Minimum Spanning TreesProblem DefinitionKruskal’s algorithmDisjoint-set Data StructureCycle DetectionUpdate Disjoint-set Data StructureTime for Disjoint-set Data Structure OperationsOverall Running TimePrim’s algorithmPriority QueueUpdating Priority QueueSlide 12Running time AnalysisProofs of Correctness (Kruskal)Slide 15Slide 16Proofs of Correctness (Prim)Minimum Spanning Trees•Definition•Algorithms–Prim–Kruskal•Proofs of correctnessProblem Definition•Input–Weighted, connected undirected graph G=(V,E)•Weight (length) function w on each edge e in E•Task–Compute a spanning tree of G of minimum total weight•Spanning tree–If there are n nodes in G, a spanning tree consists of n-1 edges such that no cycles are formedKruskal’s algorithm•Greedy approach to edge selection–Select the minimum weight edge that does not form a cycle–Implementation•Sort edges by increasing lengths–Alternatively use heap to store edges–Either way: O(E log E) time•Include minimum weight edge if it does not form a cycle.•How do we test if edge forms a cycle?•Use disjoint-set data structure (Chapter 21)–Each vertex belongs to a set containing the vertices in its current tree (initially only itself)–Find-set(A) = Find-set(B) = Find-set(E) = A–Find-set(C) = Find-set(D) = C–Find-set(F) = F–Find-set(G) = GDisjoint-set Data StructureA B CDE F G1223455610•Take minimum weight edge and test both endpoints to see if they are in same set•Min weight edge: (A,E). Same Set. Cycle.•Min weight edge: (B,C): Different Sets. No cycle.Cycle DetectionA B CDE F G1223455610•Merge sets corresponding to node B and node C.–We need the result to now be:–Find-set(A) = Find-set(B) = Find-set(E) = Find-set(C) = Find-set(D) = A (or C)–Find-set(F) = F–Find-set(G) = GUpdate Disjoint-set Data StructureA B CDE F G1223455610Time for Disjoint-set Data Structure Operations•Initialization:–Makeset(v) for all vertices v: O(V) time•Merges and Find-set Operations–O(V) merges and O(E) Find-sets–Each can be implemented in amortized (V) time where  is a very slow growing function–O(E (V)) time overall which is essentially O(E) for all practical purposesOverall Running Time•Initialization:–Makeset(v) for all vertices v: O(V) time•Merges and Find-set Operations–O(E) of them–Each can be implemented in amortized (V) time where  is a very slow growing function–O(E (V)) time overall for cycle detection•O(E log E) time for sorting edges by weight dominatesPrim’s algorithm•Different greedy approach to edge selection–Initialize connected component N to be any node v–Select the minimum weight edge connecting N to V-N–Implementation•Maintain a priority queue for the nodes in V-N based on how close they are to any node in N•When a new node v is added to N, we need to update the weight of the neighbors of v in V-N•Maintain priority queue of nodes in V-N•If we started with node D, N is now {C,D}•Priority Queue values of other nodes:–A, E, F: infinity–B: 4–G: 6Priority QueueA B CDE F G1223455610•Node B is added to N; edge (B,C) is added to T•Need to update priority queue values of A, E, F–Decrease-key operation•Priority Queue values of other nodes:–A: 1–E: 2–F: 5–G: 6Updating Priority QueueA B CDE F G1223455610•Node A is added to N; edge (A,B) is added to T•Need to update priority queue values of E–Decrease-key operation•Priority Queue values of other nodes:–E: 2 (unchanged because 2 is smaller than 3)–F: 5–G: 6Updating Priority QueueA B CDE F G1223455610Running time Analysis•Assume binary heap implementation•Build initial heap takes O(v) time•V extract-min operations for O(V log V) time•For each edge, potentially 1 decrease-key operation, so O(E log V) time•Overall: O(E log V) time which is asymptotically equivalent to our implementation of Kruskal’s algorithm•Use of fibonacci heap can improve running time to O(E + V log V) time–Decrease-key drops to O(1) amortized timeProofs of Correctness (Kruskal)•Let T’ be a minimum spanning tree for G•Let T be the tree formed by Kruskal’s algorithm that utilizes edges in T’ whenever a tie needs to be broken•Assumption: T has more weight than T’–Otherwise Kruskal’s algorithm has produced an optimal treeProofs of Correctness (Kruskal)•Let e be the smallest weight edge in T that is not in T’•Add e to T’ and consider the cycle Y that is formed•There must be some edge e’ on Y that has weight greater than e. •Explain why?–What do we know about T and T’ up to this point?Proofs of Correctness (Kruskal)•Replace e’ by e in T’•We have a new spanning tree T’’ such that the weight of T’’ is smaller than the weight of T’•This contradicts the fact that T’ was optimal.•This implies no such edge e can be found, and thus T must be optimal.Proofs of Correctness (Prim)•Let T’ be a minimum spanning tree for G•Let T be the tree formed by Prim’s algorithm that utilizes edges in T’ whenever a tie needs to be broken•Assumption: T has more weight than T’–Otherwise Prim’s algorithm has produced an optimal tree•Let e be the first edge added to T that is not in T’•Finish this