CMPS 6610/4610 – Fall 2016 Minimum Spanning Trees Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk CMPS 6610/4610 – Fall 2016 1
Minimum spanning trees Input: A connected, undirected graph G = ( V , E ) with weight function w : E R . • For simplicity, assume that all edge weights are distinct. Output: A spanning tree T — a tree that connects all vertices — of minimum weight: w ( T ) w ( u , v ) . ( u , v ) T CMPS 6610/4610 – Fall 2016 2
Example of MST 6 12 9 5 7 14 15 8 10 3 CMPS 6610/4610 – Fall 2016 3
Growing an MST Grow an MST by greedily adding one edge at a time. G ENERIC - M ST (G,w){ T while T does not form a spanning tree { // Maintain invariant that T is a subset of an MST for G Find a “safe” edge {u,v} such that T {{u,v}} is a subset of an MST for G T T {{u,v}} } return A } CMPS 6610/4610 – Fall 2016 4
Hallmark for “greedy” algorithms Greedy-choice property A locally optimal choice is globally optimal. Theorem [Cut property]. Let G = ( V , E ) and let A V . Suppose that { u , v} E is the least-weight edge connecting A to V \ A . Then, { u , v } is contained in an MST T of G . CMPS 6610/4610 – Fall 2016 5
Proof of theorem Proof. Suppose { u , v } T . Cut and paste. T : v u A { u , v } = least-weight V \ A edge connecting A to V \ A CMPS 6610/4610 – Fall 2016 6
Proof of theorem Proof. Suppose { u , v } T . Cut and paste. T : v u A { u , v } = least-weight V \ A edge connecting A to V \ A Consider the unique simple path from u to v in T . CMPS 6610/4610 – Fall 2016 7
Proof of theorem Proof. Suppose { u , v} T . Cut and paste. T : v u A { u , v} = least-weight edge V \ A connecting A to V \ A Consider the unique simple path from u to v in T . Swap { u , v} with the first edge on this path that connects a vertex in A to a vertex in V \ A . CMPS 6610/4610 – Fall 2016 8
Proof of theorem Proof. Suppose { u , v} T . Cut and paste. T : v u A { u , v} = least-weight edge V \ A connecting A to V \ A Consider the unique simple path from u to v in T . Swap { u , v} with the first edge on this path that connects a vertex in A to a vertex in V \ A . A lighter-weight spanning tree than T results. CMPS 6610/4610 – Fall 2016 9
MST algorithms • Prim’s algorithm: • Maintains one tree • Runs in time O (| E| log | V| ) with binary heaps, in time O (| E| + | V | log | V| ), with Fibonacci heaps • Kruskal’s algorithm: • Maintains a forest and uses the disjoint-set data structure • Runs in time O (| E| log | E| ) CMPS 6610/4610 – Fall 2016 10
Prim’s algorithm I DEA : Maintain V \ A as a priority queue Q . Key each vertex in Q with the weight of the least- weight edge connecting it to a vertex in A . Q V Dijkstra: while Q do key [ v ] for all v V u E XTRACT -M IN ( Q ) key [ s ] 0 for some arbitrary s V S S { u } while Q for each v Adj [ u ] do do u E XTRACT -M IN ( Q ) if d [ v ] > d [ u ] + w ( u , v ) then for each v Adj [ u ] d [ v ] d [ u ] + w ( u , v ) do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) D ECREASE -K EY [ v ] u At the end, {( v , [ v ])} forms the MST edges. CMPS 6610/4610 – Fall 2016 11
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 12
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 13
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 14
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 15
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 16
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 17
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 18
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 19
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 20
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 21
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 22
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 23
Example of Prim’s algorithm A 6 12 V \ A 9 5 7 14 15 8 0 10 3 u E XTRACT -M IN ( Q ) for each v Adj [ u ] do if v Q and w ( u , v ) < key [ v ] then key [ v ] w ( u , v ) ⊳ D ECREASE -K EY [ v ] u CMPS 6610/4610 – Fall 2016 24
Analysis of Prim Q V (| V| ) key [ v ] for all v V total key [ s ] 0 for some arbitrary s V while Q do u E XTRACT -M IN ( Q ) for each v Adj [ u ] | V | do if v Q and w ( u , v ) < key [ v ] degree ( u ) times times then key [ v ] w ( u , v ) [ v ] u Handshaking Lemma (| E| ) implicit D ECREASE -K EY ’s. Time = (| V| )· T E XTRACT- M IN + (| E| )· T D ECREASE- K EY CMPS 6610/4610 – Fall 2016 25
Analysis of Prim (continued) Time = (| V| )· T E XTRACT- M IN + (| E| )· T D ECREASE- K EY Q T E XTRACT- M IN T D ECREASE- K EY Total O (| V| 2 ) array O (| V| ) O (1) binary O (log | V| ) O (log | V| ) O (| E| log | V| ) heap Fibonacci O (log | V| ) O (1) O (| E| + | V| log | V| ) heap amortized amortized worst case CMPS 6610/4610 – Fall 2016 26
Kruskal’s algorithm I DEA (again greedy) : Repeatedly pick edge with smallest weight as long as it does not form a cycle. • The algorithm creates a set of trees (a forest ) • During the algorithm the added edges merge the trees together, such that in the end only one tree remains CMPS 6610/4610 – Fall 2016 27
Example of Kruskal’s algorithm S={ {a},{b},{c},{d},{e} MST edges a {f},{g},{h} } 6 12 a set repr. 9 5 b c d 7 14 15 8 e f g 10 3 h Every node is a single tree. CMPS 6610/4610 – Fall 2016 28
Example of Kruskal’s algorithm S={ {a},{b},{c},{d},{f} MST edges a {g}, {e, h} } 6 12 a set repr. 9 5 b c d 7 14 15 8 e f g 10 3 h Edge 3 merged two singleton trees. CMPS 6610/4610 – Fall 2016 29
Example of Kruskal’s algorithm S={ {a},{d},{f}, {g} MST edges a {e, h}, {b, c} } 6 12 a set repr. 9 5 b c d 7 14 15 8 e f g 10 3 h CMPS 6610/4610 – Fall 2016 30
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