Most efficient way to search a sorted matrix?

Question

I have an assignment to write an algorithm (not in any particular language, just pseudo-code) that receives a matrix [size: M x N] that is sorted in a way that all of it\'s

Answer 1

This is in the vein of Michal's answer (from which I will steal the nice graphic).

Matrix:

  min ..... b ..... c
    .       .       .
    .  II   .   I   . 
    .       .       .
    d .... mid .... f
    .       .       .
    .  III  .   IV  .
    .       .       .
    g ..... h ..... max

Min and max are the smallest and largest values, respectively. "mid" is not necessarily the average/median/whatever value.

We know that the value at mid is >= all values in quadrant II, and <= all values in quadrant IV. We cannot make such claims for quadrants I and III. If we recurse, we can eliminate one quadrant at each level.

Thus, if the target value is less than mid, we must search quadrants I, II, and III. If the target value is greater than mid, we must search quadrants I, III, and IV.

The space reduces to 3/4 its previous at each step:

n * (3/4)^x = 1

n = (4/3)^x

x = log_4/3(n)

Logarithms differ by a constant factor, so this is O(log(n)).

find(min, max, target)
    if min is max
        if target == min
            return min
        else
            return not found
    else if target < min or target > max
        return not found
    else
        set mid to average of min and max 
        if target == mid
            return mid
        else
            find(b, f, target), return if found
            find(d, h, target), return if found

            if target < mid
                return find(min, mid, target)
            else
                return find(mid, max, target)