[PDF] a generalization of Dijkstras shortest path algorithm PDF dijkstra.pdf

The shortest path problem on weighted directed graphs is one of the basic complexity of Dijkstra's algorithm and shows a linear behavior in the case of

Using Dijkstra's algorithm to calculate the single-source shortest path problem will always involve n delete-min operations, giving a total time complexity of O(m + n log n)

A Theorem on the Expected Complexity of Dijkstras Shortest Path

assigned to nodes in Dijkstra's algorithm for the single-source shortest path problem is Let S*(u) denote the shortest distance from s to node u in V For the

[PDF] Efficiency Evaluation of Shortest Path Algorithms - ThinkMind

In more complex executions of the shortest paths, sets of paths with the shortest distance 3) Dijkstra's algorithm using a heap: It is not possible to decrease the

[PDF] All-Pairs Shortest Paths Dijkstras Single Source Algorithm - UF CISE

Use Dijkstra's algorithm n times, once with each of the n Time complexity is Theta(n3) time • Works so on the shortest path and no intermediate vertex is

[PDF] a generalization of Dijkstras shortest path algorithm

The shortest path problem on weighted directed graphs is one of the basic complexity of Dijkstra's algorithm and shows a linear behavior in the case of

[PDF] 1 Dijkstras Algorithm for SSSP

Then, if V \ X = , there must exist a vertex u such that δ(s,u) = d and a shortest path to u that only goes through vertices in X Proof Let Y = {v ∈ V : δ(s, v) = d } be all

[PDF] Average-case analysis of shortest-paths algorithms in the vertex

Abstract We study the average-case complexity of shortest-paths problems in N oshita [23] analyzed the average-case complexity of D ij kstra 's algorithm ; [8 ] E W Dijkstra, A note on two problems in connexion with graphs, Numer Math

[PDF] Greedy Algorithms: Dijkstras Shortest Path Algorithm

CSC 373 - Algorithm Design, Analysis, and Complexity Summer 2016 This is precisely the idea behind Dijkstra's algorithm Example: Consider the graph

Lalla Mouatadid

Greedy Algorithms: Dijkstra's Shortest Path AlgorithmLetG(V;E;w) be an edge weighted graph, wherew:E!R+. Lets;tbe two vertices inG(think ofsas a

source,tas a terminal), and suppose you were asked to compute a shortest (i.e. cheapest) path betweens

andt. Notice thatGcould possibly have more than one shortest path betweensandt. Consider the graph below for instance: BothP=s;a;tandQ=s;b;tare cheapest paths fromstot.sa bt11 112

One way to solve this problem is to computeallstpaths inG, and choose the cheapest. Notice that if the

graph is unweighted (think of this as all the edges having equal weights), then starting BFS fromswould

solve this problem. Maybe we can adjust BFS to take into account the weights.

Recall in BFS vertices are pushed into a queue when visited. When visiting the neighbours of a vertexu,

the ordering in which the neighbours enter the queue is arbitrary. If now the goal is to compute the cheapest

path, then one way to modify BFS would be to push the cheapest neighbours rst. By cheapest, we mean with shortest distance.

\Modied BFS": Consider using a priority queue instead of a queue to collect unvisited vertices. Set the

priority to be the shortest distance so far. This is precisely the idea behindDijkstra's algorithm. Example: Consider the graph below for instance. Suppose we want to compute the cheapest path from s=Atot=F.AB CD EF G4 3 765
11 822
10 53

The table below keeps track of the distances computed at every iteration from the sourceAto the every

vertex in the graph. Read the table as follows: In the st iteration, the distance from the source to itself is

0, and1to any other vertex. At every iteration, choose the cheapest available vertex and try to build the

next cheapest path from said vertex. Repeat the process until all vertices have been visited. 1 2

ABCDEFGS

ii= 00111111; i= 1: A0431711;

i= 2: C04314711fACgi= 3: B0439711fAC;ABgi= 4: E04397112fAC;AB;AEgi= 5: D043971112fAC;AB;AE;BDgi= 6: F043971112fAC;AB;AE;BD;DFgi= 7: G043971112fAC;AB;AE;BD;DF;EGgNotice form the table above that at every iterationi, the setSiis the set of edges that could potentially

lead to a shortest path. Ife= (u;v) is added toSi, withu2Si;v =2Si, then the current value ofd[v] is a

shortestsvpath. Formally the algorithm is as follows:Algorithm 1Dijkstra's Shortest Path AlgorithmInput:An edge weighted connected graphG(V;E;w) wherew:E!R+and two verticess;t.

Output:A path fromstotwith minimum total cost (shortest path)

1:S=;.

2:Initialize empty priority queue.

3:foreachv2Vdo

4:p[v] = NIL.predecessor ofvn shortests;vpath so far

5:d[v] =1.priotity ofv= min distances;vso far.

6:enqueue(v).With priorityd[v] =1.

7:end for

8:d[s] = 0

9:Update queue order ofs

10:whilequeue is not emptydo.Main Loop

11:v= dequeue element with min priorityd[]

12:ifp[v]! = NILthen

13:S=S[ f(p[v];v)g

14:end if

15:foreachedge (u;v)do

16:ifuis in the queue andd[v] +w(v;u)< d[u]then

17:p[u] =v

18:d[u] =d[v] +w(v;u)

19:Update queue order ofu

20:end if

21:end for

22:end while

23:returnSProof of Correctness:

We will argue on theSi's. In particular, at every iteration, we generate subsets of edgesS1;S2;:::;Sn. We

say thatSican be extended to some collection of shortest pathsSi(this is just a tree of shortest paths)

using only edges that donothave both endpoints inSi. That is, we only add edges toSiwith at least one

endpoint in the queue.

Loop Invariant

S iis promising,and(1)

8u2Si;8v =2Si:d[u] =dist[s;v]dist[s;v]d[v] (2)

3 Wheredist[s;u] is the minimum cost ofallpaths fromstou.

Proof.By induction oni.

Base Case:S0=;is trivially promising.

Induction Hypothesis: For somei, supposeSican be extended to some shortest paths treeSi, using only edges without both endpoints inSi, and that: d[u] =dist[s;u]dist[s;v]d[v];for allu2Si;v =2Si Induction Step: ConsiderSi+1=Si[ f(u;v)gwithu2Si,voutsideSi. We have 2 possible cases: 1. ( u;v)2Si: If (u;v)2SithenSiextendsSi+1anddist[s;v] =dist[s;u]+w(u;v) since (u;v)2Si. Thus we have: d[v] =d[u] +w(u;v) =dist[s;u] +w(u;v) =dist[s;v] Moreover, sinced[v] was the cheapest of alld[] values for vertices outsideSi, it follows that8x2Si+1 and8y =2Si+1:d[x] =d[s;x]d[s;y]d[y]. 2. ( u;v)=2Si: Then consider the pathPinSifromstov.Pis optimal. Let (z;v) be the last edge on this path. Ifzwere outsideSi, then let (x;y) be the rst edge onPwithx2Si;y =2Si(why does such an edge exist?). We haved[y]d[x] +w(x;y) (becaused[y] is the smallest value ofd[t] +w(t;y);8e= (t;y);t2Si.) We thus have: d[y]d[x] +w(x;y) =dist[s;x] +w(x;y) < dist[s;x] +w(x;y) +dist[y;v] (Sincew:E!R+.) =dist[s;v] d[v] )d[y]< d[v] But this contradicts the fact thatd[v] is the smallestd[] value for all vertices outsideSi. Thereforezmust be inSi. This implies thatdist[s;v] =dist[s;z] +w(z;v) =d[z] +w(z;v). Sinced[v] is the minimum ofalld[x] +w(x;v)8x2Si, it follows that: d[v]d[z] +w(z;v) =dist[s;v] )d[v] =dist[s;v] (i.e. it is the shortest distance already.) So we can letSi+1=Sinf(z;v)g [ f(u;v)gand after the update tod[a] for alle= (v;a) witha =2Si, we still have thatd[x] =dist[s;x]dist[s;y]d[y], for allx2Si;y =2Si. Hence the loop invariant holds. When the algorithm terminates, we have:d[u] =dist[s;u];8u2V.quotesdbs_dbs19.pdfusesText_25

[PDF] [PDF] a generalization of Dijkstras shortest path algorithm

[PDF] Improved shortest path algorithms for nearly acyclic graphs - CORE