Chapter 4Shortest Path Label-Setting Algorithms

Introduction & Assumptions

Applications

Dijkstra’s Algorithm

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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)

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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

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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

Store the shortest path tree as a vector of n-1 predecessor nodes: pred(j) is the node i that satisfies above equality.

, , iji j P d j d i c

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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 s1. 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.

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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)

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

S

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Dijkstra’s Algorithm in GIDEN

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Complexity

• Node selection requires time proportional to

• Distance updates are performed for each arc emanating from node i; total of m updates in the entire algorithm

• Since

• For dense networks,– Complexity can be improved for sparse networks by cleverness

and special data structures

21 2 ... 1n n n O n

2 2, the complexity is n m O n2n m