Dijkstra Algorithm

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Dijkstra's Algorithm

Sources: S. Skiena. The Algorithm Design Manual. S. Sedgewick. Algorithms in C++ (3rd Edition)

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Problem

Two algorithms, depending on what information we want Recall: All pairs shortest path: Floyd's algorithm Single source shortest path: Dijkstra's algorithm

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Dijkstra's algorithm solves related problem to APSP (Floyd's algorithm): Single source shortest path (SSSP) Idea: For a single node, what is shortest path to any

other node? More restricted question Can answer somewhat more quickly

Dijkstra's algorithm: Greedy approach

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Basic idea: Divide vertices into three groups Tree vertices: We know the shortest path to these Fringe vertices: Adjacent to a tree vertex; we don't

know the shortest path to these yet Distant vertices: Not adjacent to a tree vertex; we

don't consider them at all Iterate n times:

Find fringe vertex with lowest distance from source Move from fringe to tree Update any vertices that are now fringe vertices

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Example

Assume we start at A

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5 Tree Fringe Distant

Vertex Dist NeighA* 0 -B 7 AC - -D 3 AE - -F - -G - -H - -I - -J - -

Vertex Dist NeighA* 0 -B 7 AC - -D 3 AE - -F - -G - -H - -I - -J - -

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Example

Add closest

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2 7

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Vertex Dist NeighA* 0 -B 7 AC - -D* 3 AE 4 DF - -G 10 DH - -I - -J - -

Vertex Dist NeighA* 0 -B 7 AC - -D* 3 AE 4 DF - -G 10 DH - -I - -J - -

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Example

Add closest

A

B

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2 7

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Vertex Dist NeighA* 0 -B 7 AC - -D* 3 AE* 4 DF 8 EG 10 DH - -I - -J - -

Vertex Dist NeighA* 0 -B 7 AC - -D* 3 AE* 4 DF 8 EG 10 DH - -I - -J - -

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Example

Add closest

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2 7

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Vertex Dist NeighA* 0 -B* 7 AC - -D* 3 AE* 4 DF 8 EG 10 DH - -I - -J - -

Vertex Dist NeighA* 0 -B* 7 AC - -D* 3 AE* 4 DF 8 EG 10 DH - -I - -J - -

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Example

Add closest

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2 7

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Vertex Dist NeighA* 0 -B* 7 AC - -D* 3 AE* 4 DF* 8 EG 9 FH - -I - -J 14 F

Vertex Dist NeighA* 0 -B* 7 AC - -D* 3 AE* 4 DF* 8 EG 9 FH - -I - -J 14 F

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Example

Add closest

A

B

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2 7

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Vertex Dist NeighA* 0 -B* 7 AC - -D* 3 AE* 4 DF* 8 EG* 9 FH - -I - -J 14 F

Vertex Dist NeighA* 0 -B* 7 AC - -D* 3 AE* 4 DF* 8 EG* 9 FH - -I - -J 14 F

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Example

Add closest

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B

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2 7

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Vertex Dist NeighA* 0 -B* 7 AC 19 JD* 3 AE* 4 DF* 8 EG* 9 FH 22 JI 23 JJ* 14 F

Vertex Dist NeighA* 0 -B* 7 AC 19 JD* 3 AE* 4 DF* 8 EG* 9 FH 22 JI 23 JJ* 14 F

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Example

Add closest

A

B

D

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F

J

H

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2 7

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Vertex Dist NeighA* 0 -B* 7 AC* 19 JD* 3 AE* 4 DF* 8 EG* 9 FH 22 JI 23 JJ* 14 F

Vertex Dist NeighA* 0 -B* 7 AC* 19 JD* 3 AE* 4 DF* 8 EG* 9 FH 22 JI 23 JJ* 14 F

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Example

Add closest

A

B

D

C

E

F

J

H

I

G

4

2 7

6

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3

4

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1

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Vertex Dist NeighA* 0 -B* 7 AC* 19 JD* 3 AE* 4 DF* 8 EG* 9 FH* 22 JI 23 JJ* 14 F

Vertex Dist NeighA* 0 -B* 7 AC* 19 JD* 3 AE* 4 DF* 8 EG* 9 FH* 22 JI 23 JJ* 14 F

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Example

Add closest

A

B

D

C

E

F

J

H

I

G

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2 7

6

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3

4

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13

1

41

5

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Vertex Dist NeighA* 0 -B* 7 AC* 19 JD* 3 AE* 4 DF* 8 EG* 9 FH* 22 JI* 23 JJ* 14 F

Vertex Dist NeighA* 0 -B* 7 AC* 19 JD* 3 AE* 4 DF* 8 EG* 9 FH* 22 JI* 23 JJ* 14 F

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Implementation

Question: How to implement? Need to be able to find fringe vertices quickly Also need to be able to find least valued fringe

vertex Solution: A priority queue, implemented with a

heap Order = distance from source vertex

Need to specify graph structure...

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Graph Structure

class Vertex data label or similar index integer: Index in G.vertices qindex integer: Index in priority queue neighbors list

Elements are (float,int) pairs First = weight of edge, second = index of neighbor

nearestv integer Used for SSSP

nearestw float Used for SSSP

class Graph vertices list of Vertex objects

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Example

Example for this small graph: G: Graph object

vertices = [ , , , ] : Vertex object

data = A, index=0 neighbors = [(7,1),(3,3)]

: Vertex object data = B, index=1 neighbors=[ (2,0) ,(4,3) ]

: Vertex object data = C, index=2 neighbors=[ (1,3) ]

: Vertex object data = D, index=3 neighbors = [ ]

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def dijkstra(G,src): #G = graphQ=[ ]for v in G.vertices:

v.nearestv = Noneif v.name == src: v.nearestw=0else: v.nearestw=NoneQ.append(tmp)

heapify(Q) #None counts as infinityfor i in range(len(Q)):

Q[i].qindex = i...continued...

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while len(Q) > 0:x=heappop(Q) #updates qindexfor nn in x.neighbors:

ew = nn[0] #edge weightni = nn[1] #neighbor indexn = G.vertices[ ni ] #neighbor itselfif n.nearestw == None or n.nearestw > x.nearestw + ew:

n.nearestv = x.indexn.nearestw = x.nearestw + ewreheapify(Q,n)

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Reheapify

Remember, priority queue is sorted based on distance from source vertex

If we change a vertex's nearest neighbor, we might move it up in the queue

So this is what reheapify does Parameters: The queue (array) and the index

node we have altered

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Reheapify

def reheapify(Q,v):vi = v.qindex #index in queuepi = (vi-1)/2 #parent's index in queuep = Q[pi] #parent node itselfwhile vi > 0 and p.nearestw > v.nearestw:

#swap queue entries for parent & vtmp = Q[pi]Q[pi] = Q[vi]Q[vi] = tmp#update qslot dataQ[pi].qslot = piQ[vi].qslot = vivi = pipi = (vi-1)/2

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heappop

def heappop(Q):x = Q[0]; y = Q.pop(); Q[0]=y; Q[0].qindex=0vi = 0; li = vi*2+1 ; ri = vi*2+2 #li,ri=left/right childrenwhile 1:

if li < len(Q): v1=Q[li].nearestwelse: v1 = infinityif ri < len(Q): v2=Q[ri].nearestwelse: v2 = infinityif v1 < Q[vi].value and v1

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Analysis

How much time for Dijkstra? Does V loop iterations Each loop iteration takes time lg(V) (to pop and

reheapify) Thus, O(V lg V) for vertex processing Also, within loop, we consider all neighbors of

popped vertex Each edge gets considered exactly once

So O(V lg V + E) time total Dense graph: EV2 Thus, O(V2 + V lg V)

Dijkstra Algorithm

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