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9/14/2007
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayn e
Sofya Raskhodnikova
Algorithm Design and Analysis
LECTURE 8
Greedy Graph Algorithms
•Shortest Paths
•Dijkstra’s
•Minimum Spanning Tree
•Kruskal’s, Prim’s
•Reverse-Delete
9/14/2007
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayn e
Review Question
•Express v∈Vdegree(v) in terms of m, the number of
edges in the (undirected) graph.
(
Handshaking Lemma:
v∈Vdegree(v) = 2|E|
for undirected graphs
)
•When a priority queue is impemented using an unsorted
array, how long do Extract-Min and Decrease-Key take?
(Answer: Extract-Min – O(n);
Decrease-Key – O(1))
9/14/2007
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayn e
Shortest Path Problem
•Input:
– Directed graph G = (V, E).
– Source node s, destination node t.
– for each edge e, length
(e)
= length of e.
•Find: shortest directed path from s to t.
s
3
t
2
6
7
4
5
23
18
2
9
14
15
5
30
20
44
16
11
6
19
6
9/14/2007
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayn e
Dijkstra's Algorithm (Greedy)
•Maintain a set of explored nodes S whose shortest path
distance d(u) from s to u is known.
•Initialize S = { s}, d(s) = 0.
•Repeatedly choose unexplored node v which minimizes
•add v to S, and set d(v) = π(v).
,)()(min)(
:),(
eudv
Suvue
+
=
∈=
π
! "# $
$
%
9/14/2007
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayn e
Dijkstra's Algorithm (Greedy)
•Maintain a set of explored nodes S whose shortest path
distance d(u) from s to u is known.
•Initialize S = { s}, d(s) = 0.
•Repeatedly choose unexplored node v which minimizes
•add v to S, and set d(v) = π(v).
! "# $
$
%
,)()(min)(
:),(
eudv
Suvue
+
=
∈=
π
9/14/2007
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayn e
Proof of Correctness
Invariant. For each node u ∈S, d(u) is the length of
the shortest path from s to u.
Proof: (by induction on |S|)
•Base case: |S| = 1 is trivial.
•Inductive hypothesis: Assume for |S| = k ≥1.
– Let v be next node added to S, and let (u,v) be the chosen edge.
– The shortest s-u path plus (u,v) is an s-v path of length π(v).
– Consider any s-v path P. We'll see that it's no shorter than π(v).
– Let x-y be the first edge in P that leaves S,
and let P' be the subpath to x.
– P + (x,y) has length d(x)+ (x,y)π(y)π(v)
%
#
$
&
&'
