Suppose G = (V, E) is a directed network. Each edge (i,j) in E has an associated ‘length’ c ij (cost, time, distance, …). Determine a path of shortest length between two specified nodes s and t. The length c(P) of a path P is the sum of its constituent arc lengths: c(P) = Σc ij Applications: transportation planning (shortest route) telecommunications (most reliable connection) scheduling (critical path) pattern recognition, computer graphics subproblem to a larger optimization problem Shortest Paths

1. Differentiated by the number of origin/destination(source/sink) vertices: from s to t from s to all other vertices to t from all other vertices from i to j 2. Differentiated by the properties of the edge length: Nonnegative edge lengths Negative edge lengths Different Types of SPP

1.Integer data 2.Directed network 3.There is a path from s to all other vertices 4.There is no negative length directed cycle Two algorithms to be studied: Label-setting – no negative length edges Label-correcting – negative length edges allowed KEY IDEA: Distance labels – D(i) = how far away vertex i is from s Assumptions & Algorithms

Let d j correspond to the distance of the shortest path from vertex s to vertex j. Certainly d s =0. Moreover, the condition d j ≤ d i + c ij (***) says that if we have a shortest path its length cannot be improved by means of an edge (i, j) not currently on the path. This condition (***) is used in both algorithms to efficiently solve the SPP. FOCUS: single source problems, first with nonnegative costs, then allow negatives. Optimality Conditions

Key observation: any subpath of a shortest path is also a shortest path When we find a shortest path from s to t, we also find a shortest path from s to any intermediate vertices. This leads to a way of representing all shortest paths from vertex s. Find a shortest path from s to some t; call it P 1. Repeat with another vertex t not on path P 1. Repeat until all vertices are used. This results in a shortest path tree for G. Namely, a directed tree in G whose paths s to j are all shortest paths from s to j. s b a ct Shortest Path Trees

Use two arrays to represent the shortest path tree. d(i) records the shortest distance to vertex i. p(i) records the predecessor of vertex i in the shortest path Example -p(i) i 0d(i)

Nonnegative edge lengths: At each step in the algorithm, temporarily labeled vertices have a distance label of D(i), which is an upper bound on d(i); permanently labeled vertices are guaranteed to have D(i) = d(i). Two sets to keep track of: S is the set of permanently labeled vertices; Ŝ = V – S is the set of temporarily labeled vertices Vertices are transferred from Ŝ to S, one at a time. Upon termination, all vertices are permanently labeled. The algorithm processes vertices in increasing distance from vertex s. At the k th iteration, the k th closest vertex to s will be permanently labeled. S S ^ s b a ct Label Setting Algorithms

If network has a topological order, examine vertices from 1 to n (reaching). If no topological order exists, examine vertices in order of increasing distance from s (Dijkstra’s Algorithm). DIJKSTRA’S start with all vertices in Ŝ; give vertex s a distance D(s) = 0; give all other vertices the distance D(j) = ∞; while set S is not full pick a vertex, i, in Ŝ with the smallest distance; move that vertex from Ŝ to S; update distances for vertices reachable from i; for each (i,j) in E(i) if D(j) > D(i) + c ij then D(j) = D(i) + c ij and pred(j) = i; Topological(Acyclic) or Not Topological

i p(i) D(i)0 ∞∞∞∞∞ i p(i)0 D(i) Example move to S

i p(i)01 D(i) i p(i)012 D(i) Example move to S

i p(i)0122 D(i) i p(i)01223 D(i) Example move to S

p(i) i D(i) Example 7

0D(i) ∞∞∞∞∞ i Example – Topological

Example – Dijkstra’s 0D(i) ∞∞∞∞∞ i

s t Example

Suppose G = (V, E) is a directed network. Each edge (i,j) in E has an associated ‘length’ c ij = positive, zero, negative d(j) = minimum length of a path from s to j D(j) = distance label, current length of some path from s to j D(j) = d(j) for all j in V iff D(j) ≤ D(i) + c ij for all (i,j) in E(i) and D(s) = 0 Label Correcting In a network with negative edge lengths, maintain a LIST of vertices with changed labels (Bellman-Ford Algorithm). Search edge list and update until no distance is changed. Maintain LIST as: Queue – FIFO Stack – LIFO Dequeue – vertex selected from top; first time vertex is added, it is placed at the bottom; and anytime after that, a vertex re-enters at top of list

Queue Stack Dequeue (Pape) List Structures enter leave enter leave enter re-enter leave

∞∞ ∞ ∞∞ 0D(i) i =List 0D(i) =List ∞∞∞3 {1} {2, 3} queue

∞∞ ∞ ∞∞ 0D(i) i =List D(i) 0 =List ∞ ∞∞3 {1} {2, 3} dequeue

A negative length cycle will cause some distance labels to decrease without limit. With label correcting algorithms, negative length cycles can be detected in a variety of ways. 1. If C is max of all |c ij |, then any D(j) falling below -nC. 2. In the queue(FIFO) implementation, more than n-1 updates for a single vertex. 3. If the graph based on predecessors fails to be a tree Negative cycle detection

1. Apply Dijkstra’s n times, once for each vertex. Good for sparse network. 2. Floyd-Warshall Algorithm finds best in phases. Good for dense network. Assume network is strongly connected. Matrix entry D(i, j) = distance from i to j. Floyd-Warshall begin set D(i, j) = c ij, pred(i, j) = i for each (i, j) in E; set D(i, i) = c ii, pred(i, i) = i for each i; for k = 1 to n do for all i,j = 1 to n do if D(i, j) > D(i, k) + D(k, j) then D(i, j) = D(i, k) + D(k, j) and pred(i, j) = pred(k, j); end All Pairs Shortest Path

Example 013∞∞ 201∞∞ ∞∞0∞2 ∞4∞0∞ ∞∞∞ k = 0 k = 1 D =pred= ∞∞ ∞0 40 ∞ k = 2 D =pred=

Example k = 5 D =pred= What is shortest path from 1 to 4? What is shortest path from 5 to 3? 4