826, Right.Count = 28, Left.Count = 8
RotateLeft @value=0,20709771951991
RotateRight @value=0,229543754006524
RotateLeft @value=0,203250563798123
LastValue = 0,218358597354199, Top.Value = 0,236504829598826, Right.Count = 28, Left.Count = 9
RotateRight @value=0,0999000217299443
RotateRight @value=0,161392807104342
LastValue = 0,0642934488431986, Top.Value = 0,236504829598826, Right.Count = 28, Left.Count = 10
RotateRight @value=0,154996359793002
RotateLeft @value=0,0999000217299443
LastValue = 0,148295871982489, Top.Value = 0,236504829598826, Right.Count = 28, Left.Count = 11
LastValue = 0,217621828065078, Top.Value = 0,236504829598826, Right.Count = 28, Left.Count = 12
RotateRight @value=0,599106418713511
LastValue = 0,553135806020878, Top.Value = 0,236504829598826, Right.Count = 29, Left.Count = 12
LastValue = 0,982277666210326, Top.Value = 0,236504829598826, Right.Count = 30, Left.Count = 12
RotateRight @value=0,8182332901369
LastValue = 0,803671114520948, Top.Value = 0,236504829598826, Right.Count = 31, Left.Count = 12
RotateRight @value=0,203250563798123
RotateRight @value=0,218358597354199
LastValue = 0,19310415405459, Top.Value = 0,236504829598826, Right.Count = 31, Left.Count = 13
LastValue = 0,0133136604043253, Top.Value = 0,236504829598826, Right.Count = 31, Left.Count = 14
RotateLeft @value=0,46832062046431
RotateRight @value=0,531275219531392
RotateRight @value=0,641024029180884
RotateRight @value=0,675667521858433
RotateRight @value=0,75742144871383
LastValue = 0,483394719419719, Top.Value = 0,236504829598826, Right.Count = 32, Left.Count = 14
RotateLeft @value=0,431767346538495
RotateRight @value=0,453910283024381
LastValue = 0,453370328738061, Top.Value = 0,236504829598826, Right.Count = 33, Left.Count = 14
LastValue = 0,762330518459124, Top.Value = 0,236504829598826, Right.Count = 34, Left.Count = 14
LastValue = 0,699010426969738, Top.Value = 0,236504829598826, Right.Count = 35, Left.Count = 14
轮换的发生并不是因为树不平衡，而是因为随机选择的优先级。例如，我们在第 13 次插入时进行了 4 次旋转。我们有一棵树在 5/7 处平衡(很好)，但达到 13/0! 随机优先级的使用似乎值得进一步研究。无论如何，很明显随机插入比有序插入导致更多的旋转。
关于c# - 为什么在排序输入上插入到我的树中比随机输入更快？，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/2437733/