March 7, Using Pattern Recognition Techniques to Derive a Formal Analysis of Why Heuristic Functions Work B. John Oommen A Joint Work with Luis G. Rueda School of Computer Science Carleton University

March 7, Optimization Problems Any arbitrary optimization problem: Instances, drawn from a finite set, X, An Objective function Some feasibility functions The aim: Find an (hopefully the unique) instance of X, which leads to a maximum (or minimum) subject to the feasibility constraints.

March 7, An Example The Traveling Salesman Problem (TSP) Consider the cities numbered from 1 to n, The salesman starts from city 1, visits every city once, and Returns to city 1. An instance of X is a permutation of cities: For example, , if five cities considered The objective function: The sum of the inter-city distances: 1  4, 4  3, 3  2, 2  5, 5  1

March 7, Heuristic Functions A Heuristic algorithm is an algorithm which attempts to find a certain instance X that maximizes the objective function It iteratively invokes a Heuristic function. The heuristic function estimates (or measures) the cost of the solution. The heuristic itself is a method that performs one or more changes to the current instance.

March 7, An Open Problem Consider a Heuristic algorithm that invokes any of Two Heuristic Functions : H 1 and H 2 used in estimating the solution to an Optimization problem If Estimation accuracy of H 1 > Estimation accuracy of H 2 Does it imply that H 1 has higher probability of leading to the optimal QEP?

March 7, Pattern Recogniton Modeling Two heuristic functions : H 1 and H 2 Probability of choosing a cost value of a Solution: two independent random variables: X 1 and X 2 Distribution -- doubly exponential: c where, and

March 7, Pattern Recogniton Modeling Our model: Error function is doubly exponential. Typical in reliability analysis and failure models. How reliable is a Solution when only estimate known? Assumptions: Mean cost of Optimal Solution: , then shift the origin by   E[X] = 0 Variances: Estimate X 1 better than Estimate of X 2

March 7, Main Result (Exponential) H 1 and H 2, two heuristic functions. X 1 and X 2, two r.v.  optimal solution obtained by H 1 and H 2 X 1 ’ and X 2 ’, other two r.v. for sub-optimal solution Let p 1 and p 2 the prob. that H 1 and H 2 respectively make the wrong decision. Shown that: then :

March 7, Proof (Graphical Sketch) For a particular x, the prob. that x leads to wrong decision by H 1 is given by: X 1 (opt) X 1 (subopt) X 2 (subopt) X 2 (opt)

March 7, Proof (Cont’d) or X 1 (opt) X 1 (subopt) X 2 (subopt) X 2 (opt) if x < c

March 7, Proof (Cont’d) The total probability that H 1 makes the wrong decision for all values of x is: Similarly, the prob. that H 2 makes the wrong decision for all values of x is:

March 7, Proof (Cont’d) Solving integrals and making p 1  p 2, we have: which, using ln x  x - 1, implies that p 1  p 2 QED where  1 = 1 c and  2 = 2 c Also  2 substituted for k  1

March 7, Second Theorem F(  1,k) can also be written in terms of  1 and k as: Suppose that  1  0 and 0  k  1, then G(  1,k)  0, and there are two solutions for G(  1,k) = 0 and Proof: Taking partial derivatives and solving: and

March 7, Graphical Analysis (Histograms) R-ACM / Eq-width R-ACM / Eq-depth T-ACM / Eq-width T-ACM / Eq-depth G >>> 0, or p 1 <<< p 2 R-ACM / T-ACM Eq-width / Eq-depth G  0, or p 1  p 2 Minimum in  1 = 0 and 0  k  1

March 7, Analysis : Normal Distn’s No integration possible for the normal pdf Shown numerically that p 1  p 2

March 7, Plot of the Function G

March 7, Estimation for Histograms is estimated as where N is the # of samples

March 7, Similarities of R-ACM and d-Exp Estimated for RACM True d-Exp

March 7, Simulations Details Simulations performed in Query Optimization: 4 independent runs per simulation. 100 random Databases per run  400 per simulation. 6 Relations, 6 Attributes per relation, 100 tuples per relation. Four independent runs on 100 databases: R-ACM vs. Traditional using: 11 bins, 50 values

March 7, Empirical Results # of times in which R-ACM yields better QEP # of times in which Eq-width yields better QEP # of times in which Eq-depth yields better QEP

March 7, Conclusions Applied PR Techniques to solve problem of relating Heuristic Function Accuracy and Solution Optimality Used a reasonable model of accuracy (doubly exponential distribution). Shown analytically how the high accuracy of heuristic function leads to a superior solutions. Numerically shown the results for normal distributions Shown that R-ACM yield better QEPs in a larger number of times than Equi-width and Equi-depth. Empirical results on randomly generated databases also shown the superiority of R-ACM. Graphically demonstrated the validity of our model.