OR 215
Network Flows
Spring 1998
M. Hartmann
MINIMUM COST SPANNING TREE PROBLEM II
Optimality Conditions
Prim’s Algorithm
Sensitivity Analysis
OPTIMALITY CONDITIONS
Theorem. (Cut optimality theorem) A spanning tree T*
is a minimum spanning tree if and only if it satisfies the
following cut optimality condition:
 For every tree arc (i,j)  T*, cij  ckl for every arc
(k,l) contained in the cut formed by deleting arc (i,j)
from T*.
Theorem. (Path optimality theorem) A spanning tree T*
is a minimum spanning tree if and only if it satisfies the
following path optimality condition:
 For every non-tree arc (k,l) of G, cij  ckl for every
arc (i,j) contained in the path of T* connecting
nodes k and l.
Recall: The path optimality condition leads to Kruskal’s
greedy algorithm.
Theorem. Let F be a subset of arcs of some minimum
cost spanning tree. Let P be the set of nodes of some
component of F. Let (i,j) be a minimum cost arc in the cut
(P,N\P). Then F+(i,j) is a subset of arcs of some minimum
cost spanning tree.
Proof: Suppose that F is a subset of the minimum cost
tree T*. If (i,j)  T*, there is nothing to prove. So suppose
that (i,j)  T*. Adding (i,j) to T* creates a cycle C, and C
has at least on arc (k,l)  (i,j) in (P,N\P). (Why?) By
assumption, cij  ckl. Also, T* satisfies the cut optimality
conditions, so cij  ckl. It follows that cij = ckl, and that
T*+(i,j)\ (k,l) is also a minimum cost spanning tree.
P
i
k
N\P
l
j
 How can we take advantage of this property?
PRIM’S ALGORITHM
Prim's algorithm grows a spanning subtree rooted at node
1, which is a subset of arcs of some minimum cost spanning tree.
i
1
P
j
N\P
To identify the arc (i,j), Prim’s algorithm maintains labels
d(j) for each node j not yet added to the spanning tree:
d(j) = minimum cost of an arc (i,j) with i  P
Prim’s algorithm also keeps track of which node i  P has
cij = d(j) by setting pred(j) := i.
algorithm PRIM;
begin
P := {1} ; T := N \ {1}; F := ;
d(1) := 0 and pred(1) := ;
d(j) := c1j and pred(j) := 1 for all (1,j)  A ,
and d(j) :=  otherwise;
while P  N do
begin
{ node selection, also called FINDMIN }
let i  T be a node with d(i) = min { d(j) : j  T };
P := P  { i }; T := T \ { i }; add (i,pred(i)) to F;
{ cost update }
for each (i,j)  A(i) do
if d(j) > cij then d(j) := cij and pred(j) := i;
end;
end
 Where have we seen this algorithm before?
algorithm DIJKSTRA;
begin
P := {1} ; T := N \ {1};
d(1) := 0 and pred(1) := ;
d(j) := c1j and pred(j) := 1 for all (1,j)  A ,
and d(j) :=  otherwise;
while P  N do
begin
{ node selection, also called FINDMIN }
select a node i  T with d(i) = min { d(j) : j  T };
P := P  { i }; T := T \ { i };
{ distance update }
for each (i,j)  A(i) do
if d(j) > d(i) + cij then
d(j) := d(i) + cij and pred(j) := i ;
end;
end
RUNNING TIME
Naïve Implementation:
Prim’s algorithm can be implemented using heaps.
Recall that the standard heap implementation has the
following running times per operation:
FIND-MIN:
DECREASE-KEY:
DELETE-MIN:
INSERT:
O(1)
O(log n)
O(log n)
O(log n)
Heap Implementation:
Note: Prim’s algorithm can also be implemented using
Fibonacci heaps in O(m+n log n) time.
EXAMPLE
The set F of arcs is "grown" starting from node 1:
10
2
35
4
20
25
1
40
d( )
30
15
3
5
1
2
3
4
5
0
35
40



10
4
2
35
20
25
1
40
d( )
30
15
3
5
1
2
3
4
5
0
35
25
10


10
2
35
4
20
25
1
40
d( )
30
15
3
5
1
2
3
4
5
0
35
20
10
30

10
2
35
4
20
25
1
15
40
d( )
30
5
3
1
2
3
4
5
0
35
20
10
15

10
2
35
4
20
25
1
15
40
d( )
30
3
5
1
2
3
4
5
0
35
20
10
15
SENSITIVITY ANALYSIS
Suppose that T* is a minimum cost spanning tree. For any
arc (i,j)  A, the cost interval of (i,j) is the range of values
of cij for which T* continues to be a minimum cost
spanning tree.
 How can we find the cost interval of an arc (i,j)  T*?
35
2
10
4
20
25
1
40
3
30
15
5
 How can we find the cost interval of an arc (i,j)  T*?
10
2
4
COMPREHENSIVE
SENSITIVITY
ANALYSIS
35
20
25
30
1
 How much time does it take to find the cost intervals for
every arc40
(i,j)  T*? 15
3
5
35
10
2
4
20
1
15
3
5
max-cost arc in path from i to j
1
2
3
4
5
1
2
3
4
5
In the homework, you will show that all of these values can
be computed in O(n2) time.
 How much time does it take to find the cost intervals for
every arc (i,j)  T*?
2
4
25
1
40
30
3
5
2
4
30
1
40
3
5
SUMMARY
1. The minimum cost spanning tree (MST) problem is an
important problem in its own right.
2. The MST problem is also an important sub-problem.
3. The greedy algorithm works.
4. Other very efficient algorithms.
5. Sensitivity analysis can be performed efficiently.