Shortest Pathsand

Dijkstra's Algorithm

CS 110: Data Structures and Algorithms

First Semester, 2010-2011

Single-Source Shortest Paths

►Given a weighted graph G and a source vertex v in G, determine the shortest paths from v to all other vertices in G►Path length: sum of all edges in path

►Useful in road map applications (e.g., Google maps or map quest) for example

Dijkstra’s Algorithm►Solves the single-source shortest

path problem►Involves keeping a table of current

shortest path lengths from a source vertex (initialize to infinity for all vertices except v, which has length 0)

Dijkstra’s Algorithm►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

Dijkstra’s Algorithm►Can make use of a priority queue to

facilitate selection of shortest lengths►Need to refine the data structure to

allow the updating of key values►Time complexity: O(n+m) or O(n2 log

n)►O(n2 ) if computation of minimum is

simplified

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

∞

∞

∞

∞∞

∞

∞

∞

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

∞

621

∞

∞946

∞

184

∞

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

∞

621

∞

∞946

∞

184

∞

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

∞

∞946

371

184

1575

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

∞

∞946

371

184

1575

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

∞

∞946

371

184

1575

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

∞

∞946

371

184

1575

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

3075

∞946

371

184

1575

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

3075

∞946

371

184

1575

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

2467

∞946

371

184

1423

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

2467

∞946

371

184

1423

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

2467

3288

946

371

184

1423

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

2467

3288

946

371

184

1423

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

2467

3288

946

371

184

1423

0

Dijkstra’s Algorithm

MIA

JFK

PVD

BOS

DFW

SFO

LAX

BWI

ORD

2704

1846

337

1235

1464

802621

2342

1391

1121

144

849

740

867

187

946

184

1090

1258

328

621

2467

2658

946

371

184

1423

0

Dijkstra’s Algorithm

Pseudo-Code: Dijkstrafunction Dijkstra( Graph g, Vertex source )

for each vertex v in gdist[v] <-- infinityprevious[v] <-- undefined

dist[source] <-- 0Q <-- the set of all nodes in Graphwhile Q is not empty

u <-- vertex in Q with smallest dist[]if dist[u] == infinity

breakremove u from Qfor each neighbor v of u

alt = dist[u] + dist_between( u, v )if alt < dist[v]

dist[v] = altprevious[v] = u

return dist[]