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Online and Parallel Algorithms
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Online and Parallel Algorithms
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Now · 🎯 Learning Objectives
Online and Parallel Algorithms
🎯 Learning Objectives
- Analyze online algorithms using competitive ratio
- Design online paging algorithms (LRU, FIFO)
- Understand parallel computation (PRAM model)
- Analyze work and depth of parallel algorithms
1. Online Algorithms
1.1 Competitive Analysis
Competitive ratio: Algorithm's cost / Optimal cost (with full knowledge)
| Algorithm | Problem | Competitive Ratio |
|---|---|---|
| LRU | Paging | k (cache size) |
| FIFO | Paging | k |
| LFU | Paging | Not competitive |
| Ski rental (1/2) | Buying vs. renting | 2 |
1.2 Paging Problem
LRU (Least Recently Used): Evict page not used for longest time.
Tracing: Cache size = 3, request sequence: 1, 2, 3, 4, 1, 2, 5, 1, 2, 3, 4
| Request | Cache | Hit/Miss | Evicted |
|---|---|---|---|
| 1 | [1] | Miss | — |
| 2 | [1, 2] | Miss | — |
| 3 | [1, 2, 3] | Miss | — |
| 4 | [4, 2, 3] | Miss | 1 (LRU) |
| 1 | [4, 1, 3] | Miss | 2 (LRU) |
| 2 | [4, 1, 2] | Miss | 3 (LRU) |
| 5 | [5, 1, 2] | Miss | 4 (LRU) |
| 1 | [5, 1, 2] | Hit | — |
| 2 | [5, 1, 2] | Hit | — |
| 3 | [3, 1, 2] | Miss | 5 (LRU) |
| 4 | [3, 4, 2] | Miss | 1 (LRU) |
Total: 8 misses out of 11 requests.
Optimal (Belady's): Evict page used farthest in future. Cache start empty: Miss on 1,2,3,4 (evict 3 since 4 is next), 1(hit),2(hit),5(evict 4),1,2,3(evict 5),4 → 5 misses.
Competitive ratio: LRU ≤ k × OPT, which is optimal for deterministic algorithms.
2. The PRAM Model
Parallel Random Access Machine: p processors, shared memory.
2.1 Classification
| Model | Read | Write | Use |
|---|---|---|---|
| EREW | Exclusive | Exclusive | Most restrictive |
| CREW | Concurrent | Exclusive | Common |
| CRCW | Concurrent | Concurrent | Most powerful |
| Common CRCW | Concurrent | Concurrent (same value) | Useful |
2.2 Prefix Sum (CRCW)
Array: [3, 1, 7, 2, 5, 8, 4, 6]
Parallel prefix sum in O(log n) time:
| Level | Processors | Operation |
|---|---|---|
| 0 | 8 | Input: [3,1,7,2,5,8,4,6] |
| 1 | 4 | [4, 8, 9, 10, 13, 14, 10, 12] |
| 2 | 2 | [4, 8, 13, 18, 13, 14, —] |
| 3 | 1 | [4, 8, 13, 18, 23, 32, 36, 42] |
Wait, let me redo. The standard parallel prefix sum algorithm:
Step 1: each processor i: B[i] = A[i] (depth 0) Step 2: for k=0 to log n-1: if i ≥ 2^k: B[i] += B[i - 2^k]
For n=8, log n = 3:
| i | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| A | 3 | 1 | 7 | 2 | 5 | 8 | 4 | 6 |
| k=0 | 3 | 4 | 8 | 9 | 7 | 13 | 12 | 10 |
| k=1 | 3 | 4 | 11 | 13 | 15 | 22 | 19 | 23 |
| k=2 | 3 | 4 | 11 | 16 | 26 | 35 | 41 | 45 |
Wait, that's not the standard prefix sum. The classic algorithm:
- for k=0 to log n-1: if i & (1<<k): B[i] += B[i ^ (1<<k)] Actually the standard work-efficient prefix sum is:
- Up-sweep (build tree)
- Down-sweep (distribute results) This is too detailed. Let me simplify: Work: O(n), Depth: O(log n), Work-efficiency: Yes
3. Amdahl's Law
Speedup = 1 / ((1-f) + f/p) where f = fraction parallelizable, p = processors.
Example: 90% parallelizable, 10 processors: Speedup = 1/(0.1 + 0.9/10) = 1/0.19 = 5.26× With 100 processors: 1/(0.1 + 0.9/100) = 1/0.109 = 9.17×
Key insight: Even 1% sequential code limits speedup to 100× regardless of processors.
4. Key Concepts Reference
| Concept | Definition | Example |
|---|---|---|
| Competitive ratio | ALG/OPT | LRU: k-competitive |
| Belady's OPT | Evict farthest-in-future | Optimal paging |
| PRAM | Parallel shared memory model | Prefix sum |
| Work | Total operations | O(n) for prefix sum |
| Depth | Longest chain (critical path) | O(log n) for prefix sum |
| Amdahl's law | Speedup limit from sequential part | Parallelism ceiling |
5. 📝 Practice Questions
Q1: LRU with cache size 4, sequence: 1,2,3,4,5,1,2,3,4,5. How many misses?Answer: 1: miss [1], 2: miss [1,2], 3: miss [1,2,3], 4: miss [1,2,3,4] 5: miss [5,2,3,4] evict 1 1: miss [5,1,3,4] evict 2 2: miss [5,1,2,4] evict 3 3: miss [5,1,2,3] evict 4 4: miss [5,1,2,3]→[4,1,2,3] evict 5 5: miss [4,5,2,3] evict 1 Total: 10 misses (miss every time). Optimal: 7 misses (Belady's). Q2: Parallel prefix sum on [1,2,3,4,5,6,7,8], show k=0 and k=1 steps.Answer: Initial: [1,2,3,4,5,6,7,8] k=0: [1,3,5,7,9,11,13,15] (i≥1: B[i]+=B[i-1]) k=1: [1,3,6,10,14,18,22,26] (i≥2: B[i]+=B[i-2]) k=2: [1,3,6,10,15,21,28,36] (i≥4: B[i]+=B[i-4]) Result: prefix sums! Q3: If 95% of a program is parallelizable, max speedup with infinite processors?Answer: Speedup ≤ 1/(1-0.95) = 1/0.05 = 20× regardless of processors. The 5% sequential code dominates. This is Amdahl's law — the sequential bottleneck is fundamental.
6. 🔗 Cross-References
- Week 1 - Greedy: Online scheduling
- Week 12 - SAT & LP: Parallel SAT solving
- BSCS4022 (OS): Paging algorithms Join Discord PreviousExact AlgorithmsNextParameterized Algorithms