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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 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)

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

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|V|-1 edges vertex of degree 1 Suppose all degrees 2. Then |E|=(1/2) deg(v) |V| Contradiction. Done. vVvV

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

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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)

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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.

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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.

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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 |G 1 |,...,|G k | be the connected components. Then |E_i| = |V_i| - 1, hence |E| = |V| - k. Thus k = 1.

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Spanning trees

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How many spanning trees ?

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Spanning trees How many spanning trees ? 4

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Spanning trees How many spanning trees ?

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Spanning trees How many spanning trees ? 8

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

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1 5 3 6 4 2 4 2 7 3 6

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1 5 3 6 4 2 4 2 7 3 6

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1 5 3 6 4 2 4 2 7 3 6

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1 5 3 6 4 2 4 2 7 3 6

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1 5 3 6 4 2 4 2 7 3 6

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Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle

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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 ?

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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

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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)

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

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

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

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Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle Need to maintain components. Find-Set(u) Union(u,v)

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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]

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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)

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Kruskal’s algorithm Find-Set(u) Union(u,v) O(1) O(n) Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle

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

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

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

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

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Union-Find 2 S[1..n] for i 1 to n do S[i] i 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]++ n=|V| O(log n)

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

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

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Kruskal’s algorithm log E log V 2 = 2 log V = O(log V) Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle O(E log E + (E+V)log V) O((E+V) log V)

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

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1 5 3 6 4 2 4 2 7 3 6

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1 5 3 6 4 2 4 2 7 3 6

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1 5 3 6 4 2 4 2 7 3 6

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1 5 3 6 4 2 4 2 7 3 6

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1 5 3 6 4 2 4 2 7 3 6

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Prim’s algorithm 1) S {1} 2) add the cheapest edge {u,v} such that u S and v S C S S {v} (until S=V)

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

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Minimum weight spanning trees P = MST output by Prim T = optimal MST P = T assuming all the edgeweights different v 1,v 2,...,v n order added to S by Prim smallest i such that an edge e E connecting S={v 1,...,v i } to S C different in T than in Prim (f)

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Minimum weight spanning trees smallest i such that an edge e E connecting S={v 1,...,v i } to S C different in T than in Prim (f) e SCSC S T

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Minimum weight spanning trees smallest i such that an edge e E connecting S={v 1,...,v i } to S C different in T than in Prim (f) e f SCSC S T+f

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Minimum weight spanning trees for i 1 to n do C[i] C[0]=0 S={} while S V do j smallest such that j S C and C[j] is minimal S S { j } for u neighbors of j do if w[{j,u}] < C[u] then C[u] w[{j,u}]; P[u] j

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Minimum weight spanning trees for i 1 to n do C[i] C[0]=0 S={} while S V do j smallest such that j S C and C[j] is minimal S S { j } for u neighbors of j do if w[{j,u}] < C[u] then C[u] w[{j,u}]; P[u] j O(n)

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Minimum weight spanning trees for i 1 to n do C[i] C[0]=0 S={} while S V do j smallest such that j S C and C[j] is minimal S S { j } for u neighbors of j do if w[{j,u}] < C[u] then C[u] w[{j,u}]; P[u] j O(n) O(V 2 + E) = O(V 2 )

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Minimum weight spanning trees for i 1 to n do C[i] C[0]=0 S={} while S V do j smallest such that j S C and C[j] is minimal S S { j } for u neighbors of j do if w[{j,u}] < C[u] then C[u] w[{j,u}]; P[u] j O(log n) O((E+V)log V) using heap

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Minimum weight spanning trees Kruskal O( (E+V) log V) Prim O( (E+V) log V)

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Minimum weight spanning trees Kruskal O( (E+V) log V) Prim O( (E+V) log V) can be made O(E + V log V) using Fibonacci heaps if edges already sorted then O(E log * V)

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