CSC300: Resizing Arrays and Amortized Analysis

Contents [0/13]

Amortized Analysis [1/13]
PerformanceOfArrays [2/13]
PerformanceOfArrays +1 [3/13]
PerformanceOfArrays +10 [4/13]
PerformanceOfArrays +100 [5/13]
PerformanceOfArrays *2 [6/13]
PerformanceOfArrays *4 [7/13]
PerformanceOfArrays *8 [8/13]
PerformanceOfArrays *1.5 [9/13]
PerformanceOfArrays *1.1 [10/13]
PerformanceOfStrings [11/13]
PerformanceOfStrings [12/13]
PerformanceOfStrings [13/13]

(Click here for one slide per page)


Amortized Analysis [1/13]

Algorithm input includes data and a sequence of operations performed by the client

Amortized analysis provides a worst-case performance guarantee on a sequence of operations

We will look at

PerformanceOfArrays [2/13]

file:PerformanceOfArrays.java [source]
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package ds1.student.orderofgrowth;

import stdlib.StdOut;
import stdlib.Stopwatch;

public class PerformanceOfArrays {
  private double[] a;
  private int N;
  public PerformanceOfArrays() {
    this.a = new double[1];
    this.N = 0;
  }
  private void resize(int capacity) {
    double[] temp = new double[capacity];
    for (int i = 0; i < N; i+=1) {
      if (SHOW) StdOut.format ("%02d ", i);
      temp[i] = a[i];
      ops += 1;
    }
    if (SHOW) StdOut.println();
    a = temp;
  }
  public void append(double item) {
    if (N == a.length) resize (1+N); //resize((int)Math.ceil (N*1.5));
    a[N] = item;
    N += 1;
    ops += 1;
  }
  
  private static long ops;
  private static boolean SHOW = true;
  public static void main(String[] args) {
    timeTrial(32);
    SHOW = false;
    
    int MIN = 256; 
    int MAX = 100_000_000;
    double prevTime = timeTrial(MIN);
    double prevOps = ops;
    double deltaSum = 0;
    int deltaNum = 0;
    for (int N = MIN*2; N<=MAX; N += N) {
      double time = timeTrial(N);
      StdOut.format ("Elapsed count f(%,13d): %,17d: %10.3f [%10.3f : %10.3f]\n", N, ops, ops / prevOps, time, time/prevTime);
      prevTime = time;
      deltaSum += ops/prevOps;
      deltaNum += 1;
      prevOps = ops;
    }
    StdOut.format ("Average delta: %.3f\n", deltaSum/deltaNum);
  }
  public static double timeTrial(int N) {
    ops = 0;
    Stopwatch s = new Stopwatch();
    PerformanceOfArrays perfOfArrays = new PerformanceOfArrays();
    for (int j=0; j<N; j+=1) {
      perfOfArrays.append(j);
    }
    return s.elapsedTime();
  }
}

PerformanceOfArrays +1 [3/13]

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  public void append(double item) {
    if (N == a.length) resize(1+N);
    a[N] = item;
    N += 1;
  }

Output 

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00 01 02 03 04 05 06 07 
00 01 02 03 04 05 06 07 08 
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00 01 02 03 04 05 06 07 08 09 10 
00 01 02 03 04 05 06 07 08 09 10 11 
00 01 02 03 04 05 06 07 08 09 10 11 12 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 
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00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 
Elapsed count f(          512):           131,328:      3.992 [     0.005 :      2.500]
Elapsed count f(        1,024):           524,800:      3.996 [     0.010 :      2.000]
Elapsed count f(        2,048):         2,098,176:      3.998 [     0.008 :      0.800]
Elapsed count f(        4,096):         8,390,656:      3.999 [     0.060 :      7.500]
Elapsed count f(        8,192):        33,558,528:      4.000 [     0.088 :      1.467]
Elapsed count f(       16,384):       134,225,920:      4.000 [     0.182 :      2.068]
Elapsed count f(       32,768):       536,887,296:      4.000 [     0.685 :      3.764]
Elapsed count f(       65,536):     2,147,516,416:      4.000 [     2.292 :      3.346]
Elapsed count f(      131,072):     8,590,000,128:      4.000 [     8.996 :      3.925]
Elapsed count f(      262,144):    34,359,869,440:      4.000 [    38.306 :      4.258]

N appends in quadratic time

Each append is amortized linear time

PerformanceOfArrays +10 [4/13]

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  public void append(double item) {
    if (N == a.length) resize(10+N);
    a[N] = item;
    N += 1;
  }

Output 


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00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 
Elapsed count f(          512):            13,824:      3.914 [     0.001 :   Infinity]
Elapsed count f(        1,024):            53,657:      3.881 [     0.003 :      3.000]
Elapsed count f(        2,048):           211,353:      3.939 [     0.003 :      1.000]
Elapsed count f(        4,096):           842,956:      3.988 [     0.011 :      3.667]
Elapsed count f(        8,192):         3,366,912:      3.994 [     0.011 :      1.000]
Elapsed count f(       16,384):        13,441,433:      3.992 [     0.063 :      5.727]
Elapsed count f(       32,768):        53,713,305:      3.996 [     0.085 :      1.349]
Elapsed count f(       65,536):       214,813,900:      3.999 [     0.348 :      4.094]
Elapsed count f(      131,072):       859,176,960:      4.000 [     1.005 :      2.888]
Elapsed count f(      262,144):     3,436,288,409:      4.000 [     3.955 :      3.935]
Elapsed count f(      524,288):    13,744,314,777:      4.000 [    17.241 :      4.359]
Elapsed count f(    1,048,576):    54,976,629,964:      4.000 [    71.803 :      4.165]

N appends in quadratic time

Out of 10 appends: 9 are constant time; 1 is linear

Each append is amortized linear time

PerformanceOfArrays +100 [5/13]

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  public void append(double item) {
    if (N == a.length) resize(100+N);
    a[N] = item;
    N += 1;
  }

Output 


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Elapsed count f(          512):             2,018:      3.610 [     0.000 :      0.000]
Elapsed count f(        1,024):             6,535:      3.238 [     0.000 :        NaN]
Elapsed count f(        2,048):            23,069:      3.530 [     0.001 :   Infinity]
Elapsed count f(        4,096):            86,137:      3.734 [     0.006 :      6.000]
Elapsed count f(        8,192):           340,374:      3.952 [     0.008 :      1.333]
Elapsed count f(       16,384):         1,353,148:      3.975 [     0.013 :      1.625]
Elapsed count f(       32,768):         5,395,896:      3.988 [     0.019 :      1.462]
Elapsed count f(       65,536):        21,550,192:      3.994 [     0.072 :      3.789]
Elapsed count f(      131,072):        86,002,883:      3.991 [     0.162 :      2.250]
Elapsed count f(      262,144):       343,877,866:      3.998 [     0.553 :      3.414]
Elapsed count f(      524,288):     1,374,719,831:      3.998 [     1.775 :      3.210]
Elapsed count f(    1,048,576):     5,498,344,562:      4.000 [     7.048 :      3.971]
Elapsed count f(    2,097,152):    21,992,308,724:      4.000 [    29.507 :      4.187]

N appends in quadratic time

Out of 100 appends: 99 are constant time; 1 is linear

Each append is amortized linear time

PerformanceOfArrays *2 [6/13]

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  public void append(double item) {
    if (N == a.length) resize(2*N);
    a[N] = item;
    N += 1;
  }

Output 

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Elapsed count f(          512):             1,023:      2.002 [     0.000 :        NaN]
Elapsed count f(        1,024):             2,047:      2.001 [     0.000 :        NaN]
Elapsed count f(        2,048):             4,095:      2.000 [     0.000 :        NaN]
Elapsed count f(        4,096):             8,191:      2.000 [     0.000 :        NaN]
Elapsed count f(        8,192):            16,383:      2.000 [     0.000 :        NaN]
Elapsed count f(       16,384):            32,767:      2.000 [     0.001 :   Infinity]
Elapsed count f(       32,768):            65,535:      2.000 [     0.003 :      3.000]
Elapsed count f(       65,536):           131,071:      2.000 [     0.007 :      2.333]
Elapsed count f(      131,072):           262,143:      2.000 [     0.002 :      0.286]
Elapsed count f(      262,144):           524,287:      2.000 [     0.003 :      1.500]
Elapsed count f(      524,288):         1,048,575:      2.000 [     0.007 :      2.333]
Elapsed count f(    1,048,576):         2,097,151:      2.000 [     0.015 :      2.143]
Elapsed count f(    2,097,152):         4,194,303:      2.000 [     0.028 :      1.867]
Elapsed count f(    4,194,304):         8,388,607:      2.000 [     0.043 :      1.536]
Elapsed count f(    8,388,608):        16,777,215:      2.000 [     0.055 :      1.279]
Elapsed count f(   16,777,216):        33,554,431:      2.000 [     0.147 :      2.673]
Elapsed count f(   33,554,432):        67,108,863:      2.000 [     0.360 :      2.449]
Elapsed count f(   67,108,864):       134,217,727:      2.000 [     0.597 :      1.658]
Average delta: 2.000

N appends in linear time

Each time we resize: constant appends double

Each append is amortized constant time

PerformanceOfArrays *4 [7/13]

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  public void append(double item) {
    if (N == a.length) resize(4*N);
    a[N] = item;
    N += 1;
  }

Output 



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Elapsed count f(          512):               853:      2.501 [     0.000 :        NaN]
Elapsed count f(        1,024):             1,365:      1.600 [     0.000 :        NaN]
Elapsed count f(        2,048):             3,413:      2.500 [     0.000 :        NaN]
Elapsed count f(        4,096):             5,461:      1.600 [     0.000 :        NaN]
Elapsed count f(        8,192):            13,653:      2.500 [     0.001 :   Infinity]
Elapsed count f(       16,384):            21,845:      1.600 [     0.001 :      1.000]
Elapsed count f(       32,768):            54,613:      2.500 [     0.002 :      2.000]
Elapsed count f(       65,536):            87,381:      1.600 [     0.003 :      1.500]
Elapsed count f(      131,072):           218,453:      2.500 [     0.007 :      2.333]
Elapsed count f(      262,144):           349,525:      1.600 [     0.006 :      0.857]
Elapsed count f(      524,288):           873,813:      2.500 [     0.008 :      1.333]
Elapsed count f(    1,048,576):         1,398,101:      1.600 [     0.010 :      1.250]
Elapsed count f(    2,097,152):         3,495,253:      2.500 [     0.037 :      3.700]
Elapsed count f(    4,194,304):         5,592,405:      1.600 [     0.029 :      0.784]
Elapsed count f(    8,388,608):        13,981,013:      2.500 [     0.100 :      3.448]
Elapsed count f(   16,777,216):        22,369,621:      1.600 [     0.061 :      0.610]
Elapsed count f(   33,554,432):        55,924,053:      2.500 [     0.429 :      7.033]
Elapsed count f(   67,108,864):        89,478,485:      1.600 [     0.288 :      0.671]
Average delta: 2.050

N appends in linear time

Each time we resize: constant appends quadruple

Each append is amortized constant time

PerformanceOfArrays *8 [8/13]

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  public void append(double item) {
    if (N == a.length) resize(8*N);
    a[N] = item;
    N += 1;
  }

Output 




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Elapsed count f(          512):               585:      1.778 [     0.000 :        NaN]
Elapsed count f(        1,024):             1,609:      2.750 [     0.000 :        NaN]
Elapsed count f(        2,048):             2,633:      1.636 [     0.000 :        NaN]
Elapsed count f(        4,096):             4,681:      1.778 [     0.000 :        NaN]
Elapsed count f(        8,192):            12,873:      2.750 [     0.001 :   Infinity]
Elapsed count f(       16,384):            21,065:      1.636 [     0.001 :      1.000]
Elapsed count f(       32,768):            37,449:      1.778 [     0.001 :      1.000]
Elapsed count f(       65,536):           102,985:      2.750 [     0.006 :      6.000]
Elapsed count f(      131,072):           168,521:      1.636 [     0.005 :      0.833]
Elapsed count f(      262,144):           299,593:      1.778 [     0.007 :      1.400]
Elapsed count f(      524,288):           823,881:      2.750 [     0.013 :      1.857]
Elapsed count f(    1,048,576):         1,348,169:      1.636 [     0.016 :      1.231]
Elapsed count f(    2,097,152):         2,396,745:      1.778 [     0.017 :      1.063]
Elapsed count f(    4,194,304):         6,591,049:      2.750 [     0.095 :      5.588]
Elapsed count f(    8,388,608):        10,785,353:      1.636 [     0.043 :      0.453]
Elapsed count f(   16,777,216):        19,173,961:      1.778 [     0.060 :      1.395]
Elapsed count f(   33,554,432):        52,728,393:      2.750 [     0.733 :     12.217]
Elapsed count f(   67,108,864):        86,282,825:      1.636 [     0.298 :      0.407]
Average delta: 2.055

N appends in linear time

Each time we resize: constant appends multiply by 8

Each append is amortized constant time

PerformanceOfArrays *1.5 [9/13]

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  public void append(double item) {
    if (N == a.length) resize((int)Math.ceil (N*1.5));
    a[N] = item;
    N += 1;
  }

Output 

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00 01 02 03 04 
00 01 02 03 04 05 06 07 
00 01 02 03 04 05 06 07 08 09 10 11 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 
Elapsed count f(          512):             1,922:      2.189 [     0.000 :        NaN]
Elapsed count f(        1,024):             3,144:      1.636 [     0.001 :   Infinity]
Elapsed count f(        2,048):             6,831:      2.173 [     0.000 :      0.000]
Elapsed count f(        4,096):            14,872:      2.177 [     0.000 :        NaN]
Elapsed count f(        8,192):            32,453:      2.182 [     0.001 :   Infinity]
Elapsed count f(       16,384):            52,782:      1.626 [     0.002 :      2.000]
Elapsed count f(       32,768):           114,681:      2.173 [     0.002 :      1.000]
Elapsed count f(       65,536):           249,859:      2.179 [     0.003 :      1.500]
Elapsed count f(      131,072):           407,564:      1.631 [     0.004 :      1.333]
Elapsed count f(      262,144):           884,271:      2.170 [     0.006 :      1.500]
Elapsed count f(      524,288):         1,924,095:      2.176 [     0.012 :      2.000]
Elapsed count f(    1,048,576):         3,148,295:      1.636 [     0.020 :      1.667]
Elapsed count f(    2,097,152):         6,821,541:      2.167 [     0.039 :      1.950]
Elapsed count f(    4,194,304):        14,824,201:      2.173 [     0.042 :      1.077]
Elapsed count f(    8,388,608):        32,305,900:      2.179 [     0.147 :      3.500]
Elapsed count f(   16,777,216):        52,653,164:      1.630 [     0.150 :      1.020]
Elapsed count f(   33,554,432):       114,275,340:      2.170 [     0.508 :      3.387]
Elapsed count f(   67,108,864):       248,730,932:      2.177 [     1.268 :      2.496]
Average delta: 2.025

N appends in linear time

Each time we resize: constant appends multiply by 1.5

Each append is amortized constant time

PerformanceOfArrays *1.1 [10/13]

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  public void append(double item) {
    if (N == a.length) resize((int)Math.ceil (N*1.1));
    a[N] = item;
    N += 1;
  }

Output 

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00 01 02 
00 01 02 03 
00 01 02 03 04 
00 01 02 03 04 05 
00 01 02 03 04 05 06 
00 01 02 03 04 05 06 07 
00 01 02 03 04 05 06 07 08 
00 01 02 03 04 05 06 07 08 09 
00 01 02 03 04 05 06 07 08 09 10 
00 01 02 03 04 05 06 07 08 09 10 11 12 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 
Elapsed count f(          512):             5,721:      2.024 [     0.001 :   Infinity]
Elapsed count f(        1,024):            11,402:      1.993 [     0.001 :      1.000]
Elapsed count f(        2,048):            22,520:      1.975 [     0.002 :      2.000]
Elapsed count f(        4,096):            48,320:      2.146 [     0.001 :      0.500]
Elapsed count f(        8,192):            94,698:      1.960 [     0.002 :      2.000]
Elapsed count f(       16,384):           185,318:      1.957 [     0.005 :      2.500]
Elapsed count f(       32,768):           362,372:      1.955 [     0.006 :      1.200]
Elapsed count f(       65,536):           772,598:      2.132 [     0.012 :      2.000]
Elapsed count f(      131,072):         1,509,413:      1.954 [     0.018 :      1.500]
Elapsed count f(      262,144):         2,948,653:      1.954 [     0.018 :      1.000]
Elapsed count f(      524,288):         6,283,714:      2.131 [     0.048 :      2.667]
Elapsed count f(    1,048,576):        12,272,649:      1.953 [     0.046 :      0.958]
Elapsed count f(    2,097,152):        23,970,316:      1.953 [     0.051 :      1.109]
Elapsed count f(    4,194,304):        46,819,577:      1.953 [     0.102 :      2.000]
Elapsed count f(    8,388,608):        99,760,500:      2.131 [     0.223 :      2.186]
Elapsed count f(   16,777,216):       194,835,921:      1.953 [     0.454 :      2.036]
Elapsed count f(   33,554,432):       380,541,255:      1.953 [     0.953 :      2.099]
Elapsed count f(   67,108,864):       743,288,834:      1.953 [     1.579 :      1.657]
Average delta: 2.002

N appends in linear time

Each time we resize: constant appends multiply by 1.1

Each append is amortized constant time

The choice of multiplier value is a tradeoff between time and space.

PerformanceOfStrings [11/13]

Strings are typically implemented as arrays of characters.

file:PerformanceOfStrings.java [source]
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package ds1.student.orderofgrowth;

import stdlib.StdOut;
import stdlib.Stopwatch;

public class PerformanceOfStrings {
  /** create a string consisting of N asterisks */
  public static String makeStringUsingConcat (int N) {
    String result = "";
    for (int i=0; i<N; i+=1) {
      result = result + "*";
      ops += result.length();
    }
    return result;
  }
  /** create a string consisting of N asterisks */
  public static String makeStringUsingBuilder (int N) {
    StringBuilder result = new StringBuilder ();
    for (int i=0; i<N; i+=1) { 
      result.append ("*");
      ops += 1;
    }
    return result.toString ();
  }
  private static long ops;
  private static String result;
  public static void main(String[] args) {
    timeTrial(32);
    StdOut.println(result);
    
    int MIN = 256; 
    int MAX = 1_000_000_000;
    double prevTime = timeTrial(MIN);
    double prevOps = ops;
    for (int N = MIN*2; N<=MAX; N += N) {
      double time = timeTrial(N);
      StdOut.format ("Elapsed count f(%,13d): %,17d: %10.3f [%10.3f : %10.3f]\n", N, ops, ops / prevOps, time, time/prevTime);
      prevTime = time;
      prevOps = ops;
    }
  }
  public static double timeTrial(int N) {
    ops = 0;
    Stopwatch sw = new Stopwatch();
    result = makeStringUsingConcat(N);
    //result = makeStringUsingBuilder(N);
    return sw.elapsedTime();
  }
}

PerformanceOfStrings [12/13]

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07
 
  public static String makeStringUsingConcat (int N) {
    String result = "";
    for (int i=0; i<N; i+=1) {
      result = result + "*";
    }
    return result;
  }

Output 

********************************
Elapsed count f(          512):           131,328:      3.992 [     0.001 :      1.000]
Elapsed count f(        1,024):           524,800:      3.996 [     0.001 :      1.000]
Elapsed count f(        2,048):         2,098,176:      3.998 [     0.004 :      4.000]
Elapsed count f(        4,096):         8,390,656:      3.999 [     0.011 :      2.750]
Elapsed count f(        8,192):        33,558,528:      4.000 [     0.048 :      4.364]
Elapsed count f(       16,384):       134,225,920:      4.000 [     0.098 :      2.042]
Elapsed count f(       32,768):       536,887,296:      4.000 [     0.180 :      1.837]
Elapsed count f(       65,536):     2,147,516,416:      4.000 [     0.574 :      3.189]
Elapsed count f(      131,072):     8,590,000,128:      4.000 [     1.794 :      3.125]
Elapsed count f(      262,144):    34,359,869,440:      4.000 [     6.947 :      3.872]
Elapsed count f(      524,288):   137,439,215,616:      4.000 [    28.864 :      4.155]
Elapsed count f(    1,048,576):   549,756,338,176:      4.000 [   125.439 :      4.346]

N concatenations in quadratic time

Each concatenation is amortized linear time

A common bug in collections (how to print an array in quadratic time):

01
02
03
04
05
06
07
 
  public static String toString () {
    String result = "";
    for (int i=0; i<N; i+=1) {
      result = result + a[i];
    }
    return result;
  }

PerformanceOfStrings [13/13]

01
02
03
04
05
06
07
 
  public static String makeStringUsingBuilder (int N) {
    StringBuilder result = new StringBuilder ();
    for (int i=0; i<N; i+=1) { 
      result.append ("*");
    }
    return result.toString ();
  }

Output 

********************************
Elapsed count f(          512):               512:      2.000 [     0.001 :   Infinity]
Elapsed count f(        1,024):             1,024:      2.000 [     0.000 :      0.000]
Elapsed count f(        2,048):             2,048:      2.000 [     0.000 :        NaN]
Elapsed count f(        4,096):             4,096:      2.000 [     0.000 :        NaN]
Elapsed count f(        8,192):             8,192:      2.000 [     0.001 :   Infinity]
Elapsed count f(       16,384):            16,384:      2.000 [     0.002 :      2.000]
Elapsed count f(       32,768):            32,768:      2.000 [     0.002 :      1.000]
Elapsed count f(       65,536):            65,536:      2.000 [     0.002 :      1.000]
Elapsed count f(      131,072):           131,072:      2.000 [     0.004 :      2.000]
Elapsed count f(      262,144):           262,144:      2.000 [     0.007 :      1.750]
Elapsed count f(      524,288):           524,288:      2.000 [     0.008 :      1.143]
Elapsed count f(    1,048,576):         1,048,576:      2.000 [     0.014 :      1.750]
Elapsed count f(    2,097,152):         2,097,152:      2.000 [     0.037 :      2.643]
Elapsed count f(    4,194,304):         4,194,304:      2.000 [     0.057 :      1.541]
Elapsed count f(    8,388,608):         8,388,608:      2.000 [     0.120 :      2.105]
Elapsed count f(   16,777,216):        16,777,216:      2.000 [     0.229 :      1.908]
Elapsed count f(   33,554,432):        33,554,432:      2.000 [     0.430 :      1.878]
Elapsed count f(   67,108,864):        67,108,864:      2.000 [     0.879 :      2.044]
Elapsed count f(  134,217,728):       134,217,728:      2.000 [     1.843 :      2.097]
Elapsed count f(  268,435,456):       268,435,456:      2.000 [     3.706 :      2.011]
Elapsed count f(  536,870,912):       536,870,912:      2.000 [     7.635 :      2.060]

N appends in linear time

Each append is amortized constant time

Java's StringBuilder is implemented with a resizing array.