1. Lecture I Introduction
Problem: Dynamic Connectivity
Solutions: correctness, complexity, improvements
Applications
ACKNOWLEDGEMENTS: Some slides in this lecture source from COS226 of Princeton University by Kevin Wayne and Bob Sedgewick

2. The problem Problem: DynamicConnectivity
Input: n sites, m operations (union or connected query)
Output: the answer of connected queries
union(4, 3)
union(3, 8)
union(6, 5)
union(9, 4)
union(2, 1) ✗ 1 2 3 4
connected(0, 7) ✔
connected(8, 9)
union(5, 0) 5 6 7 8 9
union(7, 2)
union(6, 1) ✔
connected(0, 7)
union(1, 0)
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3. More difficult example
Q. Connected(p, q) p q
A. Yes.
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4. Where are we? Problem: Dynamic Connectivity
Solutions: correctness, complexity, improvements
Applications
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5. Model the problem { 0 } { 1 4 5 } { 2 3 6 7 } { 0 } { 1 2 3 4 5 6 7 }
union(2, 5) 1 2 3 1 2 3 4 5 6 7 4 5 6 7
{ 0 } { } { }
{ 0 } { }
3 connected components
2 connected components
Connected(2, 5): Are 2 and 5 in the same connected components?
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6. 1, 2, and 7 are connected 3, 4, 8, and 9 are connected
S1: Quick find
0, 5 and 6 are connected
1, 2, and 7 are connected 3, 4, 8, and 9 are connected 1 8 2 3 4 5 6 7 9
id[]
Demo:
0-9
union (4,3) (3,8) (6,5) (9,4) (2,1)
connected (0,7) (8,9)
union (5,0) (7,2) (6,1)
connected (0,7)
union (1,0) 1 2 3 4 5 6 7 8 9
Find(p,q). Check if p and q have the same id.
Union(p,q). To merge components containing p and q, change all entries whose id equals id[p] to id[q].
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7. Quiz Assume there are N sites, then in the worst case,
quick-find need ( ) array access for union.
quick-find need ( ) array access for find.
A. N B C. N2 D. logN
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8. 3's root is 9; 5's root is 6 3 and 5 are not connected
S2: Quick union 1 9 4 2 3 6 5 7 8
id[] 3 5 4 7 1 9 6 8 2 p q
3's root is 9; 5's root is 6 3 and 5 are not connected
Find(p,q). Check if p and q have the same root. Union(p,q). To merge components containing p and q, set the id of p's root to the id of q's root.
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9. Quiz Assume there are N sites, then in the worst case,
quick-union need ( ) array access for union.
quick-union need ( ) array access for find.
A. N B C. N2 D. logN
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10. Complexity Can we do better? algorithm initialize union find
quick-find N 1
quick-union
Can we do better?
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11. S3: Weighted union
Find(p,q). Check if p and q have the same root. Union(p,q). if size[root(p)] <= size[root(q)] then id[root(p)] = id[root(q)] else id[root(q)] = id[root(p)]
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12. S3: Weighted union
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13. Quiz Assume there are N sites, then in the worst case,
weighted-QU need ( ) array access for union.
weighted-QU need ( ) array access for find.
A. N B C. N2 D. logN
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14. Complexity Proposition. Depth of any node x is at most log N. 9/4/2018
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15. Complexity algorithm initialize union connected quick-find N 1
Proposition. Depth of any node x is at most log N.
algorithm
initialize
union
connected
quick-find N 1
quick-union
weighted QU
log N
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16. S4: Path compression
Quick union with path compression. Just after computing the root of p, set the id of each node on the path to point to that root. 2 5 4 1
root 7 3 10 8 6 p 12 11 9 x
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17. S4: Path compression
Quick union with path compression. Just after computing the root of p, set the id of each node on the path to point to that root. 2 5 4 1
root p 12 11 9 7 3 10 8 6 x
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18. S4: Path compression
Quick union with path compression. Just after computing the root of p, set the id of each node on the path to point to that root. 2 5 4 1
root p 12 11 9 10 8 6 x 7 3
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19. S4: Path compression
Quick union with path compression. Just after computing the root of p, set the id of each node on the path to point to that root. 2 5 4 1
root p 12 11 9 10 8 6 7 3 x
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20. Proofs are refer to [textbook] 4.3 and [DPV]5.1.4.
Complexity
Proposition. Starting from an empty data structure,
any sequence of M union-find operations on N objects makes at most proportional to N + M log* N array accesses. N
log* N 1 2 4 16 3
65536
265536 5
Union by rank alone: n + mlogn for n sites, m operations
Path compression alone: n + f(1 + log_{2+f/n} n ) for n sites, f find-perations
Both: one complexity by Hopcroft and Ulman is n + mlog*n, another complexity by Tarjan is n + m\alpha(m,n)
Bob Tarjan
John Hopcroft
Jeffrey Ullman
Proofs are refer to [textbook] 4.3 and [DPV]5.1.4.
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21. Complexity Proposition. Starting from an empty data structure,
any sequence of M union-find operations on N objects makes at most proportional to N + M log* N array accesses.
Can we do better?
Amazing fact. No linear-time algorithm exists.
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22. Where are we? Problem: Dynamic Connectivity
Solutions: correctness, complexity, improvements
Applications
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23. Application Dynamic connectivity.
Percolation. [Coursera. AlgoPartI. PA1]
Dynamic connectivity.
Hindley-Milner polymorphic type inference.
Kruskal's minimum spanning tree algorithm.
Compiling equivalence statements in Fortran.
......
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24. Conclusion Model a problem
Solve it, then ask three questions: Is it correct? How much time does it take? Can we do better?
Applications
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