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Scenario-Based Robust Optimization of Tree Structures
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Studies robust binary search trees and Huffman trees under k frequency scenarios, giving NP-hardness, near-optimal competitive-ratio/regret algorithms, a Pareto-optimal fairness DP for uniform BSTs, and MILP-based experiments on European-language letter frequencies.
Problem
Constructing a single binary search tree or Huffman tree that performs well across several known frequency distributions (scenarios), measured by competitive ratio or regret, with fairness tradeoffs for BSTs.
Approach
Introduces scenario-based robust optimization for tree structures: a recursive BST construction achieving competitive ratio ceil(log2(k+1)) (matched by a lower bound), an aggregation of per-scenario optimal Huffman trees with regret ceil(log2 k) (near-matched by floor(log2 k)), and a DP computing Pareto-optimal BST regret points for uniform scenarios as a fairness notion.
Key results
- On language letter frequencies, R-bst achieves competitive ratio 1.215 versus 1.047 for the optimal-cost BST and regret 0.680 versus 0.151.
- R-ht achieves competitive ratio 1.091 and regret 0.364 versus optimal Huffman tree competitive ratio 1.038 and regret 0.155.
- Gurobi MILPs take 8 to 14 seconds for BSTs, while R-bst and R-ht run in under 0.1 seconds.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use R-bst instead of the optimal BST | Robust BST · 10 European languages, 26-letter alphabet | cost: 3.389 -> 3.940 | Hurt |
| Use R-bst instead of the optimal BST | Robust BST · 10 European languages, 26-letter alphabet | competitive ratio: 1.047 -> 1.215 | Hurt |
| Use R-bst instead of the optimal BST | Robust BST · 10 European languages, 26-letter alphabet | regret: 0.151 -> 0.680 | Hurt |
| Use R-ht instead of the optimal Huffman tree | Robust Huffman tree · 10 European languages, 26-letter alphabet | cost: 4.271 -> 4.425 | Hurt |
| Use R-ht instead of the optimal Huffman tree | Robust Huffman tree · 10 European languages, 26-letter alphabet | competitive ratio: 1.038 -> 1.091 | Hurt |
| Use R-ht instead of the optimal Huffman tree | Robust Huffman tree · 10 European languages, 26-letter alphabet | regret: 0.155 -> 0.364 | Hurt |
| Run Pareto-optimal algorithm PO on the generated country-pair strings | BST fairness · City-name strings from all 10-country pairs, n=30 | 0-regret and 1-regret: 27 and 28; 28 and 27 | Helped |
| Run PO across all strings with a and b keys in the two scenarios | BST fairness · All strings with a,b in {0,...,11} | 0-regret and 1-regret: at most a and b | Helped |
Surprising findings
- Experiments suggest that for a binary string with a zeros and b ones, a BST exists with 0-regret at most a and 1-regret at most b, a stronger fairness guarantee than the theory proves.
Experimental setup
Datasets: letter frequency data for ten European languages
Benchmarks: letter frequencies of ten European languages city populations from ten countries
Metrics: cost competitive ratio regret running time
Baselines: optimal per-scenario BST optimal per-scenario Huffman tree Gurobi MILP optimal trees
Builds on: Knuth optimal BST dynamic programming Huffman coding scenario-based robust optimization
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