Shortest Paths in Directed Graphs

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1 Shortet Path in Direted Graph Jonathan Turner January, 0 Thi note i adapted from Data Struture and Network Algorithm y Tarjan. Let G = (V, E) e a direted graph and let length e a real-valued funtion on E. The length of a direted path in G i defined to e the um of the length of it edge. So for example, in Figure, the length of the path, a,, g i 4. A hortet path joining a pair of vertie u and v i a 8 4 e 4 d 4 f 7 g Figure : Example of a hortet path in a graph defined a a path of minimum length among all path from u to v. So for example, the highlighte path a, e, d, f in Figure i a hortet path from a to f, ut the path a,, g, f i not. The hortet path prolem ha many appliation. For example, it i ued to determine the et way to forward paket in the internet, and y mapping appliation to determine the hortet driving ditane etween two loation. It alo arie in le oviou form, and often appear a a uprolem within other prolem. Note that we allow edge to have negative length, whih mean that hortet path may have negative length a well. Thi allow the poiility that a graph might ontain a yle of negative length, in whih ae hortet

2 path are not well-defined (ine we an get a path of aritrarily mall length y travering a negative yle multiple time). Thi mean that we have to take extra are when dealing with graph that have negative edge length. You might well wonder why we would want to allow negative edge length. While many appliation of hortet path do not require negative edge length, there are other appliation where they are ueful or even eential. Indeed, hortet path omputation are ued in algorithm for the min ot flow prolem and in thi ontext, negative edge length are an unavoidale feature of the prolem. We note that if there i no negative yle on any path from to t in a graph, then there mut e a imple path of minimum length, ine we an alway remove a imple yle of non-negative length from a path without inreaing it length. So for example, in Figure the hortet path, a,,, t an e replaed with the imple hortet path, a, t y removing the non-negative yle a,,. Thee oervation yield the following ai 4 a t theorem. Figure : Removing non-negative yle from hortet path Theorem Let G e digraph with a path from to t. There i a hortet path from to t if and only if no path from to t that ontain a negative yle. If there i a hortet path from to t, there i one that i imple. There are everal variant of the hortet path prolem. The implet i the ingle pair prolem, in whih we are given a oure vertex and a ink vertex t and our ojetive i find a hortet path from to t. In the ingle oure prolem, we are given a oure vertex, and our ojetive i to find hortet path from to every other vertex in G. In the ingle ink prolem, we are given a ink vertex t, and our ojetive i to find hortet path to t from every other vertex in G. Finally, in the all pair prolem, our ojetive i to find hortet path etween every pair of vertie. It turn out that olving the ingle pair prolem generally require olving a ingle oure (or ingle ink) prolem. Alo, the ingle oure and ingle ink prolem are

3 eentially the ame, and the all pair prolem an e olved y olving the ingle oure prolem multiple time. Hene, it make ene to fou mot of our attention on the ingle oure verion of the prolem. Solution to the ingle oure prolem an e expreed onviently uing a hortet path tree. A direted panning tree of a graph i a direted utree of the graph that inlude a direted path from the root to every vertex. A hortet path tree i a direted panning tree rooted at the oure vertex, in whih all path are hortet path. So for example, the heavy weight edge in Figure form a hortet path tree rooted at a. The following theorem a 8 4 e 4 d 4 f 7 g Figure : Shortet path tree haraterize the exitene of hortet path in term of hortet path tree. Theorem G ontain hortet path from to every other vertex if and only if it ontain a hortet path tree with root. The proof of thi will e deferred until later. Next, we derie a key property of hortet path tree. If T i a panning tree of G with root, define ditane(v) to e the length of the path from to v in T. Conider an edge (v, w) in G that i not in T and note that that if ditane(w) > ditane(v) + length(v, w), then the tree path from to w i not a hortet path (ee Figure 4). Hene, if T i a hortet path tree, then we mut have ditane(w) ditane(v) + length(v, w) for all edge (v, w) in G. What aout the onvere? That i, an we ay that if ditane(w) ditane(v) + length(v, w) for all edge (v, w) in G, that all path in T are hortet path? To prove thi, we mut how that for any path p in G from to a vertex x, ditane(x) length(p). Note that if p onit of a ingle edge (, x),

4 v w Figure 4: Shortet path tree property then length(p) = ditane() + length(, x) ditane(x) So, let try to prove the general reult uing indution on the numer of edge in p. Aume then that p ha k + edge and that for all vertie x and all path p with k edge, ditane(x ) length(p ). Now, let q e the path oniting of the firt k edge of p and let y e the lat vertex in q (ee Figure ). Then, q y x p Figure : Indutive tep etalihing hortet path tree property ditane(x) ditane(y) + length(y, x) length(q) + length(y, x) = length(p) Thi give u the following theorem. Theorem Let T i a panning tree of G with root, and let ditane(v) e the length of the path from to v in T. T i a hortet path tree if and only if, ditane(w) ditane(v) + length(v, w) for all edge (v, w) in G. The property deried in the theorem i alled the hortet path tree property and i the ai for a fundamental method for olving the hortet path prolem. Thi method eentially onvert an aritrary panning tree into a hortet path tree y finding edge that violate the hortet path tree property and modifying the tree o a to eliminate the violation. Thi 4

5 method underlie the variou peifi algorithm that are ued to find hortet path.

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