Distributions of Randomized Backtrack Search Key Properties: I Erratic behavior of mean II Distributions have “heavy tails”.

Median = 1! sample mean 3500! Erratic Behavior of Search Cost Quasigroup Completion Problem number of runs

1

75%<=30 Number backtracks Proportion of cases Solved 5%>100000

Heavy-Tailed Distributions … infinite variance … infinite mean… infinite variance … infinite mean Introduced by Pareto in the 1920’s --- “probabilistic curiosity.” Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena. Examples: stock-market, earth-quakes, weather,...

Decay of Distributions Standard --- Exponential Decay e.g. Normal: Heavy-Tailed --- Power Law Decay e.g. Pareto-Levy:

Standard Distribution (finite mean & variance) Power Law Decay Exponential Decay

Normal, Cauchy, and Levy Normal - Exponential Decay Cauchy -Power law Decay Levy -Power law Decay

Tail Probabilities (Standard Normal, Cauchy, Levy)

Example of Heavy Tailed Model (Random Walk) Random Walk: Start at position 0 Toss a fair coin: –with each head take a step up (+1) –with each tail take a step down (-1) X --- number of steps the random walk takes to return to position 0.

The record of 10,000 tosses of an ideal coin (Feller) Zero crossing Long periods without zero crossing

Random Walk Heavy-tails vs. Non-Heavy-Tails Normal (2, ) Normal (2,1) O,1%> % 2 Median=2 1-F(x) Unsolved fraction X - number of steps the walk takes to return to zero (log scale)

How to Check for “Heavy Tails”? Log-Log plot of tail of distribution should be approximately linear. Slope gives value of infinite mean and infinite variance infinite mean and infinite variance infinite variance infinite variance

Number backtracks (log) (1-F(x))(log) Unsolved fraction => Infinite mean Heavy-Tailed Behavior in QCP Domain 18% unsolved 0.002% unsolved

Formal Models of Heavy- Tailed Behavior in Combinatorial Search Chen, Gomes, Selman 2001

MotivationMotivation Research on heavy-tails has been largely based on empirical studies of run time distribution. Goal: to provide a formal characterization of tree search models and show under what conditions heavy-tailed distributions can arise. Intuition: Heavy-tailed behavior arises: from the fact that wrong branching decisions may lead the procedure to explore an exponentially large subtree of the search space that contains no solutions; the procedure is characterized by a large variability in the time to find a solution on different runs, which leads to highly different trees from run to run;

Balanced vs. Imbalanced Tree Model Balanced Tree Model: chronological backtrack search model; fixed variable ordering; random child selection with no propagation mechanisms; (show demo)

T(n) - the number of leaf nodes visited - choice at level i; (1 - bad choice; 0 -good choice) (note : there is exactly one choice of zero-one assignments to the variables for each possible value of T(n); any such assignment has probability. T(n) follows an Uniform distribution

The run time distribution of chronological backtrack search on a complete balanced tree is uniform (therefore not heavy-tailed). Both the expected run time and variance scale exponentially

Balanced Tree Model –The expected run time and variance scale exponentially, in the height of the search tree (number of variables); –The run time distribution is Uniform, (not heavy tailed ). –Backtrack search on balanced tree model has no restart strategy with exponential polynomial time. Chen, Gomes & Selman 01

How can we improve on the balanced serach tree model? Very clever search heuristic that leads quickly to the solution node - but that is hard in general; Combination of pruning, propagation, dynamic variable ordering that prune subtrees that do not contain the solution, allowing for runs that are short. ---> resulting trees may vary dramatically from run to run.

T - the number of leaf nodes visited up to and including the successful node; b - branching factor Formal Model Yielding Heavy-Tailed Behavior b = 2 (show demo)

Expected Run Time (infinite expected time) Variance (infinite variance) Tail (heavy-tailed)

Bounded Heavy-Tailed Behavior (show demo)

No Heavy-tailed behavior for Proving Optimality

Proving Optimality

Small-World Vs. Heavy-Tailed Behavior Does a Small-World topology (Watts & Strogatz) induce heavy-tail behavior? The constraint graph of a quasigroup exhibits a small-world topology (Walsh 99)

Exploiting Heavy-Tailed Behavior Heavy Tailed behavior has been observed in several domains: QCP, Graph Coloring, Planning, Scheduling, Circuit synthesis, Decoding, etc. Consequence for algorithm design: Use restarts or parallel / interleaved runs to exploit the extreme variance performance. Restarts provably eliminate heavy-tailed behavior. (Gomes et al. 97, Hoos 99, Horvitz 99, Huberman, Lukose and Hogg 97, Karp et al 96, Luby et al. 93, Rish et al. 97, Wlash 99)

XXXXX solved 10 Sequential: = 51 seconds Parallel: 10 machines second 51 x speedup Super-linear Speedups Interleaved (1 machine): 10 x 1 = 10 seconds 5 x speedup

Restarts 70% unsolved 1-F(x) Unsolved fraction Number backtracks (log) no restarts restart every 4 backtracks 250 (62 restarts) 0.001% unsolved

Example of Rapid Restart Speedup (planning) ~100 restarts Cutoff (log) Number backtracks (log) ~10 restarts

Sketch of proof of elimination of heavy tails Let’s truncate the search procedure after m backtracks. Probability of solving problem with truncated version: Run the truncated procedure and restart it repeatedly.

Y - does not have Heavy Tails