1 dijkstra’s algorithm dr. ying lu [email protected] raik 283 data structures & algorithms
TRANSCRIPT
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RAIK 283Data Structures & Algorithms
Giving credit where credit is due:» Most of slides for this lecture are based on slides
created by Dr. John Vergara at Ateneo De Manila University
» I have modified them and added new slides
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Single-Source Shortest-Paths Problem Problem: given a connected graph G with non-negative weights
on the edges and a source vertex s in G, determine the shortest paths from s to all other vertices in G.
Useful in many applications (e.g., road map applications)
1410
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6 4
5
2
9
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Single-Source Shortest-Paths
For instance, the following edges form the shortest paths from node A
H B C
G E D
F
A
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3
6 4
5
2
9
15
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Single-Source Shortest-Paths
H B C
G E D
F
A
1410
3
6 4
5
2
9
15
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• Hint: does this problem looks similar?• Can we modify Prim’s algorithm to solve this problem?
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Dijkstra’s Algorithm Solves the single-source shortest paths problem Involves keeping a table of current shortest path lengths from
source vertex (initialize to infinity for all vertices except s, which has length 0)
Repeatedly select the vertex u with shortest path length, and update other lengths by considering the path that passes through that vertex
Stop when all vertices have been selected Could use a priority queue to facilitate selection of shortest lengths
» Here, priority is defined differently from that of the Prim’s algorithm» Like Prim’s algorithm, it needs to refine data structure so that the update of
key-values in priority queue is allowed» Like Prim’s algorithm, time complexity is O( (n + m) log n )
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Dijkstra’s algorithm
ORD
BOS
PVD
JFK
BWI
MIA
DFWLAX
SFO
•
•
•
0
•
•
•
•
•
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
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•
•
•
Dijkstra’s algorithm
ORD
BOS
PVD
JFK
BWI
MIA
DFWLAX
SFO
•
•
184
0
946
621
•
•
•
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
9
Dijkstra’s algorithm
ORD
BOS
PVD
JFK
BWI
MIA
DFWLAX
SFO
•
•
184
0
946
621
•
•
•
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
JFK
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Dijkstra’s algorithm
ORD
BOS
PVD
JFK
BWI
MIA
DFWLAX
SFO
371
328
184
0
946
621
1575
•
•
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
PVD
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•
Dijkstra’s algorithm
ORD
BOS
PVD
JFK
BWI
MIA
DFWLAX
SFO
371
328
184
0
946
621
1575
•
3075
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704BOS
12
3075
1575
Dijkstra’s algorithm
ORD
BOS
PVD
JFK
BWI
MIA
DFWLAX
SFO
371
328
184
0
946
621
1423
•
2467
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
ORD
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•
Dijkstra’s algorithm (cont)
ORD
BOS
PVD
JFK
BWI
MIA
DFWLAX
SFO
371
328
184
0
946
621
1423
3288
2467
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
MIA
14
3288
Dijkstra’s algorithm (cont)
ORD
BOS
PVD
JFK
MIA
DFWLAX
SFO
371
328
184
0
946
621
1423
2658
2467
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
BWI
DFW
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Dijkstra’s algorithm (cont)
LAX
SFO
371
328
184
0
946
621
1423
2658
2467
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
ORD
BOS
PVD
JFK
BWI
MIA
DFW
SFO
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Dijkstra’s algorithm (cont)
LAX
SFO
371
328
184
0
946
621
1423
2658
2467
849
867
187144
1841258
1090
946
740621
1391
802
1121
2342
1235
1464
337
1846
2704
ORD
BOS
PVD
JFK
BWI
MIA
DFWLAX
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In-Class Exercises
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In-Class Exercises
Design a linear-time, (i.e., ( n + m)) algorithm for solving the single-source shortest-paths problem for dags (directed acyclic graphs) represented by their adjacency lists.
Hint: topologically sort the dag’s vertices first.
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In-Class Exercise
Suppose a person is making a travel plan driving from city 1 to city n, n > 1, following a route that will go through cities 2 through n –1 in between. The person knows the mileages between adjacent cities, and knows how many miles a full tank of gasoline can travel. Based on this information, the problem is to minimize the number of stops for filling up the gas tank, assuming there is exactly one gas station in each of the cities. Design a greedy algorithm to solve this problem and analyze its time complexity.
fill up the gas tank in City 1for c = 2 to n –1 do// decide whether to stop in City c for gas while approaching(2.1) if there is still enough gas to go to City (c+1) then don’t stop in City c (2.2) else
fill up the gas tank in City c
It can be proved that this greedy algorithm minimizes the number of gas stops. The time complexity is O(n) because the loop in Step (2) runs O(n) iterations in which each iteration uses O(1) time (in Steps (2.1) and (2.2)).
Proofs Techniques for Greedy Algorithms