Post on 05-Jan-2016
description
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
no cycle vertex of degree 1|V|-1 edges vertex of degree 1connected no vertex of degree 0 (unless |V|=1)
no cycle vertex of degree 1
Supose not, i.e., all vertices of degree 2, yet no cycle.
Let v1,...,vt be the longest pathvt has 2-neighbors, one differentfrom vt-1. Why cannot takev1,...,vt,u ? Cycle. Contradiction.
|V|-1 edges vertex of degree 1
Suppose all degrees 2. Then
|E|=(1/2) deg(v) |V|
Contradiction. Done.vV
connected no vertex of degree 0(unless |V|=1)
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 1connected no vertex of degree 0 (unless |V|=1)
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 connectedand has no cycle. By IH G’ has |V|-2 edges and hence G has |V|-1 edges.
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 connectedand has |V’|-1 edges. By IH G’ has no cycle. Hence G has no cycle.
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.
Spanning trees
Spanning trees
Spanning trees
Spanning trees
How many spanning trees ?
Spanning trees
How many spanning trees ?
4
Spanning trees
How many spanning trees ?
Spanning trees
How many spanning trees ?
8
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
Kruskal’s algorithm
1) sort the edges by weight2) add edges in order if they do not create cycle
Kruskal’s algorithm1) sort the edges by weight2) add edges in order if they do not create cycle
K = MST output by KruskalT = optimal MST
Is K = T ?
Kruskal’s algorithm1) sort the edges by weight2) add edges in order if they do not create cycle
K = MST output by KruskalT = optimal MST
Is K = T ?
no – e.g. if all edge-weights are the same many optima
Kruskal’s algorithm1) sort the edges by weight2) add edges in order if they do not create cycle
K = MST output by KruskalT = optimal MST
Assume all edge weightsare different. Then K=T. (In particular, unique optimum)
Kruskal’s algorithm1) sort the edges by weight2) add edges in order if they do not create cycle
G connected, e in a cycle G-e connected
K = MST output by KruskalT = optimal MST
Kruskal’s algorithm
e the edge of the smallest weightin K-T. Consider T+e.
G connected, e in a cycle G-e connected
K = MST output by KruskalT = optimal MST
Kruskal’s algorithm
e the edge of the smallest weightin K-T. Consider T+e.
Case 1: all edgeweights in C smaller that we
Case 2: one edgeweight in C larger that we
Kruskal’s algorithm
1) sort the edges by weight2) add edges in order if they do not create cycle
Need to maintain components. Find-Set(u) Union(u,v)
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]
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)
Kruskal’s algorithm Find-Set(u) Union(u,v)
O(1)
O(n)
Kruskal’s algorithm
1) sort the edges by weight2) add edges in order if they do not create cycle
Kruskal’s algorithm Find-Set(u) Union(u,v)
O(1)
O(n)
Kruskal’s algorithm
1) sort the edges by weight2) add edges in order if they do not create cycle
O(E log E + V^2)
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|
uv
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|
uv
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)
O(n)
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)
O(log n)
Kruskal’s algorithm Find-Set(u) Union(u,v)
O(log n)
O(log n)
Kruskal’s algorithm
1) sort the edges by weight2) add edges in order if they do not create cycle
Kruskal’s algorithm Find-Set(u) Union(u,v)
O(log n)
O(log n)
Kruskal’s algorithm
1) sort the edges by weight2) add edges in order if they do not create cycle
O(E log E + (E+V)log V)
Kruskal’s algorithmlog E log V2 = 2 log V = O(log V)
Kruskal’s algorithm
1) sort the edges by weight2) add edges in order if they do not create cycle
O(E log E + (E+V)log V)
O((E+V) log V)
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
1
53
6
4
2
4
27
3
6
Minimum weight spanning trees
Prim’s algorithm 1) S {1}2) add the cheapest edge {u,v} such that uS and vSC
S S{v} (until S=V)
Minimum weight spanning trees
1) S {1}2) add the cheapest edge {u,v} such that uS and vSC
S S{v} (until S=V)
P = MST output by PrimT = optimal MST
Is P = T ?
assume all the edgeweights different
Minimum weight spanning treesP = MST output by PrimT = 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)
Minimum weight spanning trees
smallest i such that an edge e E connecting S={v1,...,vi} to SC different in T than in Prim (f)
e
SC
ST
Minimum weight spanning trees
smallest i such that an edge e E connecting S={v1,...,vi} to SC different in T than in Prim (f)
e
f
SC
ST+f
Minimum weight spanning trees
for i 1 to n do C[i] C[0]=0
S={}while S V do j smallest such that jSC 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
Minimum weight spanning trees
for i 1 to n do C[i] C[0]=0
S={}while S V do j smallest such that jSC 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)
Minimum weight spanning trees
for i 1 to n do C[i] C[0]=0
S={}while S V do j smallest such that jSC 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(V2 + E) = O(V2)
Minimum weight spanning trees
for i 1 to n do C[i] C[0]=0
S={}while S V do j smallest such that jSC 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
Minimum weight spanning trees
Kruskal O( (E+V) log V)
Prim O( (E+V) log V)
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)