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.
w(T ) 
 w(u, v)
( u ,v )T
• A minimum spanning tree (MST) is a spanning tree of minimum weight.
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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• 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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• 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 UnionFind
• 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)
}
Θ(E log E) for sorting the edges
}
O(V log V + E) for a sequence of E
return A
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
S
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Prim’s Algorithm
Prim(G=(V,E),w,r) {
for each (u in V) {
key[u] = infinity
pred[u] = NIL
}
key[r] = 0;
PQ = {V} //add all vertices to Priority Queue PQ based on keys
while (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] = u
key[v]=w(u,v)
PQ.DecreaseKey(v, key[v]) //reorganizes the PQ
}
}
}