quantrs2-tytan 0.2.0

High-level quantum annealing interface inspired by Tytan for the QuantRS2 framework
Documentation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
#![allow(clippy::pedantic, clippy::unnecessary_wraps)]
#![allow(unused_must_use)]
//! Comprehensive tests for AI-assisted quantum optimization.

use quantrs2_tytan::ai_assisted_optimization::*;
use scirs2_core::ndarray::{Array1, Array2};
use std::collections::HashMap;
use std::time::Duration;

#[test]
fn test_ai_optimizer_creation() {
    let config = AIOptimizerConfig::default();
    let _optimizer = AIAssistedOptimizer::new(config);

    // Test that optimizer is created successfully (if it didn't panic, it succeeded)
}

#[test]
fn test_ai_optimization_small_problem() {
    let config = AIOptimizerConfig {
        parameter_optimization_enabled: true,
        reinforcement_learning_enabled: false, // Disabled for faster testing
        auto_algorithm_selection_enabled: true,
        structure_recognition_enabled: true,
        quality_prediction_enabled: true,
        learning_rate: 0.01,
        batch_size: 16,
        max_training_iterations: 100,
        convergence_threshold: 1e-4,
        replay_buffer_size: 1000,
    };

    let mut optimizer = AIAssistedOptimizer::new(config);

    // Create a small test QUBO problem
    let mut qubo = Array2::zeros((6, 6));

    // Create a structured problem with two blocks
    for i in 0..3 {
        for j in 0..3 {
            if i != j {
                qubo[[i, j]] = -1.0; // Strong coupling within first block
            }
        }
    }
    for i in 3..6 {
        for j in 3..6 {
            if i != j {
                qubo[[i, j]] = -1.0; // Strong coupling within second block
            }
        }
    }

    // Weak coupling between blocks
    qubo[[2, 3]] = 0.1;
    qubo[[3, 2]] = 0.1;

    let result = optimizer.optimize(&qubo, Some(0.9), Some(Duration::from_secs(30)));
    assert!(result.is_ok());

    let optimization_result = result.unwrap();

    // Verify basic result structure
    assert_eq!(optimization_result.problem_info.size, 6);
    assert!(!optimization_result.recommended_algorithm.is_empty());
    assert!(optimization_result.confidence >= 0.0);
    assert!(optimization_result.confidence <= 1.0);

    // Verify structure recognition worked
    assert!(!optimization_result
        .problem_info
        .structure_patterns
        .is_empty());
    let has_block_pattern = optimization_result
        .problem_info
        .structure_patterns
        .iter()
        .any(|p| matches!(p, StructurePattern::Block { .. }));
    assert!(has_block_pattern);

    // Verify quality prediction
    assert!(optimization_result.predicted_quality.expected_quality >= 0.0);
    assert!(optimization_result.predicted_quality.expected_quality <= 1.0);
    assert!(
        optimization_result.predicted_quality.confidence_interval.0
            <= optimization_result.predicted_quality.expected_quality
    );
    assert!(
        optimization_result.predicted_quality.expected_quality
            <= optimization_result.predicted_quality.confidence_interval.1
    );

    // Verify alternatives are provided
    assert!(!optimization_result.alternatives.is_empty());

    println!("Small problem AI optimization completed successfully");
    println!(
        "Recommended algorithm: {}",
        optimization_result.recommended_algorithm
    );
    println!(
        "Predicted quality: {:.3}",
        optimization_result.predicted_quality.expected_quality
    );
    println!("Confidence: {:.3}", optimization_result.confidence);
}

#[test]
fn test_problem_feature_extraction() {
    let mut config = AIOptimizerConfig::default();
    let mut optimizer = AIAssistedOptimizer::new(config);

    // Test different problem types

    // 1. Dense problem
    let mut dense_qubo = Array2::ones((5, 5));
    let dense_features = optimizer.extract_problem_features(&dense_qubo).unwrap();
    assert_eq!(dense_features[0], 5.0); // Size
    assert!(dense_features[4] > 0.9); // High density

    // 2. Sparse problem
    let mut sparse_qubo = Array2::zeros((8, 8));
    sparse_qubo[[0, 1]] = 1.0;
    sparse_qubo[[1, 0]] = 1.0;
    sparse_qubo[[2, 3]] = 1.0;
    sparse_qubo[[3, 2]] = 1.0;

    let sparse_features = optimizer.extract_problem_features(&sparse_qubo).unwrap();
    assert_eq!(sparse_features[0], 8.0); // Size
    assert!(sparse_features[4] < 0.5); // Low density

    // 3. Symmetric problem
    let mut symmetric_qubo = Array2::zeros((4, 4));
    symmetric_qubo[[0, 1]] = 2.0;
    symmetric_qubo[[1, 0]] = 2.0;
    symmetric_qubo[[2, 3]] = -1.0;
    symmetric_qubo[[3, 2]] = -1.0;

    let symmetric_features = optimizer.extract_problem_features(&symmetric_qubo).unwrap();
    assert_eq!(symmetric_features[8], 1.0); // Symmetric flag should be 1.0

    println!("Feature extraction tests passed");
}

#[test]
fn test_structure_recognition() {
    let mut recognizer = ProblemStructureRecognizer::new();

    // Test grid structure recognition
    let mut grid_qubo = Array2::zeros((9, 9)); // 3x3 grid

    // Add grid edges
    for i in 0..3 {
        for j in 0..3 {
            let idx = i * 3 + j;

            // Right neighbor
            if j < 2 {
                let right_idx = i * 3 + (j + 1);
                grid_qubo[[idx, right_idx]] = -1.0;
                grid_qubo[[right_idx, idx]] = -1.0;
            }

            // Bottom neighbor
            if i < 2 {
                let bottom_idx = (i + 1) * 3 + j;
                grid_qubo[[idx, bottom_idx]] = -1.0;
                grid_qubo[[bottom_idx, idx]] = -1.0;
            }
        }
    }

    let grid_patterns = recognizer.recognize_structure(&grid_qubo).unwrap();
    let has_grid = grid_patterns
        .iter()
        .any(|p| matches!(p, StructurePattern::Grid { .. }));
    assert!(has_grid);

    // Test block structure recognition
    let mut block_qubo = Array2::zeros((6, 6));

    // Two blocks of size 3
    for block in 0..2 {
        let start = block * 3;
        let end = (block + 1) * 3;

        for i in start..end {
            for j in start..end {
                if i != j {
                    block_qubo[[i, j]] = -2.0;
                }
            }
        }
    }

    let block_patterns = recognizer.recognize_structure(&block_qubo).unwrap();
    let has_block = block_patterns
        .iter()
        .any(|p| matches!(p, StructurePattern::Block { .. }));
    assert!(has_block);

    // Test chain structure recognition
    let mut chain_qubo = Array2::zeros((5, 5));
    for i in 0..4 {
        chain_qubo[[i, i + 1]] = -1.0;
        chain_qubo[[i + 1, i]] = -1.0;
    }

    let chain_patterns = recognizer.recognize_structure(&chain_qubo).unwrap();
    let has_chain = chain_patterns
        .iter()
        .any(|p| matches!(p, StructurePattern::Chain { .. }));
    assert!(has_chain);

    println!("Structure recognition tests passed");
}

#[test]
fn test_algorithm_selection() {
    let mut config = AIOptimizerConfig::default();
    let mut selector = AutomatedAlgorithmSelector::new(&config);

    // Test selection for different problem characteristics

    // Small problem
    let mut small_features = Array1::from(vec![50.0, 0.0, 1.0, 1.0, 0.3, 5.0, 1.0, 3.0, 1.0, 0.5]);
    let small_patterns = vec![];
    let small_alg = selector
        .select_algorithm(&small_features, &small_patterns)
        .unwrap();
    assert_eq!(small_alg, "BranchAndBound"); // Should prefer exact methods for small problems

    // Dense problem
    let mut dense_features = Array1::from(vec![200.0, 0.0, 1.0, 1.0, 0.9, 5.0, 1.0, 8.0, 1.0, 2.0]);
    let dense_patterns = vec![];
    let dense_alg = selector
        .select_algorithm(&dense_features, &dense_patterns)
        .unwrap();
    assert_eq!(dense_alg, "SimulatedAnnealing"); // Should prefer SA for dense problems

    // Tree-structured problem
    let mut tree_features = Array1::from(vec![100.0, 0.0, 1.0, 1.0, 0.2, 3.0, 1.0, 2.5, 1.0, 1.0]);
    let tree_patterns = vec![StructurePattern::Tree {
        depth: 5,
        branching_factor: 2.0,
    }];
    let tree_alg = selector
        .select_algorithm(&tree_features, &tree_patterns)
        .unwrap();
    assert_eq!(tree_alg, "DynamicProgramming"); // Should prefer DP for tree structures

    // Large general problem
    let mut large_features =
        Array1::from(vec![1000.0, 0.0, 2.0, 1.5, 0.4, 8.0, 2.0, 6.0, 1.0, 3.0]);
    let large_patterns = vec![];
    let large_alg = selector
        .select_algorithm(&large_features, &large_patterns)
        .unwrap();
    assert_eq!(large_alg, "GeneticAlgorithm"); // Should prefer GA for large problems

    println!("Algorithm selection tests passed");
}

#[test]
fn test_parameter_optimization() {
    let mut config = AIOptimizerConfig::default();
    let mut optimizer_net = ParameterOptimizationNetwork::new(&config);

    // Test parameter optimization for different algorithms
    let mut features = Array1::from(vec![100.0, 0.0, 1.0, 1.0, 0.5, 5.0, 1.0, 4.0, 1.0, 1.5]);

    // Test Simulated Annealing parameters
    let sa_params = optimizer_net
        .optimize_parameters(&features, "SimulatedAnnealing", Some(0.9))
        .unwrap();
    assert!(sa_params.contains_key("initial_temperature"));
    assert!(sa_params.contains_key("cooling_rate"));
    assert!(sa_params.contains_key("min_temperature"));

    let initial_temp = sa_params["initial_temperature"];
    let cooling_rate = sa_params["cooling_rate"];
    let min_temp = sa_params["min_temperature"];

    assert!(initial_temp > 0.0);
    assert!(cooling_rate > 0.0 && cooling_rate < 1.0);
    assert!(min_temp > 0.0 && min_temp < initial_temp);

    // Test Genetic Algorithm parameters
    let ga_params = optimizer_net
        .optimize_parameters(&features, "GeneticAlgorithm", Some(0.85))
        .unwrap();
    assert!(ga_params.contains_key("population_size"));
    assert!(ga_params.contains_key("mutation_rate"));
    assert!(ga_params.contains_key("crossover_rate"));

    let pop_size = ga_params["population_size"];
    let mutation_rate = ga_params["mutation_rate"];
    let crossover_rate = ga_params["crossover_rate"];

    assert!(pop_size > 0.0);
    assert!((0.0..=1.0).contains(&mutation_rate));
    assert!((0.0..=1.0).contains(&crossover_rate));

    println!("Parameter optimization tests passed");
}

#[test]
fn test_quality_prediction() {
    let mut config = AIOptimizerConfig::default();
    let mut predictor = SolutionQualityPredictor::new(&config);

    // Test quality prediction for different scenarios
    let mut features = Array1::from(vec![50.0, 0.0, 1.0, 1.0, 0.3, 3.0, 1.0, 2.5, 1.0, 1.0]);
    let mut params = HashMap::new();

    // Test predictions for different algorithms
    let algorithms = vec!["SimulatedAnnealing", "GeneticAlgorithm", "TabuSearch"];

    for algorithm in algorithms {
        let prediction = predictor
            .predict_quality(&features, algorithm, &params)
            .unwrap();

        // Verify prediction structure
        assert!(prediction.expected_quality >= 0.0);
        assert!(prediction.expected_quality <= 1.0);
        assert!(prediction.confidence_interval.0 <= prediction.expected_quality);
        assert!(prediction.expected_quality <= prediction.confidence_interval.1);
        assert!(prediction.optimal_probability >= 0.0);
        assert!(prediction.optimal_probability <= 1.0);
        assert!(prediction.expected_convergence_time > Duration::ZERO);

        println!(
            "Algorithm {}: quality = {:.3}, confidence interval = [{:.3}, {:.3}]",
            algorithm,
            prediction.expected_quality,
            prediction.confidence_interval.0,
            prediction.confidence_interval.1
        );
    }

    println!("Quality prediction tests passed");
}

#[test]
fn test_difficulty_assessment() {
    let mut config = AIOptimizerConfig::default();
    let mut optimizer = AIAssistedOptimizer::new(config);

    // Test different difficulty levels

    // Easy problem (small, sparse)
    let mut easy_qubo = Array2::zeros((10, 10));
    easy_qubo[[0, 1]] = 1.0;
    easy_qubo[[1, 0]] = 1.0;

    let easy_features = optimizer.extract_problem_features(&easy_qubo).unwrap();
    let easy_patterns = vec![];
    let easy_assessment = optimizer
        .assess_difficulty(&easy_qubo, &easy_features, &easy_patterns)
        .unwrap();

    assert!(easy_assessment.difficulty_score < 0.5);
    assert!(!easy_assessment.recommended_resources.gpu_recommended);
    assert!(
        !easy_assessment
            .recommended_resources
            .distributed_recommended
    );

    // Hard problem (large, dense, high frustration)
    let size = 100;
    let mut hard_qubo = Array2::ones((size, size));
    for i in 0..size {
        hard_qubo[[i, i]] = 0.0; // Remove self-loops
    }

    let hard_features = optimizer.extract_problem_features(&hard_qubo).unwrap();
    let hard_patterns = vec![];
    let hard_assessment = optimizer
        .assess_difficulty(&hard_qubo, &hard_features, &hard_patterns)
        .unwrap();

    assert!(hard_assessment.difficulty_score > 0.5);
    assert!(hard_assessment.recommended_resources.cpu_cores >= 4);
    assert!(hard_assessment.expected_solution_time > Duration::from_secs(1));

    println!("Difficulty assessment tests passed");
    println!(
        "Easy problem difficulty: {:.3}",
        easy_assessment.difficulty_score
    );
    println!(
        "Hard problem difficulty: {:.3}",
        hard_assessment.difficulty_score
    );
}

#[test]
fn test_training_components() {
    let config = AIOptimizerConfig {
        max_training_iterations: 10, // Reduced for testing
        batch_size: 8,
        ..Default::default()
    };
    let mut optimizer = AIAssistedOptimizer::new(config);

    // Create mock training data
    let mut training_data = Vec::new();

    for i in 0..20 {
        let size = 10 + i * 5;
        let mut features = Array1::from(vec![
            size as f64,
            0.0,
            1.0,
            1.0,
            0.3,
            3.0,
            1.0,
            2.5,
            1.0,
            1.0,
        ]);

        let mut algorithm_scores = HashMap::new();
        algorithm_scores.insert("SimulatedAnnealing".to_string(), 0.8);
        algorithm_scores.insert("GeneticAlgorithm".to_string(), 0.75);
        algorithm_scores.insert("TabuSearch".to_string(), 0.85);

        training_data.push(TrainingExample {
            features,
            optimal_algorithm: "TabuSearch".to_string(),
            algorithm_scores,
            metadata: ProblemMetadata {
                problem_type: "Test".to_string(),
                size,
                density: 0.3,
                source: "Generated".to_string(),
                difficulty_level: DifficultyLevel::Medium,
            },
        });
    }

    // Test training
    let training_results = optimizer.train(&training_data, 0.2);
    assert!(training_results.is_ok());

    let results = training_results.unwrap();

    // Verify training results structure
    if let Some(param_results) = results.parameter_optimizer_results {
        assert!(param_results.final_loss >= 0.0);
        assert!(param_results.training_time > Duration::ZERO);
    }

    if let Some(selector_results) = results.algorithm_selector_results {
        assert!(selector_results.accuracy >= 0.0);
        assert!(selector_results.accuracy <= 1.0);
        assert!(!selector_results.cross_validation_scores.is_empty());
    }

    if let Some(predictor_results) = results.quality_predictor_results {
        assert!(predictor_results.r2_score >= -1.0); // R² can be negative for bad models
        assert!(predictor_results.mae >= 0.0);
        assert!(predictor_results.rmse >= 0.0);
    }

    println!("Training components tests passed");
}

#[test]
fn test_comprehensive_optimization_workflow() {
    let config = AIOptimizerConfig {
        parameter_optimization_enabled: true,
        reinforcement_learning_enabled: false, // Skip RL for faster testing
        auto_algorithm_selection_enabled: true,
        structure_recognition_enabled: true,
        quality_prediction_enabled: true,
        learning_rate: 0.01,
        batch_size: 16,
        max_training_iterations: 50,
        convergence_threshold: 1e-4,
        replay_buffer_size: 1000,
    };

    let mut optimizer = AIAssistedOptimizer::new(config);

    // Test workflow on different problem types
    let test_cases = vec![
        ("Small Dense", create_small_dense_qubo()),
        ("Medium Sparse", create_medium_sparse_qubo()),
        ("Large Structured", create_large_structured_qubo()),
    ];

    for (name, qubo) in test_cases {
        println!("\nTesting workflow on: {name}");

        let result = optimizer.optimize(&qubo, Some(0.8), Some(Duration::from_secs(60)));
        assert!(result.is_ok(), "Optimization failed for {name}");

        let optimization_result = result.unwrap();

        // Verify comprehensive results
        assert!(!optimization_result.recommended_algorithm.is_empty());
        assert!(optimization_result.confidence > 0.0);
        assert!(optimization_result.predicted_quality.expected_quality > 0.0);
        assert!(!optimization_result.alternatives.is_empty());

        // Verify problem type inference
        assert!(!optimization_result.problem_info.problem_type.is_empty());

        // Verify difficulty assessment is reasonable
        let difficulty = &optimization_result.problem_info.difficulty_assessment;
        assert!(difficulty.difficulty_score >= 0.0);
        assert!(difficulty.difficulty_score <= 1.0);
        assert!(difficulty.expected_solution_time > Duration::ZERO);

        println!("  Algorithm: {}", optimization_result.recommended_algorithm);
        println!("  Confidence: {:.3}", optimization_result.confidence);
        println!(
            "  Predicted quality: {:.3}",
            optimization_result.predicted_quality.expected_quality
        );
        println!("  Difficulty: {:.3}", difficulty.difficulty_score);
        println!(
            "  Problem type: {}",
            optimization_result.problem_info.problem_type
        );
    }

    println!("\nComprehensive optimization workflow tests passed");
}

#[test]
fn test_reinforcement_learning_components() {
    let mut config = AIOptimizerConfig::default();
    let mut rl_agent = SamplingStrategyAgent::new(&config);

    // Test basic RL agent structure
    assert_eq!(rl_agent.q_network().state_encoder.embedding_dim, 64);
    assert_eq!(rl_agent.q_network().action_decoder.action_dim, 10);
    assert_eq!(rl_agent.replay_buffer().max_size, config.replay_buffer_size);
    assert_eq!(rl_agent.training_stats().episodes, 0);

    // Test training (simplified)
    let training_data = vec![]; // Empty for this test
    let rl_results = rl_agent.train(&training_data);
    assert!(rl_results.is_ok());

    let results = rl_results.unwrap();
    assert!(results.episodes > 0);
    assert!(results.total_steps > 0);
    assert!(!results.loss_history.is_empty());

    println!("Reinforcement learning components tests passed");
}

#[test]
fn test_activation_functions() {
    use ActivationFunction::*;

    // Test that activation functions can be created and used
    let activations = vec![
        ReLU,
        Tanh,
        Sigmoid,
        LeakyReLU { alpha: 0.01 },
        ELU { alpha: 1.0 },
        Swish,
    ];

    for activation in activations {
        // Just test that they can be created and cloned
        let _cloned = activation.clone();
        println!("Activation function: {activation:?}");
    }

    println!("Activation functions tests passed");
}

#[test]
fn test_ensemble_methods() {
    use EnsembleMethod::*;

    // Test different ensemble methods
    let methods = vec![Voting, Bagging, Boosting, WeightedAverage, DynamicSelection];

    for method in methods {
        let _cloned = method.clone();
        println!("Ensemble method: {method:?}");
    }

    // Test stacking with nested box
    let stacking = Stacking {
        meta_learner: Box::new(RegressionModel::LinearRegression),
    };
    let _cloned = stacking;

    println!("Ensemble methods tests passed");
}

// Helper functions to create test problems

fn create_small_dense_qubo() -> Array2<f64> {
    let mut qubo = Array2::ones((8, 8));
    for i in 0..8 {
        qubo[[i, i]] = -2.0; // Encourage variables to be 1
    }
    qubo
}

fn create_medium_sparse_qubo() -> Array2<f64> {
    let mut qubo = Array2::zeros((20, 20));

    // Add sparse connections
    for i in 0..19 {
        qubo[[i, i + 1]] = -1.0;
        qubo[[i + 1, i]] = -1.0;
    }

    // Add some random connections
    qubo[[0, 10]] = 0.5;
    qubo[[10, 0]] = 0.5;
    qubo[[5, 15]] = -0.5;
    qubo[[15, 5]] = -0.5;

    qubo
}

fn create_large_structured_qubo() -> Array2<f64> {
    let size = 50;
    let mut qubo = Array2::zeros((size, size));

    // Create block structure with 5 blocks of 10 variables each
    for block in 0..5 {
        let start = block * 10;
        let end = (block + 1) * 10;

        for i in start..end {
            for j in start..end {
                if i != j {
                    qubo[[i, j]] = -1.0; // Strong intra-block coupling
                }
            }
        }
    }

    // Add weak inter-block coupling
    for i in 0..size - 1 {
        if i % 10 == 9 {
            // Connect last element of each block to first of next
            qubo[[i, i + 1]] = 0.1;
            qubo[[i + 1, i]] = 0.1;
        }
    }

    qubo
}