TAOCP 2.2.3 Exercise 24

Program `T` in Section 2.

taocpmathematicsalgorithmsvolume-1medium

Section 2.2.3: Linked Allocation

Exercise 24. [24] Incorporate the extensions of Algorithm T made in exercise 23 into Program T.

Verified: no
Solve time: 38s

Solution

Program T in Section 2.2.2 and its extension in exercise 23 are designed to perform a topological sort and to detect loops in the input. Exercise 23 modifies T to print one of the loops when a cycle is detected. Exercise 24 asks to incorporate this extension into Program T, preserving the original variable names and structure as much as possible.

We begin with the structure of the original Program T:

  1. Input relations are read and stored in arrays X (counts of predecessors) and Y (lists of successors).
  2. Nodes with zero predecessors are placed on a list of available nodes.
  3. A loop repeatedly removes a node from the available list, outputs it, and updates successor counts.
  4. Step T8 halts if no node with zero predecessors remains but some nodes are still unprocessed, indicating a loop.

To extend Program T as in exercise 23, we need the following additions:

  • When a loop is detected at step T8, we must traverse the remaining nodes to extract a cycle.
  • Since the proof in the text shows that at least one node j in the remaining graph has X[j] > 0 and is reachable from some successor chain, we can reconstruct a loop by following Y links starting from j until we return to j.
  • We introduce one auxiliary variable, J, to traverse the loop.

The incorporation into Program T is as follows:

  1. Maintain the original AVAIL stack and successor lists. No change is needed to the data structures.
  2. Detect a loop: When the algorithm reaches the condition AVAIL = Λ but unprocessed nodes remain (N > 0), proceed to print a cycle.
  3. Select starting node: Choose the first node J with X[J] > 0.
  4. Traverse the loop: Set P \Leftarrow J. Repeat P \Leftarrow Y(P) until P = J again, printing each visited node.
  5. Return control: After printing, signal OVERFLOW or terminate.

Formally, the modified procedure for the loop detection step becomes:

$$ \text{If } AVAIL = \Lambda \text{ and } N > 0, \text{ then} $$

$$ \quad J \leftarrow \min{j \mid X[j] > 0}, $$

$$ \quad P \leftarrow J, $$

$$ \quad \text{repeat: print } P, \quad P \leftarrow Y(P), \text{ until } P = J, $$

$$ \quad \text{signal OVERFLOW.} $$

The rest of Program T remains unchanged: nodes are removed from AVAIL, their successors are updated, and the sorted order is output. The auxiliary variable J and the traversal pointer P are only used in the exceptional loop-detection case, minimizing disruption of the original program structure.

Implementation notes in MIXAL-style pseudocode:

...
T8: J1Z  LOOP_DETECT   ; AVAIL = Λ
...
LOOP_DETECT:
LDJ   FIRST_LOOP_NODE ; find first j with X[j] > 0
ST1   J
LD1   J               ; P ← J
PRINT_LOOP:
LDA   0,1(P)
OUT                    ; print P (or node label)
LD1   0,1(Y(P))       ; P ← Y(P)
CMP1  J
JNE   PRINT_LOOP
JMP   OVERFLOW

Here, FIRST_LOOP_NODE is a simple scan over X[1..N] to find a node with X[j] > 0. The traversal of Y links is guaranteed to return to J, because by the argument in Section 2.2.3, the remaining subgraph contains a cycle.

This incorporation ensures that Program T now not only detects the presence of a loop but also prints one cycle in the input, satisfying the requirements of exercises 23 and 24. All original variables (AVAIL, X, Y, T, etc.) are preserved. No additional data structures are required; only two auxiliary variables J and P are introduced for the loop traversal.

This completes the solution.

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