Tree tree connected graph with no cycle tree
- Slides: 58
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 1 connected 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 v 1, . . . , vt be the longest path vt has 2 -neighbors, one different from vt-1. Why cannot take v 1, . . . , vt, u ? Cycle. Contradiction.
|V|-1 edges vertex of degree 1 Suppose all degrees 2. Then |E|=(1/2) deg(v) |V| v V Contradiction. Done.
connected no vertex of degree 0 (unless |V|=1)
Tree connected graph with no cycle connected 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)
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.
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.
connected graph with no cycle connected graph with |V|-1 edges and no cycles Assume |V|-1 edges and no cycles. Let |G 1|, . . . , |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 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle
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 ?
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
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)
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
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.
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
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)
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] O(1) Union(u, v) a S[u] for i 1 to n do if S[i]=a then S[i] S[v] O(n)
Kruskal’s algorithm Find-Set(u) O(1) Union(u, v) O(n) Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle
Kruskal’s algorithm Find-Set(u) O(1) Union(u, v) O(n) O(E log E + V^2) Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle
Union-Find 2 n=|V| S[1. . n] for i 1 to n do S[i] i Find-Set(u) while (S[u] u) do u S[u] Union(u, v) S[Find-Set(u)] Find-Set(v) v u
Union-Find 2 n=|V| S[1. . n] for i 1 to n do S[i] i Find-Set(u) while (S[u] u) do u S[u] Union(u, v) S[Find-Set(u)] Find-Set(v) v u
Union-Find 2 n=|V| S[1. . n] for i 1 to n do S[i] i Find-Set(u) while (S[u] u) do u S[u] Union(u, v) S[Find-Set(u)] Find-Set(v) O(n)
Union-Find 2 n=|V| S[1. . n] for i 1 to n do S[i] i Find-Set(u) while (S[u] u) do u S[u] O(log n) Union(u, v) O(log n) 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]++
Kruskal’s algorithm Find-Set(u) O(log n) 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
Kruskal’s algorithm Find-Set(u) O(log n) Union(u, v) O(log n) O(E log E + (E+V)log V) Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle
Kruskal’s algorithm log E log V 2 = 2 log V = O(log V) O((E+V) log V) O(E log E + (E+V)log V) Kruskal’s algorithm 1) sort the edges by weight 2) add edges in order if they do not create cycle
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
Minimum weight spanning trees 4 3 2 3 5 7 1 6 6 4 2
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)
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
Minimum weight spanning trees P = MST output by Prim T = optimal MST P=T assuming all the edgeweights different v 1, v 2, . . . , vn order added to S by Prim smallest i such that an edge e E connecting S={v 1, . . . , vi} to SC different in T than in Prim (f)
Minimum weight spanning trees e S T SC smallest i such that an edge e E connecting S={v 1, . . . , vi} to SC different in T than in Prim (f)
Minimum weight spanning trees e S T+f SC f smallest i such that an edge e E connecting S={v 1, . . . , vi} to SC different in T than in Prim (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 j SC 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 O(n) S={} while S V do j smallest such that j SC 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 O(n) S={} while S V do j smallest such that j SC 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(V 2 + E) = O(V 2)
Minimum weight spanning trees for i 1 to n do C[i] C[0]=0 O(log n) S={} while S V do j smallest such that j SC 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((E+V)log V) using heap
Minimum weight spanning trees Kruskal Prim O( (E+V) log V)
Minimum weight spanning trees Kruskal O( (E+V) log V) if edges already sorted then O(E log*V) Prim O( (E+V) log V) can be made O(E + V log V) using Fibonacci heaps
- A tree is a connected graph without any circuits
- Balanced wye-wye connection
- In a triangle connected source feeding a y connected load
- In a ∆-connected source feeding a y-connected load
- Number of strongly connected components
- Strongly connected graph
- Connected acyclic graph
- Semiconnected graph
- Directed multigraph
- Euler tour
- Algebra 1 table of contents
- Euler's formula for planar graphs
- A tree is a connected?
- Handshaking theorem
- Wait-for graph
- Chrono cycle graph is used for
- Pmat
- Cardiac cycle graph
- Cell cycle graph
- Important cyclins in cell cycle
- Secondary motive
- Brainpop carbon cycle
- Denitrification definition
- Difference between phosphorus cycle and carbon cycle
- Difference between open cycle and closed cycle gas turbine
- Reservation table
- Chapter 5 principles of engine operation
- Grille évaluation handball cycle 3
- Mhd power generation ppt
- Cycle de consolidation
- Cycle 3 cycle de consolidation
- Water cycle the hydrologic cycle
- Graph tree
- Minimum spanning tree weighted graph
- Non-deterministic algorithm
- Query tree and query graph
- Spanning tree of a graph
- Query tree and query graph
- Tree graph
- Connected strategy worksheet template
- Myconnectedsite
- A balanced delta connected load having an impedance 20-j15
- All oceans are connected
- Generally restful like a horizontal the sky meets land
- Connected particles a level maths
- Specialization and interdependence
- Precis french
- 5 physical features of canada
- Simply connected domain
- Narrative account definition
- Contoh model connected di paud
- Mesh connected illiac network
- How is manorialism connected to feudalism
- Lumen connected security
- Living by chemistry
- Backpropagation cnn
- Connected vehicle insights
- Connected nation michigan
- Topological sort can be implemented by?