Negative weights using Dijkstra's Algorithm -

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i trying understand why dijkstra's algorithm not work negative weights. reading example on shortest paths, trying figure out following scenario:

    2 a-------b  \     / 3 \   / -2    \ /     c

from website:

assuming edges directed left right, if start a, dijkstra's algorithm choose edge (a,x) minimizing d(a,a)+length(edge), namely (a,b). sets d(a,b)=2 , chooses edge (y,c) minimizing d(a,y)+d(y,c); choice (a,c) , sets d(a,c)=3. never finds shortest path b, via c, total length 1.

i can not understand why using following implementation of dijkstra, d[b] not updated 1 (when algorithm reaches vertex c, run relax on b, see d[b] equals 2, , therefore update value 1).

dijkstra(g, w, s)  {    initialize-single-source(g, s)    s ← Ø    q ← v[g]//priority queue d[v]    while q ≠ Ø       u ← extract-min(q)       s ← s u {u}       each vertex v in adj[u]          relax(u, v) }  initialize-single-source(g, s) {    each vertex v  v(g)       d[v] ← ∞       π[v] ← nil    d[s] ← 0 }  relax(u, v) {    //update if found strictly shortest path    if d[v] > d[u] + w(u,v)        d[v] ← d[u] + w(u,v)       π[v] ← u       update(q, v) }

thanks,

meir

the algorithm have suggested indeed find shortest path in graph, not graphs in general. example, consider graph:

figure of graph

assume edges directed left right in example,

your algorithm work follows:

  1. first, set d(a) zero , other distances infinity.
  2. you expand out node a, setting d(b) 1, d(c) zero, , d(d) 99.
  3. next, expand out c, no net changes.
  4. you expand out b, has no effect.
  5. finally, expand d, changes d(b) -201.

notice @ end of this, though, d(c) still 0, even though shortest path c has length -200. algorithm fails accurately compute distances in cases. moreover, if store pointers saying how each node start node a, you'd end taking wrong path c a.


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