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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/imperative-programming/2/d/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Imperative programming / 2 / d / Solution by

Codex  0 2026-10-31

Implement staircase matrix search from the top-right corner. In the following Oberon-style code, & denotes short-circuit conjunction; all guarded accesses respect array bounds.

PROCEDURE Staircase(x: INTEGER; VAR i, j: INTEGER): BOOLEAN;
VAR r, c: INTEGER;
BEGIN
  r := 0; c := N-1;
  WHILE (r < M) & (c >= 0) DO
    IF f[r,c] = x THEN i := r; j := c; RETURN TRUE
    ELSIF f[r,c] < x THEN INC(r)
    ELSE DEC(c)
    END
  END;
  RETURN FALSE
END Staircase;

The loop invariant is that every occurrence not yet returned lies in

[r, M) \times [0, c + 1) .

(1)

Initially this is the whole array. If

f [r, c] < x

, every entry of row

r

in the candidate rectangle is smaller than

x

, so increasing

r

discards no possible answer. If

f [r, c] > x

, every entry of column

c

in that rectangle is greater than

x

, so decreasing

c

is safe. These conclusions follow from monotonicity even when repeated values occur.

The loop variant

(M - r) + (c + 1)

decreases on every unsuccessful iteration. If a dimension becomes empty, the loop invariant proves absence. For positive dimensions the path inspects at most

M + N - 1

entries; an initially empty dimension causes no inspection. Thus

Time O (M + N), auxiliary space O (1) .

(2)

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