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Tree

Tree. tree = connected graph with no cycle tree = connected graph with |V|-1 edges tree = graph with |V|-1 edges and no cycles. Tree. tree = connected graph with no cycle tree = connected graph with |V|-1 edges tree = graph with |V|-1 edges and no cycles.

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Tree

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  1. Tree tree = connected graph with no cycle tree = connected graph with |V|-1 edges tree = graph with |V|-1 edges and no cycles

  2. Tree tree = connected graph with no cycle tree = connected graph with |V|-1 edges tree = graph with |V|-1 edges and no cycles no cycle  vertex of degree  1 |V|-1 edges  vertex of degree  1 connected  no vertex of degree 0 (unless |V|=1)

  3. no cycle  vertex of degree  1 Supose not, i.e., all vertices of degree  2, yet no cycle. Let v1,...,vt be the longest path vt has 2-neighbors, one different from vt-1. Why cannot take v1,...,vt,u ? Cycle. Contradiction.

  4. |V|-1 edges  vertex of degree  1 Suppose all degrees  2. Then |E|=(1/2)  deg(v)  |V| Contradiction. Done. vV

  5. connected  no vertex of degree 0 (unless |V|=1)

  6. Tree connected graph with no cycle connected graph with |V|-1 edges graph with |V|-1 edges and no cycles  no cycle  vertex of degree  1 |V|-1 edges  vertex of degree  1 connected  no vertex of degree 0 (unless |V|=1)

  7. connected graph with no cycle connected graph with |V|-1 edges   Induction on |V|. Base case |V|=1. Let G be connected with no cycles. Then G has vertex v of degree 1. Let G’ = G – v. Then G’ is connected and has no cycle. By IH G’ has |V|-2 edges and hence G has |V|-1 edges.

  8. connected graph with no cycle connected graph with |V|-1 edges   Induction on |V|. Base case |V|=1. Let G be connected |V|-1 edges. Then G has vertex v of degree 1. Let G’ = G – v. Then G’ is connected and has |V’|-1 edges. By IH G’ has no cycle. Hence G has no cycle.

  9. connected graph with no cycle connected graph with |V|-1 edges graph with |V|-1 edges and no cycles  Assume |V|-1 edges and no cycles. Let |G1|,...,|Gk| be the connected components. Then |E_i| = |V_i| - 1, hence |E| = |V| - k. Thus k = 1.

  10. Spanning trees

  11. Spanning trees

  12. Spanning trees

  13. Spanning trees How many spanning trees ?

  14. Spanning trees How many spanning trees ? 4

  15. Spanning trees How many spanning trees ?

  16. Spanning trees How many spanning trees ? 8

  17. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  18. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  19. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  20. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  21. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  22. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  23. Minimum weight spanning trees • Kruskal’s algorithm • sort the edges by weight • add edges in order • if they do not create cycle

  24. Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle K = MST output by Kruskal T = optimal MST Is K = T ?

  25. Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle K = MST output by Kruskal T = optimal MST Is K = T ? no – e.g. if all edge-weights are the same  many optima

  26. Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle K = MST output by Kruskal T = optimal MST Assume all edge weights are different. Then K=T. (In particular, unique optimum)

  27. Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle K = MST output by Kruskal T = optimal MST G connected, e in a cycle  G-e connected

  28. Kruskal’s algorithm K = MST output by Kruskal T = optimal MST G connected, e in a cycle  G-e connected e the edge of the smallest weight in K-T. Consider T+e.

  29. Kruskal’s algorithm e the edge of the smallest weight in K-T. Consider T+e. Case 1: all edgeweights in C smaller that we Case 2: one edgeweight in C larger that we

  30. Kruskal’s algorithm • sort the edges by weight • add edges in order • if they do not create cycle Need to maintain components. Find-Set(u) Union(u,v)

  31. Union-Find S[1..n] for i  1 to n do S[i] i Find-Set(u) return S[u] Union(u,v) a  S[u] for i 1 to n do if S[i]=a then S[i]  S[v]

  32. Union-Find S[1..n] for i  1 to n do S[i] i Find-Set(u) return S[u] Union(u,v) a  S[u] for i 1 to n do if S[i]=a then S[i]  S[v] O(1) O(n)

  33. Kruskal’s algorithm O(1) Find-Set(u) Union(u,v) O(n) • Kruskal’s algorithm • sort the edges by weight • add edges in order • if they do not create cycle

  34. Kruskal’s algorithm O(1) Find-Set(u) Union(u,v) O(n) O(E log E + V^2) • Kruskal’s algorithm • sort the edges by weight • add edges in order • if they do not create cycle

  35. Union-Find 2 S[1..n] for i  1 to n do S[i] i n=|V| Find-Set(u) while (S[u] u) do u S[u] S[u] Union(u,v) S[Find-Set(u)] Find-Set(v) v u

  36. Union-Find 2 S[1..n] for i  1 to n do S[i] i n=|V| Find-Set(u) while (S[u] u) do u S[u] S[u] Union(u,v) S[Find-Set(u)] Find-Set(v) v u

  37. Union-Find 2 S[1..n] for i  1 to n do S[i] i n=|V| Find-Set(u) while (S[u] u) do u S[u] S[u] Union(u,v) S[Find-Set(u)] Find-Set(v) O(n) O(n)

  38. Union-Find 2 S[1..n] for i  1 to n do S[i] i n=|V| Find-Set(u) while (S[u] u) do u S[u] S[u] Union(u,v)u Find-Set(u); v Find-Set(v); if D[u]<D[v] then S[u] v if D[u]>D[v] then S[v] u if D[u]=D[v] then S[u]  v; D[v]++ O(log n) O(log n)

  39. Kruskal’s algorithm O(log n) Find-Set(u) Union(u,v) O(log n) • Kruskal’s algorithm • sort the edges by weight • add edges in order • if they do not create cycle

  40. Kruskal’s algorithm O(log n) Find-Set(u) Union(u,v) O(log n) O(E log E + (E+V)log V) • Kruskal’s algorithm • sort the edges by weight • add edges in order • if they do not create cycle

  41. Kruskal’s algorithm log E  log V2 = 2 log V = O(log V) O((E+V) log V) O(E log E + (E+V)log V) • Kruskal’s algorithm • sort the edges by weight • add edges in order • if they do not create cycle

  42. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  43. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  44. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  45. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  46. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  47. Minimum weight spanning trees 4 3 2 3 5 7 1 4 6 6 2

  48. Minimum weight spanning trees Prim’s algorithm 1) S  {1} 2) add the cheapest edge {u,v} such that uS and vSC S S{v} (until S=V)

  49. Minimum weight spanning trees 1) S  {1} 2) add the cheapest edge {u,v} such that uS and vSC S S{v} (until S=V) P = MST output by Prim T = optimal MST Is P = T ? assume all the edgeweights different

  50. Minimum weight spanning trees P = MST output by Prim T = optimal MST P = T assuming all the edgeweights different v1,v2,...,vn order added to S by Prim smallest i such that an edge e E connecting S={v1,...,vi} to SC different in T than in Prim (f)

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