Review of MST Algorithms Disjoint-Set Union Amortized Analysis CS 332: Algorithms Review of MST Algorithms Disjoint-Set Union Amortized Analysis David Luebke 1 7/19/2019

Review: MST Algorithms In a connected, weighted, undirected graph, will the edge with the lowest weight be in the MST? Why or why not? Yes: If T is MST of G, and A  T is a subtree of T, and (u,v) is the min-weight edge connecting A to V-A, then (u,v)  T The lowest-weight edge must be in the tree (A=) David Luebke 2 7/19/2019

Review: MST Algorithms What do the disjoint sets in Kruskal’s algorithm represent? A: Parts of the graph we have connected up together so far David Luebke 3 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 4 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1? David Luebke 5 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 6 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2? 19 9 14 17 8 25 5 21 13 1 David Luebke 7 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 8 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5? 21 13 1 David Luebke 9 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 10 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8? 25 5 21 13 1 David Luebke 11 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 12 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9? 14 17 8 25 5 21 13 1 David Luebke 13 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 14 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13? 1 David Luebke 15 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 16 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14? 17 8 25 5 21 13 1 David Luebke 17 7/19/2019

Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 18 7/19/2019

Review: Shortest-Path Algorithms How does the Bellman-Ford algorithm work? How can we do better for DAGs? Under what conditions can we use Dijkstra’s algorithm? David Luebke 19 7/19/2019

Review: Running Time of Kruskal’s Algorithm Expensive operations: Sort edges: O(E lg E) O(V) MakeSet()’s O(E) FindSet()’s O(V) Union()’s Upshot: Comes down to efficiency of disjoint-set operations, particularly Union() David Luebke 20 7/19/2019

Disjoint Set Union So how do we represent disjoint sets? Naïve implementation: use a linked list to represent elements, with pointers back to set: MakeSet(): O(1) FindSet(): O(1) Union(A,B): “Copy” elements of A into set B by adjusting elements of A to point to B: O(A) How long could n Union()s take? David Luebke 21 7/19/2019

Disjoint Set Union: Analysis Worst-case analysis: O(n2) time for n Union’s Union(S1, S2) “copy” 1 element Union(S2, S3) “copy” 2 elements … Union(Sn-1, Sn) “copy” n-1 elements O(n2) Improvement: always copy smaller into larger How long would above sequence of Union’s take? Worst case: n Union’s take O(n lg n) time Proof uses amortized analysis David Luebke 22 7/19/2019

Amortized Analysis of Disjoint Sets If elements are copied from the smaller set into the larger set, an element can be copied at most lg n times Worst case: Each time copied, element in smaller set 1st time resulting set size  2 2nd time  4 … (lg n)th time  n David Luebke 23 7/19/2019

Amortized Analysis of Disjoint Sets Since we have n elements each copied at most lg n times, n Union()’s takes O(n lg n) time Therefore we say the amortized cost of a Union() operation is O(lg n) This is the aggregate method of amortized analysis: n operations take time T(n) Average cost of an operation = T(n)/n David Luebke 24 7/19/2019

Amortized Analysis: Accounting Method Charge each operation an amortized cost Amount not used stored in “bank” Later operations can used stored money Balance must not go negative Book also discusses potential method But we won’t worry about it here David Luebke 25 7/19/2019

The End David Luebke 26 7/19/2019