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6.006 Introduction to Algorithms
Spring 2008
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Lecture 18
Shortest Paths III: Dijkstra
6.006 Spring 2008
Lecture 18: Shortest Paths IV - Speeding up
Dijkstra
Lecture Overview
• Single-source single-target Dijkstra
• Bidirectional search
• Goal directed search - potentials and landmarks
Readings
Wagner, Dorothea, and Thomas Willhalm. "Speed-Up Techniques for Shortest-Path
Computations." In Lecture Notes in Computer Science: Proceedings of the 24th Annual
Symposium on Theoretical Aspects of Computer Science. Berlin/Heidelberg, MA: Springer,
2007. ISBN: 9783540709176. Read up to section 3.2.
DIJKSTRA single-source, single-target
Initialize()
Q ← V [G]
while Q �= φ
do u ← EXTRACT MIN(Q) (stop if u = t!)
for each vertex v � Adj[u]
do RELAX(u, v, w)
Observation: If only shortest path from s to t is required, stop when t is removed from
Q, i.e., when u = t
DIJKSTRA Demo
4
B
19
15
11
A
13
7
5
C
A C E B D
7 12 18 22
D
D B E C A
4 13 15 22
E
E C A D B
5 12 13 16
Figure 1: Dijkstra Demonstration with Balls and String
1
Lecture 18
Shortest Paths III: Dijkstra
6.006 Spring 2008
Bi-Directional Search
Note: Speedup techniques covered here do not change worst-case behavior, but reduce the
number of visited vertices in practice.
t
S
backward search
forward search
Figure 2: Bi-directional Search
Bi-D Search
Alternate forward search from s
backward search from t
(follow edges backward)
df (u) distances for forward search
db (u) distances for backward search
Algorithm terminates when some vertex w has been processed, i.e., deleted from the queue
of both searches, Qf and Qb
u
u’
3
3
3
s
t
5
5
w
Figure 3: Bi-D Search
2
Lecture 18
Shortest Paths III: Dijkstra
6.006 Spring 2008
Subtlety: After search terminates, find node x with minimum value of df (x) + db (x). x may
not be the vertex w that caused termination as in example to the left!
Find shortest path from s to x using Πf and shortest path backwards from t to x using Πb .
Note: x will have been deleted from either Qf or Qb or both.
u
Forward
3
df (s) = 0
df (u’) = 6
5
w
u
Backward
3
s
t
5
t
df (t) = 10
df (s) = 0
db (s) = 10
df (w) = 5
db(t) = 0
5
db (w) = 5
w
3
df (u) = 3
s
df (s) = 0
u’
3
5
df (w) = 5
3
u’
3
db(u) = 6
s
5
u
3
3
t
db(u’) = 3
u
Backward
3
5
t db(t) = 0
5
w db (w) = 5
u’
3
w
Forward
5
df (w) = 5
df (u) = 3
s
s
df (u’) = 6
u
3
db(u’) = 3 3
5
w
u’
3
3
t
5
df (s) = 0
u
Backward
3
df (u) = 3
s
Forward
u’
3
u’
3
3
df (u) = 3
db(u) = 6
5
db(u’) = 3
df (u’) = 6
t
5
w
db(t) = 0
df (t) = 10
db (w) = 5
df (w) = 5
deleted from both queues
so terminate!
Figure 4: Forward and Backward Search
Minimum value for df (x) + db (x) over all vertices that have been processed in at least one
search
df (u) + db (u) = 3 + 6 = 9
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Lecture 18
Shortest Paths III: Dijkstra
6.006 Spring 2008
df (u� ) + db (u� ) = 6 + 3 = 9
df (w) + db (w) = 5 + 5 = 10
Goal-Directed Search or A∗
Modify edge weights with potential function over vertices.
w (u, v) = w (u, v) − λ(u) + λ(v)
Search toward target:
v’
v
increase
go uphill
5
5
decrease
go downhill
Figure 5: Targeted Search
Correctness
w(p) = w(p) − λt (s) + λt (t)
So shortest paths are maintained in modified graph with w weights.
s
p
t
p’
Figure 6: Modifying Edge Weights
To apply Dijkstra, we need w(u, v) ≥ 0 for all (u, v).
Choose potential function appropriately, to be feasible.
Landmarks
(l)
Small set of landmarks LCV . For all u�V, l�L, pre-compute δ(u, l).
Potential λt (u) =
δ(u, l) = δ(t, l) for each l.
(l)
CLAIM: λt is feasible.
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Lecture 18
Shortest Paths III: Dijkstra
6.006 Spring 2008
Feasibility
(l)
(l)
w(u, v) = w(u, v) − λt (u) + λt (v)
= w(u, v) − δ(u, l) + δ(t, l) + δ(v, l) − δ(t, l)
= w(u, v) − δ(u, l) + δ(v, l) ≥ 0
λt (u) =
(l)
max λt (u)
l�L
is also feasible
5
by the Δ -inequality