Minimum Spanning Trees and Kruskal’s Algorithm

CLRS 23

Minimum Spanning Trees (MST)

• A common problem in communications networks and circuit design: connecting a set of nodes by a network of minimum total length

• Represent the problem as an undirected graph where edge weights denote wire lengths

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Formally,

• Given a connected undirected graph G=(V,E), a spanning tree is an acyclic subset of edges T that connects all the vertices together.

• Assuming each edge (u,v) of G has a numeric weight (or cost) w(u,v), the cost of a spanning tree T is the sum of the weights of the edges in T.

• A minimum spanning tree (MST) is a spanning tree of minimum weight.

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Minimum Spanning Trees (MST)

• Never pays to have cycles• Resulting connection graph is connected, undirected and

acyclic, hence a free tree• Not unique

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Total weight = 37

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Total weight = 37

Generic MST Algorithm

• Two main algorithms for computing MSTs– Kruskal’s

– Prim’s

• Both greedy algorithms

• The greedy strategy captured by a generic aproach:– Grow the MST one edge at a time

– which edge to select?

Generic MST Algorithms

• Maintain a set of edges : A• Loop invariant: A is a subset of some MST• Safe edge (u,v): A U {(u,v)} is also a subset of an MST

Generic-MST(G, w)

A = empty set

while A does not form a spanning tree do

find an edge (u,v) that is safe for A

A = A U {(u,v)}

return A

How to recognize safe edges?

Definitions first:• Cut: Let S be a subset of vertices V. A cut (S, V-S) is a

partition of vertices into two disjoint sets S and V-S.

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S

V-S

• We say that an edge crosses the cut (S, V-S) if one of its endpoints is in S and the other is in (V-S)

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V-S

• We say that a cut respects a set A of edges if no edge in A crosses the cut

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• An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.

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How to recognize safe edges?

Theorem: Let G=(V,E) be a connected undirected graph with a real valued weight function w defined on E. Let A be a subset of E that is included in some MST for G. Let (S, V-S) be any cut of G that respects A and let (u,v) be a light edge crossing (S, V-S). Then, edge (u,v) is safe for A.

Proof

• Suppose that no MST contains (u,v). • Let T be an MST of G.

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T

• Add (u,v) to T, creating a cycle.• There must be at least one more edge on this cycle that crosses the

cut: (x,y)

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• The edge (x,y) is not in A, since the cut respects A• Remove (x,y) and restore the spanning tree T’• w(T’) = w(T) + w(u,v) – w(x,y) < w(T)

– Conradicts the assumption that T is an MST.

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Kruskal’s Algorithm

• Add edges to A in increasing order of weights.– If the edge being considered introduces a cycle, skip it.

– Otherwise add to A.

• Notice that the edges in A make a forest, and they keep getting merged together.

Correctness

• Say (u, v) is the edge is going to be added next, and it does not introduce a cycle in A.

• Let A’ denote the tree of A that contains vertex u. Consider the cut (A’, V-A’).

• Every edge crossing the cut is not in A, so this cut respects A and (u,v) is the light edge crossing it.

• Thus, (u,v) is safe..

How to efficiently detect if adding (u,v) creates a cycle in A

• Can be done by a data structure called Disjoint Set Union-Find

• Supports three operations:– CreateSet(u): create a set containing a single item u

– FindSet(u): return the set that contains item u

– Union(u,v): merge the set containing u with the set containing v

• Suffices to know: takes O(n log n + m) time to carry out any sequence of m union-find operations on a set of size n.

Kruskal(G=(V,E), w) {

A = empty set

for each (u in V)

CreateSet(u) //create a set for each vertex

Sort E in increasing order by weight

for each ((u,v) from the sorted list E) {

if (FindSet(u) != FindSet(v)) {

// u and v are in different trees

Add(u,v) to A

Union(u,v)

}

}

return A

}

Θ(E log E) for sorting the edges

O(V log V + E) for a sequence of E union find operations

Total : O(E log E) since (E >= V-1)

Prim’s Algorithm

• Edges in A always form a tree (partial MST)• Start from a vertex r and grow until the tree spans all vertices• Let S denote the set of vertices which are on this partial MST

– A is the set of edges connecting the vertices in S

• At each stage the light edge crossing (S, V-S) is added to A

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Prim’s Algorithm

Prim(G=(V,E),w,r) {for each (u in V) {

key[u] = infinitypred[u] = NIL

}key[r] = 0;PQ = {V} //add all vertices to Priority Queue PQ based on keyswhile (PQ is not empty) do {

u = PQ.Extract-Min()for each (v in Adj[u])

if (v is in PQ and w(u,v) < key[v]) then {pred[v] = ukey[v]=w(u,v)PQ.DecreaseKey(v, key[v]) //reorganizes the

PQ}

}}