CSC300 / CSC402: Union Find

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In these lectures I will use some notes from the textbook:

• Notes: Union Find
• Demo: Quick Find
• Demo: Quick Union
• Demo: Weighed Quick Union

file:TestUF.java [source]

001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 052 053 054 055 056 057 058 059 060 061 062 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 103 104 package algs15; import java.util.Scanner; import stdlib.*; public class TestUF { public static enum Order { ZERO_I, I_ZERO, I_MINUS, MINUS_I, RANDOM } public static void main(String[] args) { show(8, "1 4, 4 5, 2 6, 2 3, 3 7, 6 3, 5 2"); //notes from textbook show(10, "4 3, 3 8, 6 5, 9 4, 2 1, 8 9, 5 0, 7 2, 6 1, 1 0, 6 7"); //textbook tinyUF.txt show(10, "0 1, 0 2, 0 3, 0 4, 0 5, 0 6, 0 7, 0 8, 0 9"); // ZERO_I show(10, "1 0, 2 0, 3 0, 4 0, 5 0, 6 0, 7 0, 8 0, 9 0"); // I_ZERO show(10, "1 0, 2 1, 3 2, 4 3, 5 4, 6 5, 7 6, 8 7, 9 8"); // I_MINUS show(10, "0 1, 1 2, 2 3, 3 4, 4 5, 5 6, 6 7, 7 8, 8 9"); // MINUS_I show(16, "0 1, 2 3, 4 5, 6 7, 8 9, 10 11, 12 13, 14 15," + "0 2, 4 6, 8 10, 12 14," + "0 4, 8 12," + "0 8"); showRandom(12); double prevUnion = 0; double prevConnected = 0; for (int N = 128; N<=MAX; N += N) { timeTrial(N, Order.RANDOM); StdOut.format("N=%,13d Union=%7.3f [%8.3f] Connected=%7.3f [%8.3f]\n", N, timeUnion, timeUnion/prevUnion, timeConnected, timeConnected/prevConnected); prevUnion = timeUnion; prevConnected = timeConnected; } } private static int MAX=50_000_000; private static UF getUF (int N) { MAX=500_000; return new QuickFindUF(N); // (262_144,RANDOM) Union ~ 36 Connected ~ .006 //MAX=500_000; return new QuickUnionUF(N); // (262_144,RANDOM) Union ~ 17 Connected ~ 18 //return new WeightedUF(N); //(33_554_432,RANDOM) Union ~ 10 Connected ~ 10 //return new CompressionUF(N); //(33_554_432,RANDOM) Union ~ 16 Connected ~ 7 //return new XWeightedHalvingUF(N); //(33_554_432,RANDOM) Union ~ 9 Connected ~ 6 //return new XWeightedCompressionUF(N); //(33_554_432,RANDOM) Union ~ 9 Connected ~ 6 } private static double timeUnion; private static double timeConnected; private static void timeTrial(int N, Order order) { UF ufTime = getUF(N); SHOW_COMPRESSION_STEPS = false; Stopwatch sw1 = new Stopwatch(); for (int i=1; i<N; i+= 1) { int p = StdRandom.uniform(N); int q = StdRandom.uniform(N); switch (order) { case ZERO_I: ufTime.union (0, i); break; case I_ZERO: ufTime.union (i, 0); break; case I_MINUS: ufTime.union (i, i-1); break; case MINUS_I: ufTime.union (i-1, i); break; case RANDOM: ufTime.union (p, q); break; } } timeUnion = sw1.elapsedTime(); Stopwatch sw2 = new Stopwatch(); for (int i=1; i<N; i+= 1) { int p = StdRandom.uniform(N); int q = StdRandom.uniform(N); ufTime.connected(p, q); } timeConnected = sw2.elapsedTime(); } public static boolean SHOW_COMPRESSION_STEPS=false; private static void showRandom (int N) { SHOW_COMPRESSION_STEPS = true; UF uf = getUF(N); uf.toGraphviz(); StdOut.format(" %2d%s\n", uf.count(), uf); for (int i=1; i<4*N; i++) { int p = StdRandom.uniform(N); int q = StdRandom.uniform(N); if (uf.connected(p, q)) { StdOut.format("%2d %2d: connected\n", p, q); } else { uf.union(p, q); uf.toGraphviz(); StdOut.format("%2d %2d: %2d%s\n", p, q, uf.count(), uf); } } StdOut.println(); } private static void show (int N, String input) { SHOW_COMPRESSION_STEPS = true; Scanner s = new Scanner(input); s.useDelimiter("[\\s,]\\s*"); // use comma or space as delimiter UF uf = getUF(N); uf.toGraphviz(); StdOut.format(" %2d%s\n", uf.count(), uf); while (s.hasNext()) { int p = s.nextInt(); int q = s.nextInt(); if (uf.connected(p, q)) { StdOut.format("%2d %2d: connected\n", p, q); } else { uf.union(p, q); uf.toGraphviz(); StdOut.format("%2d %2d: %2d%s\n", p, q, uf.count(), uf); } } StdOut.println(); s.close(); } }

  public int find(int p) {








    return id[p];
  }

  public void union(int p, int q) {
    int pid = find(p); // loser
    int qid = find(q); // champion
    if (pid == qid) return;
    for (int i = 0; i < id.length; i++)
      if (id[i] == pid) id[i] = qid;
    count--;
  }

Output

       10[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
 4  3:  9[0, 1, 2, 3, 3, 5, 6, 7, 8, 9]
 3  8:  8[0, 1, 2, 8, 8, 5, 6, 7, 8, 9]
 6  5:  7[0, 1, 2, 8, 8, 5, 5, 7, 8, 9]
 9  4:  6[0, 1, 2, 8, 8, 5, 5, 7, 8, 8]
 2  1:  5[0, 1, 1, 8, 8, 5, 5, 7, 8, 8]
 8  9: connected
 5  0:  4[0, 1, 1, 8, 8, 0, 0, 7, 8, 8]
 7  2:  3[0, 1, 1, 8, 8, 0, 0, 1, 8, 8]
 6  1:  2[1, 1, 1, 8, 8, 1, 1, 1, 8, 8]
 1  0: connected
 6  7: connected

N=          128 Union=  0.002 [Infinity] Connected=  0.000 [     NaN]
N=          256 Union=  0.001 [   0.500] Connected=  0.001 [Infinity]
N=          512 Union=  0.005 [   5.000] Connected=  0.000 [   0.000]
N=        1,024 Union=  0.004 [   0.800] Connected=  0.001 [Infinity]
N=        2,048 Union=  0.008 [   2.000] Connected=  0.000 [   0.000]
N=        4,096 Union=  0.010 [   1.250] Connected=  0.000 [     NaN]
N=        8,192 Union=  0.047 [   4.700] Connected=  0.000 [     NaN]
N=       16,384 Union=  0.163 [   3.468] Connected=  0.001 [Infinity]
N=       32,768 Union=  0.667 [   4.092] Connected=  0.001 [   1.000]
N=       65,536 Union=  2.857 [   4.283] Connected=  0.002 [   2.000]
N=      131,072 Union= 11.939 [   4.179] Connected=  0.002 [   1.000]
N=      262,144 Union= 46.604 [   3.904] Connected=  0.005 [   2.500]

Union is linear, Connected is constant

  public int find(int p) {
    int root = p;
    while (root != id[root])
      root = id[root];





    return root;
  }

  public void union(int p, int q) {
    int pid = find(p); // loser
    int qid = find(q); // champion
    if (pid == qid) return;
    id[pid] = qid;

    count--;
  }

Output

       10[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
 4  3:  9[0, 1, 2, 3, 3, 5, 6, 7, 8, 9]
 3  8:  8[0, 1, 2, 8, 3, 5, 6, 7, 8, 9]
 6  5:  7[0, 1, 2, 8, 3, 5, 5, 7, 8, 9]
 9  4:  6[0, 1, 2, 8, 3, 5, 5, 7, 8, 8]
 2  1:  5[0, 1, 1, 8, 3, 5, 5, 7, 8, 8]
 8  9: connected
 5  0:  4[0, 1, 1, 8, 3, 0, 5, 7, 8, 8]
 7  2:  3[0, 1, 1, 8, 3, 0, 5, 1, 8, 8]
 6  1:  2[1, 1, 1, 8, 3, 0, 5, 1, 8, 8]
 1  0: connected
 6  7: connected

N=          128 Union=  0.001 [Infinity] Connected=  0.000 [     NaN]
N=          256 Union=  0.000 [   0.000] Connected=  0.000 [     NaN]
N=          512 Union=  0.000 [     NaN] Connected=  0.000 [     NaN]
N=        1,024 Union=  0.000 [     NaN] Connected=  0.001 [Infinity]
N=        2,048 Union=  0.001 [Infinity] Connected=  0.002 [   2.000]
N=        4,096 Union=  0.001 [   1.000] Connected=  0.003 [   1.500]
N=        8,192 Union=  0.004 [   4.000] Connected=  0.014 [   4.667]
N=       16,384 Union=  0.023 [   5.750] Connected=  0.108 [   7.714]
N=       32,768 Union=  0.120 [   5.217] Connected=  0.610 [   5.648]
N=       65,536 Union=  0.529 [   4.408] Connected=  2.552 [   4.184]
N=      131,072 Union=  3.390 [   6.408] Connected= 18.097 [   7.091]
N=      262,144 Union= 17.161 [   5.062] Connected= 90.506 [   5.001]

Union is linear, Connected is linear

  public int find(int p) {
    int root = p;
    while (root != id[root])
      root = id[root];





    return root;
  }

  public void union(int p, int q) {
    int pid = find(p); // champion, unless q is bigger
    int qid = find(q); // loser, unless p is smaller
    if (pid == qid) return;
    if (sz[qid] > sz[pid]) { id[pid] = qid; sz[qid] += sz[pid]; }
    else                   { id[qid] = pid; sz[pid] += sz[qid]; }
    count--;
  }

Output

       10[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
 4  3:  9[0, 1, 2, 4, 4, 5, 6, 7, 8, 9]
 3  8:  8[0, 1, 2, 4, 4, 5, 6, 7, 4, 9]
 6  5:  7[0, 1, 2, 4, 4, 6, 6, 7, 4, 9]
 9  4:  6[0, 1, 2, 4, 4, 6, 6, 7, 4, 4]
 2  1:  5[0, 2, 2, 4, 4, 6, 6, 7, 4, 4]
 8  9: connected
 5  0:  4[6, 2, 2, 4, 4, 6, 6, 7, 4, 4]
 7  2:  3[6, 2, 2, 4, 4, 6, 6, 2, 4, 4]
 6  1:  2[6, 2, 6, 4, 4, 6, 6, 2, 4, 4]
 1  0: connected
 6  7: connected

N=          128 Union=  0.001 [Infinity] Connected=  0.000 [     NaN]
N=          256 Union=  0.000 [   0.000] Connected=  0.000 [     NaN]
N=          512 Union=  0.000 [     NaN] Connected=  0.000 [     NaN]
N=        1,024 Union=  0.000 [     NaN] Connected=  0.000 [     NaN]
N=        2,048 Union=  0.000 [     NaN] Connected=  0.001 [Infinity]
N=        4,096 Union=  0.001 [Infinity] Connected=  0.000 [   0.000]
N=        8,192 Union=  0.001 [   1.000] Connected=  0.001 [Infinity]
N=       16,384 Union=  0.002 [   2.000] Connected=  0.002 [   2.000]
N=       32,768 Union=  0.003 [   1.500] Connected=  0.002 [   1.000]
N=       65,536 Union=  0.004 [   1.333] Connected=  0.003 [   1.500]
N=      131,072 Union=  0.008 [   2.000] Connected=  0.008 [   2.667]
N=      262,144 Union=  0.022 [   2.750] Connected=  0.016 [   2.000]
N=      524,288 Union=  0.052 [   2.364] Connected=  0.043 [   2.687]
N=    1,048,576 Union=  0.189 [   3.635] Connected=  0.127 [   2.953]
N=    2,097,152 Union=  0.499 [   2.640] Connected=  0.427 [   3.362]
N=    4,194,304 Union=  0.841 [   1.685] Connected=  0.743 [   1.740]
N=    8,388,608 Union=  2.091 [   2.486] Connected=  1.724 [   2.320]
N=   16,777,216 Union=  4.793 [   2.292] Connected=  4.576 [   2.654]
N=   33,554,432 Union=  9.952 [   2.076] Connected=  9.855 [   2.154]

Both operations logarithmic

  public int find(int p) {
    int root = p;
    while (root != id[root])
      root = id[root];
    while (id[p] != root) {
      int newp = id[p];
      id[p] = root;
      p = newp;
    }
    return root;
  }

  public void union(int p, int q) {
    int pid = find(p); // loser
    int qid = find(q); // champion
    if (pid == qid) return;
    id[pid] = qid;

    count--;
  }

Output

       10[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
 4  3:  9[0, 1, 2, 3, 3, 5, 6, 7, 8, 9]
 3  8:  8[0, 1, 2, 8, 3, 5, 6, 7, 8, 9]
 6  5:  7[0, 1, 2, 8, 3, 5, 5, 7, 8, 9]
 4  8>  7[0, 1, 2, 8, 8, 5, 5, 7, 8, 9]
 9  4:  6[0, 1, 2, 8, 8, 5, 5, 7, 8, 8]
 2  1:  5[0, 1, 1, 8, 8, 5, 5, 7, 8, 8]
 8  9: connected
 5  0:  4[0, 1, 1, 8, 8, 0, 5, 7, 8, 8]
 7  2:  3[0, 1, 1, 8, 8, 0, 5, 1, 8, 8]
 6  0>  3[0, 1, 1, 8, 8, 0, 0, 1, 8, 8]
 6  1:  2[1, 1, 1, 8, 8, 0, 0, 1, 8, 8]
 1  0: connected
 6  1>  2[1, 1, 1, 8, 8, 0, 1, 1, 8, 8]
 6  7: connected

N=          128 Union=  0.001 [Infinity] Connected=  0.000 [     NaN]
N=          256 Union=  0.001 [   1.000] Connected=  0.000 [     NaN]
N=          512 Union=  0.000 [   0.000] Connected=  0.000 [     NaN]
N=        1,024 Union=  0.000 [     NaN] Connected=  0.001 [Infinity]
N=        2,048 Union=  0.000 [     NaN] Connected=  0.000 [   0.000]
N=        4,096 Union=  0.000 [     NaN] Connected=  0.001 [Infinity]
N=        8,192 Union=  0.002 [Infinity] Connected=  0.001 [   1.000]
N=       16,384 Union=  0.003 [   1.500] Connected=  0.001 [   1.000]
N=       32,768 Union=  0.004 [   1.333] Connected=  0.001 [   1.000]
N=       65,536 Union=  0.004 [   1.000] Connected=  0.003 [   3.000]
N=      131,072 Union=  0.011 [   2.750] Connected=  0.005 [   1.667]
N=      262,144 Union=  0.026 [   2.364] Connected=  0.011 [   2.200]
N=      524,288 Union=  0.040 [   1.538] Connected=  0.021 [   1.909]
N=    1,048,576 Union=  0.136 [   3.400] Connected=  0.084 [   4.000]
N=    2,097,152 Union=  0.487 [   3.581] Connected=  0.285 [   3.393]
N=    4,194,304 Union=  1.331 [   2.733] Connected=  0.534 [   1.874]
N=    8,388,608 Union=  2.383 [   1.790] Connected=  1.201 [   2.249]
N=   16,777,216 Union=  6.689 [   2.807] Connected=  2.797 [   2.329]
N=   33,554,432 Union= 16.546 [   2.474] Connected=  7.270 [   2.599]

Both operations logarithmic

  public int find(int p) {
    int root = p;
    while (root != id[root])
      root = id[root];
    while (id[p] != root) {
      int newp = id[p];
      id[p] = root;
      p = newp;
    }
    return root;
  }

  public void union(int p, int q) {
    int pid = find(p); // loser
    int qid = find(q); // champion
    if (pid == qid) return;
    if (sz[qid] > sz[pid]) { id[pid] = qid; sz[qid] += sz[pid]; }
    else                   { id[qid] = pid; sz[pid] += sz[qid]; }
    count--;
  }

Output

       10[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
 4  3:  9[0, 1, 2, 4, 4, 5, 6, 7, 8, 9]
 3  8:  8[0, 1, 2, 4, 4, 5, 6, 7, 4, 9]
 6  5:  7[0, 1, 2, 4, 4, 6, 6, 7, 4, 9]
 9  4:  6[0, 1, 2, 4, 4, 6, 6, 7, 4, 4]
 2  1:  5[0, 2, 2, 4, 4, 6, 6, 7, 4, 4]
 8  9: connected
 5  0:  4[6, 2, 2, 4, 4, 6, 6, 7, 4, 4]
 7  2:  3[6, 2, 2, 4, 4, 6, 6, 2, 4, 4]
 6  1:  2[6, 2, 6, 4, 4, 6, 6, 2, 4, 4]
 1  6>  2[6, 6, 6, 4, 4, 6, 6, 2, 4, 4]
 1  0: connected
 7  6>  2[6, 6, 6, 4, 4, 6, 6, 6, 4, 4]
 6  7: connected

N=          128 Union=  0.001 [Infinity] Connected=  0.000 [     NaN]
N=          256 Union=  0.001 [   1.000] Connected=  0.000 [     NaN]
N=          512 Union=  0.000 [   0.000] Connected=  0.000 [     NaN]
N=        1,024 Union=  0.000 [     NaN] Connected=  0.001 [Infinity]
N=        2,048 Union=  0.000 [     NaN] Connected=  0.000 [   0.000]
N=        4,096 Union=  0.000 [     NaN] Connected=  0.001 [Infinity]
N=        8,192 Union=  0.001 [Infinity] Connected=  0.001 [   1.000]
N=       16,384 Union=  0.003 [   3.000] Connected=  0.002 [   2.000]
N=       32,768 Union=  0.004 [   1.333] Connected=  0.002 [   1.000]
N=       65,536 Union=  0.003 [   0.750] Connected=  0.003 [   1.500]
N=      131,072 Union=  0.009 [   3.000] Connected=  0.007 [   2.333]
N=      262,144 Union=  0.018 [   2.000] Connected=  0.010 [   1.429]
N=      524,288 Union=  0.055 [   3.056] Connected=  0.026 [   2.600]
N=    1,048,576 Union=  0.172 [   3.127] Connected=  0.073 [   2.808]
N=    2,097,152 Union=  0.434 [   2.523] Connected=  0.255 [   3.493]
N=    4,194,304 Union=  0.991 [   2.283] Connected=  0.482 [   1.890]
N=    8,388,608 Union=  1.682 [   1.697] Connected=  0.937 [   1.944]
N=   16,777,216 Union=  4.240 [   2.521] Connected=  2.861 [   3.053]
N=   33,554,432 Union=  8.646 [   2.039] Connected=  5.419 [   1.894]

Even faster

  public int find(int p) {
    int root = p;
    while (root != id[root]) {
      id[root] = id[id[root]];
      root = id[root];
    }



    return root;
  }

  public void union(int p, int q) {
    int pid = find(p); // loser
    int qid = find(q); // champion
    if (pid == qid) return;
    if (sz[qid] > sz[pid]) { id[pid] = qid; sz[qid] += sz[pid]; }
    else                   { id[qid] = pid; sz[pid] += sz[qid]; }
    count--;
  }

Output

       10[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
 4  3:  9[0, 1, 2, 4, 4, 5, 6, 7, 8, 9]
 3  8:  8[0, 1, 2, 4, 4, 5, 6, 7, 4, 9]
 6  5:  7[0, 1, 2, 4, 4, 6, 6, 7, 4, 9]
 9  4:  6[0, 1, 2, 4, 4, 6, 6, 7, 4, 4]
 2  1:  5[0, 2, 2, 4, 4, 6, 6, 7, 4, 4]
 8  9: connected
 5  0:  4[6, 2, 2, 4, 4, 6, 6, 7, 4, 4]
 7  2:  3[6, 2, 2, 4, 4, 6, 6, 2, 4, 4]
 6  1:  2[6, 2, 6, 4, 4, 6, 6, 2, 4, 4]
 1  6>  2[6, 6, 6, 4, 4, 6, 6, 2, 4, 4]
 1  0: connected
 7  6>  2[6, 6, 6, 4, 4, 6, 6, 6, 4, 4]
 6  7: connected

N=          128 Union=  0.001 [Infinity] Connected=  0.000 [     NaN]
N=          256 Union=  0.000 [   0.000] Connected=  0.000 [     NaN]
N=          512 Union=  0.000 [     NaN] Connected=  0.000 [     NaN]
N=        1,024 Union=  0.000 [     NaN] Connected=  0.000 [     NaN]
N=        2,048 Union=  0.000 [     NaN] Connected=  0.001 [Infinity]
N=        4,096 Union=  0.001 [Infinity] Connected=  0.001 [   1.000]
N=        8,192 Union=  0.001 [   1.000] Connected=  0.001 [   1.000]
N=       16,384 Union=  0.003 [   3.000] Connected=  0.002 [   2.000]
N=       32,768 Union=  0.004 [   1.333] Connected=  0.001 [   0.500]
N=       65,536 Union=  0.004 [   1.000] Connected=  0.004 [   4.000]
N=      131,072 Union=  0.011 [   2.750] Connected=  0.006 [   1.500]
N=      262,144 Union=  0.032 [   2.909] Connected=  0.015 [   2.500]
N=      524,288 Union=  0.059 [   1.844] Connected=  0.022 [   1.467]
N=    1,048,576 Union=  0.132 [   2.237] Connected=  0.063 [   2.864]
N=    2,097,152 Union=  0.327 [   2.477] Connected=  0.191 [   3.032]
N=    4,194,304 Union=  0.727 [   2.223] Connected=  0.459 [   2.403]
N=    8,388,608 Union=  1.998 [   2.748] Connected=  1.056 [   2.301]
N=   16,777,216 Union=  4.947 [   2.476] Connected=  2.554 [   2.419]
N=   33,554,432 Union= 10.013 [   2.024] Connected=  6.093 [   2.386]

Also faster

Suppose that we have three elements [0, 1, 2] and that we union 0 and 1, and then 1 and 2.

When we union 0 and 1, let's suppose that 1 is the champion, so the array is [1, 1, 2]. Pictorially, we have:

What can happen after we union 1 and 2?

• For quick find, the answer 2,2,2 is possible, but this outcome is not possible for quick union, or weighted union.

  

• For quick union, the answer 1,2,2 is possible, but this outcome is not possible for quick find, or weighted union.

  

• For weighted union, the only possible outcome is 1,1,1. This outcome is possible under all three algorithms.