[PDF] Lecture 16: Shortest Paths III - Dijkstra and Special Cases

DIJKSTRA Demo Dijkstra's Algorithm For each edge (u, v) ϵ E, assume w(u, v) ≥ 0, maintain a set S of vertices whose final shortest path weights have been

Dijkstra's Algorithm Readings CLRS, Sections 24 2-24 3 Review d[v] is the length of the current shortest path from starting vertex s Through a process of

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Algorithm terminates when some vertex w has been processed, i e , deleted from the queue of both searches, Qf and Qb Subtlety: After search terminates, find

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5 avr 2011 · Single source shortest path problem • Problem: Given a digraph G = (V, E) with non- negative edge-weight function w, and a node s, find

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We achieve a O(E + V lg V ) bound on Dijkstra's algorithm using Fibonacci heaps All-pairs shortest paths • given edge-weighted graph, G = (V, E,w)

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Two algorithms: Dijkstra O(V lg V + E) assumes non-negative edge weights Bellman Ford O(V E) is a general algorithm Application • Find shortest path from

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4 nov 2011 · Normally we'd be thinking Dijkstra; we have nonnegative edge weights and we only want a single-source shortest path Dijkstra's algorithm, as

[PDF] Lecture 16: Shortest Paths III - Dijkstra and Special Cases - courses

DIJKSTRA Demo Dijkstra's Algorithm For each edge (u, v) ϵ E, assume w(u, v) ≥ 0, maintain a set S of vertices whose final shortest path weights have been

[PDF] Reach for A : an Efficient Point-to-Point Shortest Path Algorithm

Bidirectional Dijkstra's algorithm • A ∗ search • ALT Algorithm • Definition of reach • Reach-based algorithm • Reach for A ∗ • Demo Reach for A∗ MIT

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Two algorithms: Dijkstra O(V lg V + E) assumes non-negative edge weights Bellman Ford O(V E) is a Find shortest path from CalTech to MIT – See “ CalTech

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analyze the performance of Dijkstra's algorithm using various data structures (i e set 18 math mit edu/~rothvoss/18 304 3PM/Presentations/1-Melissa pdf

Lecture 16: Shortest Paths III - Dijkstra and

Special Cases

Lecture Overview•Shortest paths in DAGs

•Shortest paths in graphs without negative edges •Dijkstra"s Algorithm

ReadingsCLRS, Sections 24.2-24.3

DAGs:

Can"t have negative cycles because there are no cycles!1.Topologically sort the DAG. Path fromutovimplies thatuis beforevin the linear

ordering2.One pass over vehicles in topologically sorted order relaxing each edge that leaves each vertex

Θ(V+E) time

Example:rstxyz

∞0∞∞∞∞ 35
27-1
6 4 1 -2

2Figure 1:Shortest Path using Topological Sort.Vertices sorted left to right in topological order

Processr: stays∞. All vertices to the left ofswill be∞by definitionProcesss:t:∞ →2x:∞ →6 (see top of Figure 2)1

Lecture 16 Shortest Paths III: Dijkstra 6.006 Fall 2009rstxyz ∞026∞∞ 35
27-1
6 4 1 -2 2 rstxyz ∞02653 35
27-1
6 4 1 -2 2 process t, x, yFigure 2:Preview of Dynamic ProgrammingDIJKSTRA Demo

Dijkstra"s Algorithm

For each edge (u,v)? E, assumew(u,v)≥0, maintain a setSof vertices whose final

shortest path weights have been determined. Repeatedly selectu ? V-Swith minimumshortest path estimate, addutoS, relax all edges out ofu.

Pseudo-codeDijkstra (G,W,s) //uses priority queue Q

Initialize (G,s)

S←φ

Q←V[G] //Insert intoQ

whileQ?=φ dou←EXTRACT-MIN(Q) //deletesufromQ

S=S? {u}

for each vertexv ?Adj[u] do RELAX (u,v,w)←this is an implicit DECREASEKEY operation 2 Lecture 16 Shortest Paths III: Dijkstra 6.006 Fall 2009B C A D E 519
11 715
4 13

A C E B D

7 12 18 22 D B E C A 4 13 15 22 E C A D B 5 12 13 16 Figure 3:Dijkstra Demonstration with Balls and String.Recall

RELAX(u,v,w)

ifd[v]> d[u] +w(u,v) thend[v]←d[u] +w(u,v)

Π[v]←u

ExampleStrategy: Dijkstra is a greedy algorithm: choose closest vertex inV-Sto add to setSCorrectness: Each time a vertexuis added to setS, we haved[u] =δ(s,u)3

Lecture 16 Shortest Paths III: Dijkstra 6.006 Fall 2009

Complexity

θ(v) inserts into priority queue

θ(v) EXTRACTMIN operations

θ(E) DECREASEKEY operations

Array impl:

θ(v) time for extra min

θ(1) for decrease key

Total:θ(V.V+E.1) =θ(V2+E) =θ(V2)Binary min-heap:

θ(lgV) for extract min

θ(lgV) for decrease key

Total:θ(VlgV+ElgV)Fibonacci heap (not covered in 6.006):

θ(lgV) for extract min

θ(1) for decrease key

amortized cost

Total:θ(VlgV+E)4

Lecture 16 Shortest Paths III: Dijkstra 6.006 Fall 2009B C A0 D E 2 2 10 1 38
49
7 S = { } { A B C D E } = Q

S = { A } 0

S = { A, C }

0 10 3 after relaxing

edges from AS = { A, C } 0 7 3 11 5 after relaxing edges from C

S = { A, C, E }

0 7 3 11 5

S = { A, C , E, B}

0 7 3 9 5 after relaxing edges from B Figure 4:Dijkstra Execution5quotesdbs_dbs17.pdfusesText_23

[PDF] [PDF] Lecture 16: Shortest Paths III - Dijkstra and Special Cases - courses

Lecture 16: Shortest Paths III - Dijkstra and

Special Cases

Lecture Overview•Shortest paths in DAGs

ReadingsCLRS, Sections 24.2-24.3

Θ(V+E) time

Example:rstxyz

2Figure 1:Shortest Path using Topological Sort.Vertices sorted left to right in topological order

Dijkstra"s Algorithm

Initialize (G,s)

S←φ

Q←V[G] //Insert intoQ

S=S? {u}

A C E B D

7 12 18 22 D B E C A 4 13 15 22 E C A D B 5 12 13 16 Figure 3:Dijkstra Demonstration with Balls and String.Recall

RELAX(u,v,w)

Π[v]←u

Complexity

θ(v) inserts into priority queue

θ(v) EXTRACTMIN operations

θ(E) DECREASEKEY operations

Array impl:

θ(v) time for extra min

θ(1) for decrease key

θ(lgV) for extract min

θ(lgV) for decrease key

θ(lgV) for extract min

θ(1) for decrease key

Total:θ(VlgV+E)4

S = { A } 0

S = { A, C }

S = { A, C, E }

S = { A, C , E, B}