Search ALgorithm

Table of Contents

1 Branch & Bound algorithm

1.1 problem

The problem is to minimize a function f(x) of variables \(x_1,...,x_n\) over a region of feasible solutions S. \[min_{x\in S} f(x)\] The solutions state space S is formed as a rooted tree. The key to this algorithm is the efficient estimation of lower or upper bound. The problem is NP-hard.

1.2 notation

f(x) is called objective function. a function g(x) is the lower bound, defines on S with the property that g(x) ≤ f(x) for all x ∈ S.

1.3 Algorithm

  1. use a heuristic, find a solution xh. Store its value B ← f(xh). B is the global best solution so far. If no solution found, init B to ∞
  2. init a queue with the root ??
  3. loop until the queue is empty
    1. take a node N off the queue
    2. if N represents a single candidate solution x (N is a leaf?) and f(x) < B, then B = f(x).
    3. Else, branch on N to produce new nodes \(N_1,...,N_i\). For each new node:
      1. if g(Ni) > B, do nothing.
      2. else store Ni onto the queue

1.4 natual langauge description

The problem is to minimize (or maximize) the objective function f(x) over \(x_1,..,x_n\). The feasible solution search state space is a tree. The initial best known value is B=f(xh) or ∞ if no solution xh found by heuristic. From the root, everytime branch into two or more branches. For those branches, compute the lower bound. If the lower bound is larger than current best, then do not need to go into these branch. Thus we can eliminate the computation of this branch.

The assumption is the lower (or upper) bound is efficient to compute. Every time branch may or may not overlap, as long as the optimal solution is inside at least one branch.

2 A* algorithm

2.1 Problem

from an initial node, find the least-cost path to one goal node (out of one or more possible goals).

2.2 noatation

\[f(n) = g(n) + h(n)\]

where n is current node. f(n) is the cost function. g(n) is the known cost of getting from initial node to n. h(n) is a heuristic esitimate of the cost to get from n to any goal node. h(n) must be admissible, i.e. it never overestimates the actual cost, i.e. it is always less then or equal to the actual cost.

2.3 algorithm

from initial node, it maintains a priority queue of nodes. The lower \(f(n)\), the higher its priority. At each step, the node with lowest \(f(x)\) is removed, and \(f\) and \(g\) of its neighbors are updated. Add these neighbors into the queue. The algorithm terminates when one goal node has a lower \(f\) value than any node in the queue.

2.4 natual language description

from the start point, try all neighbors, and remember both the actual cost from the initial node, and the estimate from this node to one goal. Repeat trying neighbors until reach goal nodes. Stop when the goal nodes has the lowest cost function value.