Resource-Allocation Graph

 

Resource-Allocation Graph

Deadlocks can be precisely described and analyzed using a resource-allocation graph. This graph provides a visual and formal way to represent how threads and resources interact in a system and helps determine whether a deadlock is possible or has occurred.

1. What Is a Resource-Allocation Graph?

A resource-allocation graph is a directed graph defined as:

  • G = (V, E)

    • V: Set of vertices (nodes)

    • E: Set of directed edges

2. Types of Nodes

The set of vertices V is divided into two types:

1. Thread Nodes (T)

T={T1,T2,…,Tn}T = \{T_1, T_2, \dots, T_n\}

  • Represent active threads in the system

  • Drawn as circles

2. Resource Nodes (R)

R={R1,R2,…,Rm}R = \{R_1, R_2, \dots, R_m\}

  • Represent resource types

  • Drawn as rectangles

  • Each instance of a resource type is shown as a dot inside the rectangle

3. Types of Edges

Request Edge (Ti → Rj)

  • Indicates that thread Ti has requested resource Rj

  • Thread is waiting for that resource

  • Drawn from a thread node to a resource node

Assignment Edge (Rj → Ti)

  • Indicates that resource Rj has been allocated to thread Ti

  • Drawn from a specific resource instance (dot) to a thread node




4. Graph Evolution During Execution

  1. When a thread requests a resource:

    • A request edge is added

  2. When the resource is granted:

    • The request edge is replaced by an assignment edge

  3. When the thread releases the resource:

    • The assignment edge is removed

5. Interpreting Thread States from the Graph

From a resource-allocation graph, we can directly infer:

  • Which resources a thread holds

  • Which resources a thread is waiting for

  • Whether deadlock is possible

6. Cycles and Deadlock

Case 1: No Cycle in the Graph

✔ No deadlock exists

If the resource-allocation graph has no cycles, then no thread in the system is deadlocked.

Case 2: Cycle Exists in the Graph

A cycle indicates potential deadlock, but the outcome depends on the number of resource instances.

7. Single Instance per Resource Type

If each resource type has exactly one instance:

  • A cycle is necessary and sufficient for deadlock

  • Every thread in the cycle is deadlocked

✔ Cycle ⇔ Deadlock

8. Multiple Instances per Resource Type

If some resource types have multiple instances:

  • A cycle is necessary but not sufficient for deadlock

  • Deadlock may or may not exist

This is because:

  • Another thread may release a resource

  • The cycle can be broken dynamically

9. Example: Deadlock with Multiple Instances

When thread T3 requests resource R2 and no instances are available:

  • A new request edge is added

  • Multiple cycles appear

  • All involved threads are deadlocked

Each thread is waiting for a resource held by another thread in the cycle.




10. Example: Cycle Without Deadlock

A cycle may exist, but:

  • Another thread (e.g., T4) can release a resource

  • That resource breaks the cycle

  • System continues normally

✔ Cycle ≠ Deadlock (always)




11. Key Observations 

ConditionMeaning
No cycle    Deadlock impossible
Cycle + single instance    Deadlock guaranteed
Cycle + multiple instances    Deadlock possible

12. Why Resource-Allocation Graphs Matter

They help us:

  • Understand how deadlocks form

  • Visually detect deadlock conditions

  • Form the basis for:

    • Deadlock detection algorithms

    • Deadlock avoidance techniques

 Summary 

A resource-allocation graph models threads and resources using directed edges.
If the graph has no cycles, the system is not deadlocked.
If a cycle exists, deadlock may occur—definitely so when each resource type has only one instance.

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