Breadcrumbs Section. Click here to navigate to respective pages.

Chapter

Chapter

# ( 1 )

DOI link for ( 1 )

( 1 ) book

# ( 1 )

DOI link for ( 1 )

( 1 ) book

## ABSTRACT

The first term represents the case that a is not visible, which occurs with probability 1 — pna. In this case the only choice is to remain at n and acquire another image, which incurs cost lnn and results in an expected length of lnn + Eng. The second term represents the case that a is visible, which occurs with probability pna. In this case the expected length of the shortest path is the smaller of lna + Eag and lnn + Eng, corresponding to the options of going to a or staying at n. Recall that we are assuming that the goal is reachable from any node, and thus Eag < oo. Given that the edge n —> a is the only edge leaving n, we know that all paths from n to g have to go through a, and thus that Eng > lna + Eag· This allows us to solve Equation 2 for Eng, yielding:

Now, let us consider a node n with two outgoing edges n —» a and n —» b. The relation for the expected length of the shortest path from n to g can be expressed analogously, except that now there are four cases, de pending on which of a and b are visible. Using the shorthand p to denote (1 — p), we obtain the following relation for Eng:

Unlike in the case for only a single edge, it is no longer possible to explicitly solve these equations for Eng because of the minimum expressions, which recur sively relate the unknown quantities Eig. For example, to determine the value of the minimum expressions in Equation 4, we need to know the ordering among K a, Kb, and K n, where Ki denotes lni + Eig. While one can still deduce that K n cannot be the smallest of the three, each of the remaining 4 orderings is possible:

Note that if such orderings were known for all nodes, Equations 4 and 5 would simplify to linear equations involving the unknowns Eng. The expected lengths of the shortest paths could then be computed simply by solving a system of N — 1 linear equations, where N is the total number of landmarks (including the goal). The presence of the minima, however, prohibits this approach.