1. Searching and optimization

2.  Applications and techniques  Branch-and-bound search  Genetic algorithms  Successive refinement  Hill climbing

3.  Characterized by looking for a solution to a problem that has many potential solutions  An exhaustive search is not feasible for many problems ◦ Directed search is used instead  May not necessarily search for the optimal (best) solution ◦ Non-optimal solutions may be acceptable

4.  Examples of problems: ◦ Travelling salesperson problem ◦ 0/1 knapsack problem ◦ n-queens problem ◦ 8-puzzle ◦ 15-puzzle  All of these have an extremely large number of permutations ◦ Searches can be very time-consuming

5.  Examples of real world applications: ◦ Financial forecasting ◦ Airline fleet and crew assignment ◦ VLSI chip layout  Examples of search and optimization techniques: ◦ Branch-and-bound search ◦ Dynamic programming ◦ Hill climbing ◦ Simulated annealing ◦ Genetic algorithms

6.  Uses a state space tree ◦ Root node is the starting point ◦ Transition from one level to the next represents a choice being made towards the solution ◦ Can be considered as dividing the problem into sub-problems ◦ Each node represents a sub-problem  How the tree is constructed depends on the problem ◦ Called a dynamic tree  If the choice at each node just consists of selecting either true or false, a binary tree is created ◦ Called a static tree

7.  The state space tree can be explored using various methods ◦ Depth-first search ◦ Breath-first search ◦ Best-first search  Searching all nodes is expensive ◦ Consider only exploring paths that will likely lead to the best solution (i.e., best-first search) ◦ Avoiding unpromising paths prunes the state space tree

8.  A bounding (or cut-off) function may be used to determine whether or not a given path is promising  When a node is reached, the bounding function produces an upper or lower bound  This can be compared to the value of the best path that has been found so far ◦ Determine if it is worth it to continue searching along this path or not

9.  Terminology ◦ Live node: A node that has been reached, but not all of its children have been explored ◦ E-node: (Expanded node.) A live node in which its children are currently being explored ◦ Dead node: A node where all of its children have been explored  As the search proceeds, a list of live nodes is maintained in a queue ◦ Called an open list  A list of dead nodes may also be maintained ◦ Called a closed list ◦ Used to check for duplicates

10.  Parallelization ◦ Simplest approach...allocate a chuck of the state space tree to each processor and let them search their portion independently ◦ Possible to parallelize the evaluation of the bounding function  However, there are some issues

11.  The size of the state space tree may not be known in advance ◦ Issue of load balancing  The current lower or upper bound (generated by the bounding function) should be known by all processors ◦ Must be updated so that all processors can prune their state space tree quickly

12.  Another approach is to allow the open list queue to be accessed concurrently (i.e., in shared memory) ◦ When a processor chooses an item from the queue, that item must be locked so that another processor doesn’t choose it as well ◦ However, this will limit the potential speedup

13.  Speedup anomalies ◦ Acceleration anomaly: Super-linear speedup can be achieved if one processor finds a solution extremely quickly ◦ Deceleration anomaly: A solution may be positioned such that it cannot be found in 1/p of the sequential time ◦ Detrimental anomaly: There is evidence that shows that the speedup factor could be less than 1

14.  Attempts to simulate the natural evolution of a population of individuals  Chromosomes contain information that characterizes a living being ◦ Evolution works at the chromosome level  The chromosomes of offspring are generated from the chromosomes of the offspring’s parents ◦ This blending is called crossover  Sometimes, there may be a random change an individuals chromosome pattern ◦ This is called mutation

15.  With a low mutation rate, there is a tendency for fit individuals to pass on their traits to the next generation ◦ Unfit individuals tend to die out  With a high mutation rate, individuals of the next generation are very randomly different from the individuals of the previous generation ◦ Can cause instability or non-convergence

16.  Basically, you have a population of solutions. Crossover and mutation occur. A new generation of solutions is produced and this process repeats  Algorithm: ◦ Create an initial population of solutions (individuals) ◦ Evaluate each solution to determine how “fit” they are ◦ Characterize the solutions from most fit to least fit ◦ Select a subset of the population (favouring more fit solutions) ◦ Use the subset to produce a new generation of offspring (using crossover) ◦ Apply random mutations to some of offspring ◦ Repeat this process until some termination condition is satisfied

17.  Representation of an individual (their chromosomes) ◦ Commonly, binary strings (e.g., 000101011110) ◦ More recently, floating points, gray codes, and integer strings ◦ Bits in the binary string have some sort of meaning  For example, the first bit might represent the sign of a number and the remaining bits might represent the number itself  An initial population is generated using a random number generator  Individuals are evaluated according to some function ◦ Could produce a number, with higher numbers representing that an individual is very fit

18.  Constraints might be placed on a solution ◦ For example, if an individual falls outside of a range of values, it could be discarded or “repaired” by changing some bits  The number of individuals in a population affects: ◦ How quickly a good solution will be found ◦ How much computation needs to be done per generation

19.  Selection process ◦ In nature, there is a bias towards the most fit individuals ◦ However, there may be problems where there are many local optimum solutions ◦ Just selecting the most fit individuals could gather around a single local optimum rather than the global optimum ◦ One approach is tournament selection ◦ A set of individuals from the population is selected ◦ The most fit individual wins the tournament is selected as a parent to generate offspring ◦ Many of these tournaments are played until the whole population has participated

20.  The individuals chosen by the selection process are paired up as parents and produce offspring, using crossover and mutation  Single-point crossover ◦ Given two parents, A and B, their chromosomes are split into two at some boundary ◦ The rightmost portions of the parents’ chromosomes are then swapped to generate two children  Other types of crossover: Multi-point crossover, uniform crossover

21.  Mutation ◦ Generally, the mutation rate is kept small ◦ Mutation occurs by simply changing one or more bits in a child’s chromosome  Other variations that can be incorporated into the algorithm: ◦ Carry over some of the most fit individuals from the previous generation into the next generation ◦ Randomly generate new individuals for the next generation ◦ Vary the population size from one generation to the next

22.  Possible termination conditions: ◦ Stop after a number of generations  Cannot tell if you have an optimal solution or if the population has moved far away from it ◦ Consider the degree of improvement from one generation to the next ◦ Consider the similarity of all the individuals in the population

23.  Two approaches to parallelization: ◦ Let each processor control its own population and allow some migration to occur between populations ◦ Let each processor do a portion of each step of the algorithm on a shared population

24.  Isolated subpopulations ◦ After a certain number of generations, the processors share their best individuals ◦ Immigrants must be selected, send, received, then integrated into the population ◦ Sending and receiving is done easily with message passing  There are a couple models of migration to consider ◦ The island models allow individuals to move to any other subpopulation  Models nature better, but requires more communication ◦ The stepping-stone model only allows moving between neighbouring populations

25.  Common population ◦ The steps of the algorithm can easily be parallelized, but under a message passing system, there would be a huge amount of communication ◦ This is more feasible with shared memory

26.  Related to solving problems like finding the maximum value of a function  Consider a graph of a function ◦ A grid is placed with some amount of spacing between grid lines ◦ At each of these grid lines, a point is examined ◦ The k best points are kept  A finer grid spacing is then used ◦ Around each of the chosen best points, points on the grid lines are examined ◦ The best points are kept and the process repeats until some condition is met

27.  This is simple, but requires more computations than genetic algorithms ◦ Though, still much better than an exhaustive search  Parallelization is straightforward ◦ Divide up the points to be examined amongst all the processors ◦ Each processor sends its best points to the master processor ◦ The master processor determines which of those points are the best, then distributes them to the slave processors ◦ The points to be examined are divided up again and the process repeats

28.  Consider a blindfolded hiker placed in a terrain having many local maximums  The hiker’s task is to find the global maximum  Simple algorithm the hiker uses: ◦ Don’t go downhill  If the hiker always moves uphill, it will always reach the top of a maximum  However, if there are many local maximums, it’s unlikely the hiker will stop on the global maximum

29.  So...randomly place thousands of blindfolded hikers on the terrain  Still no guarantee, but this increases the chances that one of them will stop on the global maximum  Hill climbing is a Monte Carlo search technique  Again, this relates to problems like finding the global maximum of a function

30.  Parallelization ◦ The area can be divided up amongst all the processors ◦ Each processors generates a random number of starting points and moves all of those points ◦ Each processor reports the maximum they found to the master processor ◦ The master processor returns the largest maximum from the findings

31.  Some hikers might take longer than others to reach a maximum ◦ A tiny hill versus a steep mountain ◦ Some processors might finish earlier than others  A work-pool approach considers this ◦ Divide the area up into even smaller segments ◦ As processors finish a segment, they report their maximum to the master processor and choose another segment from the work-pool