Deadlocks Andy Wang Operating Systems COP 4610 CGS
- Slides: 74
Deadlocks Andy Wang Operating Systems COP 4610 / CGS 5765
Deadlocks ¡ Deadlocks: Occur when threads are waiting for resources with circular dependencies l Often involve nonpreemptable resources, which cannot be taken away from its current thread without failing the computation (e. g. , storage allocated to a file)
Deadlocks that involve preemptable resources (e. g. , CPU) can usually be resolved by reallocation of resources ¡ Starvation: a thread waits indefinitely ¡ A deadlock implies starvation ¡
An Example of Deadlocks Thread A Thread B P(x); P(y); P(x); ¡ A deadlock won’t always happen with this code, but it might
Deadlocks, Everywhere… ¡ Can happen with any kind of resource l ¡ Among multiple resources Cannot be resolved for each resource independently l l l A thread can grab all the memory The other grabs all the disk space Each thread may need to wait for the other to release
Deadlocks, Everywhere… Round-Robin CPU scheduling cannot prevent deadlocks (or starvation) from happening ¡ Can occur whenever there is waiting… ¡
A Classic Example of Deadlocks Dinning lawyers (philosophers) ¡ Each needs two chopsticks to eat ¡
Dining Lawyers ¡ If each first grabs the chopstick on their right before the on their left, and all grab at the same time, we have a deadlock (Personally, I prefer to starve than share chopsticks…)
A Dining Lawyer Implementation semaphore chopstick[5] = {1, 1, 1}; lawyer(int j) { while (TRUE) { P(chopstick[j]); P(chopstick[(j + 1) % 5]; // eat V(chopstick[(j + 1) % 5]; V(chopstick[j]); } }
A Dining Lawyer Implementation // chopstick[5] = {0, 0, 0}; lawyer(int j) { while (TRUE) { P(chopstick[j]); P(chopstick[(j + 1) % 5]; // eat V(chopstick[(j + 1) % 5]; V(chopstick[j]); } }
A Dining Lawyer Implementation // chopstick[5] = {0, 0, 0}; lawyer(int j) { while (TRUE) { P(chopstick[j]); P(chopstick[(j + 1) % 5]; // eat V(chopstick[(j + 1) % 5]; V(chopstick[j]); } }
Conditions for Deadlocks ¡ Four necessary (but not sufficient) conditions l l Limited access (lock-protected resources) No preemption (if someone has the resource, it cannot be taken away) Wait while holding (holding a resource while requesting and waiting for the next resource) Circular chain of requests
Deadlock Prevention Techniques ¡ When encountering a deadlock l ¡ All four conditions must be true To prevent deadlocks l Remove one of the four conditions
Deadlock Prevention Techniques 1. Infinite resources (buy a very large disk)
Deadlock Prevention Techniques 2. No sharing (independent threads)
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) easier said than done…
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) semaphore chopstick[5] = {1, 1, 1}, s = 1; lawyer(int j) { while (TRUE) { P(s); P(chopstick[j]); P(chopstick[(j + 1) % 5]; // eat V(chopstick[(j + 1) % 5]; V(chopstick[j]); V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) 1 semaphore chopstick[5] = {1, 1 0, 1, 1}, s = 1 0; lawyer(int j) { while (TRUE) { P(s); P(chopstick[j]); P(chopstick[(j + 1) % 5]; // eat V(chopstick[(j + 1) % 5]; V(chopstick[j]); V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) 3 1 // chopstick[5] = {1, 0, 0, 1, 1}, s = 0; lawyer(int j) { while (TRUE) { P(s); P(chopstick[j]); P(chopstick[(j + 1) % 5]; // eat V(chopstick[(j + 1) % 5]; V(chopstick[j]); V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) // chopstick[5] = {1, 1, 1}, s = 1; lawyer(int j) { while (TRUE) { P(s); P(chopstick[j]); P(chopstick[(j + 1) % 5]; V(s); // eat P(s); V(chopstick[(j + 1) % 5]; V(chopstick[j]); V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) 1 // chopstick[5] = {1, 1 0, 1, 1}, s = 1 0 1; lawyer(int j) { while (TRUE) { P(s); P(chopstick[j]); P(chopstick[(j + 1) % 5]; V(s); // eat P(s); V(chopstick[(j + 1) % 5]; V(chopstick[j]); V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) 1 3 // chopstick[5] = {1, 0, 0, 1 0}, s = 1 0 1; lawyer(int j) { while (TRUE) { P(s); P(chopstick[j]); P(chopstick[(j + 1) % 5]; V(s); // eat P(s); V(chopstick[(j + 1) % 5]; V(chopstick[j]); V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) 1 3 // chopstick[5] = {1, 0, 0}, s = 1 0; lawyer(int j) { while (TRUE) { 2 P(s); P(chopstick[j]); // deadlock P(chopstick[(j + 1) % 5]; V(s); // eat P(s); V(chopstick[(j + 1) % 5]; V(chopstick[j]); V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) 1 3 // chopstick[5] = {1, 0, 0}, s = 0; lawyer(int j) { while (TRUE) { 2 P(s); P(chopstick[j]); P(chopstick[(j + 1) % 5]; V(s); // eat P(s); // deadlock V(chopstick[(j + 1) % 5]; V(chopstick[j]); V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) int counter[5] = {1, 1, 1}; semaphore chopstick[5] = {1, 1, 1}, s = 1; lawyer(int j) { while (TRUE) { P(s); // if both counters j and (j + 1) % 5 > 0, decrement counters // and grab chopstick[j] and chopstick[(j + 1) % 5] V(s); // if holding both chopsticks, eat P(s); // release chopsticks and increment counters as needed V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) // counter[5] = {1, 0, 0, 1, 1}; chopstick[5] = {1, 0, 0, 1, 1}, s = 1; lawyer(int j) { while (TRUE) { P(s); 1 // if both counters j and (j + 1) % 5 > 0, decrement counters // and grab chopstick[j] and chopstick[(j + 1) % 5] V(s); // if holding both chopsticks, eat (1, 2) P(s); // release chopsticks and increment counters as needed V(s); } }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) // counter[5] = {1, 0, 0, 1, 1}; chopstick[5] = {1, 0, 0, 1, 1}, s = 1; lawyer(int j) { while (TRUE) { P(s); 1 // if both counters j and (j + 1) % 5 > 0, decrement counters // and grab chopstick[j] and chopstick[(j + 1) % 5] V(s); // if holding both chopsticks, eat (1, 2) 2 P(s); // release chopsticks and increment counters as needed V(s); } () }
Deadlock Prevention Techniques 3. Allocate all resources at the beginning (if you need 2 chopsticks, grab both at the same time) // counter[5] = {1, 0, 0}; chopstick[5] = {1, 0, 0}, s = 1; lawyer(int j) { while (TRUE) { P(s); 1 3 // if both counters j and (j + 1) % 5 > 0, decrement counters // and grab chopstick[j] and chopstick[(j + 1) % 5] V(s); // if holding both chopsticks, eat (1, 2) (3, 4) P(s); // release chopsticks and increment counters as needed V(s); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {1, 1, 1}; lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {1 0, 1, 1}; 0 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 1 0, 1, 1, 1}; 01 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1, 1}; 012 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0, 1}; 0123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1}; 01234 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1}; 1234 0 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1}; 234 01 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1}; 34 012 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1 0}; 4 0123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0}; 4 012 3 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0 1}; 4 012 3 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0, 0 1, 1}; 4 012 3 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0, 1}; 4 012 3 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1}; 4 01 2 3 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0, 0 1, 1}; 4 01 2 3 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0 1, 1, 1}; 4 01 23 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1, 1}; 4 01 23 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0, 1, 1}; 4 0 1 23 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 0 1, 1, 1}; 4 0 1 23 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0 1, 1, 1, 1}; 4 0 123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 1 0, 1, 1, 1}; 4 0 123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0, 1, 1, 1}; 4 0 123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 0 1, 1, 1, 1}; 4 0 123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0 1, 1, 1}; 4 0123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {1 0, 1, 1}; 4 0123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
Deadlock Prevention Techniques 4. Make everyone use the same ordering in accessing resource (All threads must call P(x) before P(y) // chopstick[5] = {0, 1, 1 0}; 4 0123 lawyer(int j) { while (TRUE) { P(chopstick[min(j, (j + 1) % 5)]); P(chopstick[max(j, (j + 1) % 5)]); // eat V(chopstick[max(j, (j + 1) % 5)]); V(chopstick[min(j, (j + 1) % 5)]); } }
More Deadlock Prevention Methods 5. No waiting (phone company) 6. Preempt resources (copying memory content to disk) 7. Banker’s algorithm 8. A combination of techniques
Banker’s Algorithm ¡ The idea of Banker’s algorithm: l l Allows the sum of requested resources > total resources As long as, there is some way for all threads to finish without getting into any deadlocks
Banker’s Algorithm ¡ Banker’s algorithm: l l l A thread states its maximum resource needs in advance The OS allocates resource dynamically as needed. A thread waits if granting its request would lead to deadlocks A request can be granted if some sequential ordering of threads is deadlock free
Example 1 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 30 MB P 2 10 MB P 3 120 MB 80 MB free RAM (MB) 46 time
Example 1 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 30 MB P 2 20 MB P 3 120 MB 80 MB free RAM (MB) 46 36 P 2 time
Example 1 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 30 MB P 2 0 MB P 3 120 MB 80 MB free RAM (MB) 46 36 P 2 56 time
Example 1 ¡ Total RAM 256 MB Allocated Still needed P 1 110 MB P 2 0 MB P 3 120 MB 80 MB free RAM (MB) 46 36 P 2 56 26 P 1 time
Example 1 ¡ Total RAM 256 MB Allocated Still needed P 1 0 MB P 2 0 MB P 3 120 MB 80 MB free RAM (MB) 136 46 36 P 2 56 26 P 1 time
Example 1 ¡ Total RAM 256 MB Allocated Still needed P 1 0 MB P 2 0 MB P 3 200 MB free RAM (MB) 136 46 36 P 2 56 26 P 1 56 P 3 time
Example 1 ¡ Total RAM 256 MB Allocated Still needed P 1 0 MB P 2 0 MB P 3 0 MB 256 free RAM (MB) 136 46 36 P 2 56 26 P 1 56 P 3 time
Example 1 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 30 MB P 2 10 MB P 3 120 MB 80 MB 256 free RAM (MB) 56 The system is initially in a safe state, since we can run P 2, P 1, and then P 3 time 136 46 36 P 2 56 26 P 1
Example 2 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 60 MB P 2 10 MB P 3 120 MB 80 MB free RAM (MB) 46 time
Example 2 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 60 MB P 2 20 MB P 3 120 MB 80 MB free RAM (MB) 46 36 P 2 time
Example 2 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 60 MB P 2 0 MB P 3 120 MB 80 MB free RAM (MB) 46 36 P 2 56 time
Example 2 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 60 MB P 2 0 MB P 3 120 MB 80 MB free RAM (MB) 46 36 P 2 56 time
Example 2 ¡ Total RAM 256 MB Allocated Still needed P 1 80 MB 60 MB P 2 10 MB P 3 120 MB 80 MB 46 The system is initially in an unsafe state, since we cannot find an execution sequence for P 1, P 2, and P 3 free RAM (MB) 36 P 2 56 time
Deadlock Detection and Recovery Scan the resource allocation graph ¡ Detect circular chains of requests ¡ Recover from the deadlock ¡
Resource Allocation Graph waiting for Thread A Resource Y Resource X owned by Thread B waiting for
Once A Cycle is Detected… ¡ Some possible actions l Kill a thread and force it to give up resources ¡ l Remaining system may be in an inconsistent state Rollback actions of a deadlocked thread Not always possible (a file maybe halfway through modifications) ¡ Need checkpointing, or taking snapshots of system states from time to time ¡
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