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Chapter 4 Shortest Path Label-Setting Algorithms Introduction & Assumptions Applications Dijkstra’s Algorithm Problem Definition & Assumptions Problem: Given a network G = (N, A) in which each arc (i, j) has an associated length or cost cij, let node s be the source. The length of a directed path is the sum of the lengths of the arcs in the path. For every node i s, find a shortest length directed path from s to i. Assumptions: – All arc lengths are integers (if rational but not integer, multiply them by a suitably large number) – The network contains a directed path from node s to every node i s (if not, add an arc (s, i) with very large cost) – The network does not contain a negative cycle (otherwise see Ch. 5) – The network is directed (transform undirected arcs w/positive costs as in Ch. 2; undirected arcs w/negative costs will create neg. cycles) 2 Types of Shortest Path Problems Single-source shortest path One node to all others with nonnegative arc lengths – Chapter 4 Variations: maximum capacity path, maximum reliabiltiy path One node to all others with arbitrary arc lengths – Chapter 5 All-pairs shortest path: every node to every other node – Chapter 5 String model for shortest path from s to t: Arcs = strings, knots = nodes; hold s and t and pull tight. Shortest paths will be taut: for i and j on a shortest path connected by arc (i, j), distance s-i plus cij distance s-j Associated “dual” maximization problem: pulling s and t as far apart as possible 3 Single-Source Shortest Paths Solution is a shortest-path tree rooted at s. Property 4.1. If the path s = i1 – i2 – … – ih = k is a shortest path from s to k, then for every q = 2, 3, …, h-1, the subpath s = i1 – i2 – … – iq is a shortest path from the source node to iq. Property 4.2. Let the vector d represent the shortest path distances. Then a directed path P from s to k is a shortest path if and only if for i, j P, d j d i cij Store the shortest path tree as a vector of n-1 predecessor nodes: pred(j) is the node i that satisfies above equality. 4 Acyclic Networks: Reaching Examine the nodes in topological order; perform a breadth-first search to find a shortest-path tree. Reaching Algorithm: 0. d(s) 0, d(j) for j s, i s 1. If A(i) is empty, then stop. Otherwise, to examine node i, scan the arcs in A(i). If for any arc (i, j), d(j) d(i) + cij , then set d(j) = d(i) + cij . 2. Set i to the next node in topological order and return to 1. Solves shortest path problem on acyclic networks in O(m) time. 5 Dijkstra’s Algorithm Shortest paths from source node to all other nodes with nonnegative arc lengths (cycles permitted) Output: d(i) is the distance from s to i along a shortest path pred(i) is the predecessor of i along a shortest path Intermediate: S = set of permanently labeled nodes (L in GIDEN) S set of temporarily labeled nodes (P in GIDEN) GIDEN also has set U for unlabeled nodes At each iteration, one node moves to S from S 6 Dijkstra’s Algorithm in GIDEN 7 Complexity • Node selection requires time proportional to n n 1 n 2 ... 1 O n2 • Distance updates are performed for each arc emanating from node i; total of m updates in the entire algorithm • Since n2 m, the complexity is O n2 • For dense networks, n 2 m – Complexity can be improved for sparse networks by cleverness and special data structures 8